Mercurial > repos > public > sbplib
changeset 279:d27297a9826c feature/beams
Merged with default.
| author | Jonatan Werpers <jonatan@werpers.com> |
|---|---|
| date | Thu, 08 Sep 2016 08:47:16 +0200 |
| parents | 9eff7b58c5f7 (diff) 6a5e94bb5e13 (current diff) |
| children | e1a05acc1b5d |
| files | |
| diffstat | 116 files changed, 5411 insertions(+), 1932 deletions(-) [+] |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+blockmatrix/fromMatrix.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,24 @@ +function bm = fromMatrix(A, div) + d1 = div{1}; + d2 = div{2}; + [n, m] = size(A); + if n ~= sum(d1) || m ~= sum(d2) + error('blockmatrix:fromMatrix:NonMatchingDim','The dimensions in div does not sum to the dimensions in A.'); + end + + bm = cell(length(d1), length(d2)); + I = 1; + for i = 1:length(d1) + J = 1; + for j = 1:length(d2) + Asub = A(I:(I + d1(i)-1), J:(J + d2(j)-1)); + if nnz(Asub) == 0 + bm{i,j} = []; + else + bm{i,j} = Asub; + end + J = J + d2(j); + end + I = I + d1(i); + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+blockmatrix/fromMatrixTest.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,64 @@ +function tests = fromMatrixTest() + tests = functiontests(localfunctions); +end + +function testErrorNonMatchingDim(testCase) + in = { + {magic(5), {[1 2 3], [4]}}, + {magic(5), {[1 1 1 1 1 1], [5]}}, + {magic(5), {[5], [1 1 1 1 1 1]}}, + {ones(4,2),{[2 3],[2]}}, + {ones(4,2),{[2 2],[3]}}, + }; + + for i = 1:length(in) + testCase.verifyError(@()blockmatrix.fromMatrix(in{i}{:}),'blockmatrix:fromMatrix:NonMatchingDim'); + end +end + +function testFromMatrix(testCase) + cases = { + { + {[],{[],[]}}, + {} + }, + { + { + magic(3), + {[3],[3]} + }, + {magic(3)} + }, + { + { + magic(3), + {[1 1 1],[1 1 1]} + }, + mat2cell(magic(3),[1 1 1],[1 1 1]) + }, + { + { + [17 24 1 8 15; 23 5 7 14 16; 4 6 13 20 22; 10 12 19 21 3; 11 18 25 2 9], + {[1 4],[2 3]} + }, + { + [17 24], [1 8 15]; + [23 5; 4 6; 10 12; 11 18], [7 14 16; 13 20 22; 19 21 3; 25 2 9]; + }; + }, + }; + for i = 1:length(cases) + out = convertToFull(blockmatrix.fromMatrix(cases{i}{1}{:})); + expected = cases{i}{2}; + testCase.verifyEqual(out,expected); + end +end + +function C = convertToFull(C) + [N,M] = size(C); + for i = 1:N + for j = 1:M + C{i,j} = full(C{i,j}); + end + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+blockmatrix/getDivision.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,37 @@ +function div = getDivision(bm) + if ~blockmatrix.isBlockmatrix(bm) + error('blockmatrix:getDivision:NotABlockmatrix', 'Input is not a blockmatrix'); + end + + if isempty(bm) + div = {[],[]}; + return + end + + div = {row_height(bm),col_width(bm)}; +end + + +function m = col_width(C) + m = zeros(1,size(C,2)); + for j = 1:size(C,2) + for i = 1:size(C,1) + if isempty(C{i,j}) + continue + end + m(j) = size(C{i,j},2); + end + end +end + +function n = row_height(C) + n = zeros(1,size(C,1)); + for i = 1:size(C,1) + for j = 1:size(C,2) + if isempty(C{i,j}) + continue + end + n(i) = size(C{i,j},1); + end + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+blockmatrix/getDivisionTest.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,69 @@ +function tests = getDivisionTest() + tests = functiontests(localfunctions); +end + +function testError(testCase) + cases = { + magic(3), + {[2 2 2];{1,2}}, + {[2 2 2];[1 2]}, + {[2; 2; 2], [1; 2]}, + }; + + for i =1:length(cases) + testCase.verifyError(@()blockmatrix.getDivision(cases{i}), 'blockmatrix:getDivision:NotABlockmatrix') + end +end + +function testGetDivision(testCase) + cases = { + { + {}, + {[],[]}; + }, + { + { + [2 2; 2 1], [1; 2]; + [2 2], [1] + }, + {[2 1], [2 1]} + }, + { + { + [2 2; 2 1], []; + [2 2], [1] + }, + {[2 1], [2 1]} + }, + { + { + [2 2; 2 1], []; + [2 2], [] + }, + {[2 1], [2 0]} + }, + { + { + [2 2; 2 1], [1; 2]; + [], [] + }, + {[2 0], [2 1]} + }, + { + { + [2 2; 2 1]; + [2 2] + }, + {[2 1], 2} + }, + }; + + for i = 1:length(cases) + in = cases{i}{1}; + out = blockmatrix.getDivision(in); + expected = cases{i}{2}; + testCase.verifyEqual(out, expected); + end +end + +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+blockmatrix/isBlockmatrix.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,59 @@ +function b = isBlockmatrix(bm) + if ~iscell(bm) + b = false; + return + end + + % Make sure all blocks are numerica matrices + for i = 1:length(bm) + if ~isnumeric(bm{i}) + b = false; + return + end + end + + [N,M] = size(bm); + % Make sure column dimensions agree + for i = 1:N + d = []; + for j = 1:M + d_ij = size(bm{i,j},1); + if d_ij == 0 + continue + end + + if isempty(d) + d = d_ij; + continue + end + + if d ~= d_ij + b = false; + return + end + end + end + + % Make sure row dimensions agree + for j = 1:M + d = []; + for i = 1:N + d_ij = size(bm{i,j},2); + if d_ij == 0 + continue + end + + if isempty(d) + d = d_ij; + continue + end + + if d ~= d_ij + b = false; + return + end + end + end + + b = true; +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+blockmatrix/isBlockmatrixTest.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,67 @@ +function tests = isBlockmatrixTest() + tests = functiontests(localfunctions); +end + +function testIsBlockmatrix(testCase) + cases = { + { + magic(3), + false % Must be a cell array + } + { + {[2 2 2];{1,2}}, + false % All elements of the cell matrix must be regular matrices + }, + { + {[2 2 2];[1 2]}, + false % Row dimensions must match + }, + { + {[2; 2; 2], [1; 2]}, + false % Column dimensions must match + }, + { + {}, + true % An empty matrix is a matrix too + }, + { + { + [2 2; 2 1], [1; 2]; + [2 2], [1] + }, + true % A simple valid one + }, + { + { + [2 2; 2 1], []; + [2 2], [1] + }, + true % Empty blocks assumed to be zero and match dimensions + }, + { + { + [2 2; 2 1], []; + [2 2], [] + }, + true % Empty blocks allowed. + }, + { + { + [2 2; 2 1], [1; 2]; + [], [] + }, + true % Empty blocks allowed. + }, + { + blockmatrix.zero({[1 2 3],[2 3]}), + true % A zero block matrix is a block matrix + }, + }; + + for i = 1:length(cases) + in = cases{i}{1}; + out = blockmatrix.isBlockmatrix(in); + expected = cases{i}{2}; + testCase.verifyEqual(out, expected, sprintf('Should return %d for %s', expected, toString(in))); + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+blockmatrix/isDivision.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,34 @@ +function b = isDivision(div) + % Make sure it is a cellarray + if ~iscell(div) + b = false; + return + end + + % Make sure it has the right shape + if numel(div) ~= 2 + b = false; + return + end + + if ~isDivisionVector(div{1}) || ~isDivisionVector(div{2}) + b = false; + return + end + + b = true; +end + +function b = isDivisionVector(v) + if isempty(v) + b = false; + return + end + + if any(v <= 0) + b = false; + return + end + + b = true; +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+blockmatrix/isDivisionTest.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,23 @@ +function tests = isDivisionTest() + tests = functiontests(localfunctions); +end + +function testIsDivision(testCase) + cases = { + {{[2 2 2],[1 2]} ,true}, + {{[1 2],[1 0]} ,false}, + {{[0 2],[1 1]} ,false}, + {{[1 2],[]} ,false}, + {{[1 2],[1]} ,true}, + {{[1 2],[1], [1 2 3]} ,false}, + {{[1 2 3]} ,false}, + {[1 2] ,false}, + }; + + for i = 1:length(cases) + in = cases{i}{1}; + out = blockmatrix.isDivision(in); + expected = cases{i}{2}; + testCase.verifyEqual(out, expected, sprintf('Should return %d for %s', expected, toString(in))); + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+blockmatrix/toMatrix.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,26 @@ +function A = toMatrix(bm) + if ~blockmatrix.isBlockmatrix(bm) + error('blockmatrix:toMatrix:NotABlockmatrix', 'Input is not a blockmatrix'); + end + + div = blockmatrix.getDivision(bm); + n = div{1}; + m = div{2}; + + N = sum(n); + M = sum(m); + + A = sparse(N,M); + + n_ind = [0 cumsum(n)]; + m_ind = [0 cumsum(m)]; + + for i = 1:size(bm,1) + for j = 1:size(bm,2) + if isempty(bm{i,j}) + continue + end + A(n_ind(i)+1:n_ind(i+1),m_ind(j)+1:m_ind(j+1)) = bm{i,j}; + end + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+blockmatrix/toMatrixTest.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,62 @@ +function tests = toMatrixTest() + tests = functiontests(localfunctions); +end + +function testError(testCase) + testCase.verifyError(@()blockmatrix.toMatrix([]), 'blockmatrix:toMatrix:NotABlockmatrix') +end + +function testToMatrix(testCase) + cases = { + { + {}, + [], + }, + { + {1, 2; 3, 4}, + [1,2; 3,4], + } + { + { + [2 2; 2 1], [1; 2]; + [2 2], [1] + }, + [2 2 1; + 2 1 2; + 2 2 1], + }, + { + { + [2 2; 2 1], []; + [2 2], [1] + }, + [2 2 0; + 2 1 0; + 2 2 1], + }, + { + { + [2 2; 2 1], []; + [2 2], [] + }, + [2 2; + 2 1; + 2 2], + }, + { + { + [2 2; 2 1], [1; 2]; + [], [] + }, + [2 2 1; + 2 1 2], + }, + }; + + for i = 1:length(cases) + in = cases{i}{1}; + out = full(blockmatrix.toMatrix(in)); + expected = cases{i}{2}; + testCase.verifyEqual(out, expected); + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+blockmatrix/zero.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,20 @@ +% Creates a block matrix according to the division with zeros everywhere. +function bm = zero(div) + if ~blockmatrix.isDivision(div); + error('div is not a valid division'); + end + + n = div{1}; + m = div{2}; + + N = length(n); + M = length(m); + + bm = cell(N,M); + + for i = 1:N + for j = 1:M + bm{i,j} = sparse(n(i),m(j)); + end + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+blockmatrix/zeroTest.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,52 @@ +function tests = zeroTest() + tests = functiontests(localfunctions); +end + +function testZero(testCase) + cases = { + { + {[],[]}, + {}, + }, + { + {0,0}, + {[]}; + }, + { + {1,1}, + {0}; + }, + { + {2,1}, + {[0; 0]}; + }, + { + {1,2}, + {[0 0]}; + }, + { + {[1 2],2}, + {[0 0];[0 0; 0 0]}; + }, + { + {[1 2],[2 1]}, + {[0 0],[0];[0 0; 0 0],[0; 0]}; + }, + }; + + for i = 1:length(cases) + out = convertToFull(blockmatrix.zero(cases{i}{1})); + expected = cases{i}{2}; + testCase.verifyEqual(out,expected); + end +end + + +function C = convertToFull(C) + [N,M] = size(C); + for i = 1:N + for j = 1:M + C{i,j} = full(C{i,j}); + end + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/Cartesian.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,189 @@ +classdef Cartesian < grid.Structured + properties + n % Number of points in the grid + d % Number of dimensions + m % Number of points in each direction + x % Cell array of vectors with node placement for each dimension. + h % Spacing/Scaling + end + + % General d dimensional grid with n points + methods + % Creates a cartesian grid given vectors conatining the coordinates + % in each direction + function obj = Cartesian(varargin) + obj.d = length(varargin); + + for i = 1:obj.d + obj.x{i} = varargin{i}; + obj.m(i) = length(varargin{i}); + end + + obj.n = prod(obj.m); + if obj.n == 0 + error('grid:Cartesian:EmptyGrid','Input parameter gives an empty grid.') + end + + obj.h = []; + end + % n returns the number of points in the grid + function o = N(obj) + o = obj.n; + end + + % d returns the spatial dimension of the grid + function o = D(obj) + o = obj.d; + end + + function m = size(obj) + m = obj.m; + end + + % points returns a n x d matrix containing the coordianets for all points. + % points are ordered according to the kronecker product with X*Y*Z + function X = points(obj) + X = zeros(obj.n, obj.d); + + for i = 1:obj.d + if iscolumn(obj.x{i}) + c = obj.x{i}; + else + c = obj.x{i}'; + end + + m_before = prod(obj.m(1:i-1)); + m_after = prod(obj.m(i+1:end)); + + X(:,i) = kr(ones(m_before,1),c,ones(m_after,1)); + end + end + + % matrices returns a cell array with coordinates in matrix form. + % For 2d case these will have to be transposed to work with plotting routines. + function X = matrices(obj) + + if obj.d == 1 % There is no 1d matrix data type in matlab, handle special case + X{1} = reshape(obj.x{1}, [obj.m 1]); + return + end + + X = cell(1,obj.d); + for i = 1:obj.d + s = ones(1,obj.d); + s(i) = obj.m(i); + + t = reshape(obj.x{i},s); + + s = obj.m; + s(i) = 1; + X{i} = repmat(t,s); + end + end + + function h = scaling(obj) + if isempty(obj.h) + error('grid:Cartesian:NoScalingSet', 'No scaling set') + end + + h = obj.h; + end + + % Restricts the grid function gf on obj to the subgrid g. + % Only works for even multiples + function gf = restrictFunc(obj, gf, g) + m1 = obj.m; + m2 = g.m; + + % Check the input + if prod(m1) ~= numel(gf) + error('grid:Cartesian:restrictFunc:NonMatchingFunctionSize', 'The grid function has to few or too many points.'); + end + + if ~all(mod(m1-1,m2-1) == 0) + error('grid:Cartesian:restrictFunc:NonMatchingGrids', 'Only integer downsamplings are allowed'); + end + + % Calculate stride for each dimension + stride = (m1-1)./(m2-1); + + % Create downsampling indecies + I = {}; + for i = 1:length(m1) + I{i} = 1:stride(i):m1(i); + end + + gf = reshapeRowMaj(gf, m1); + gf = gf(I{:}); + gf = reshapeRowMaj(gf, prod(m2)); + end + + % Projects the grid function gf on obj to the grid g. + function gf = projectFunc(obj, gf, g) + error('grid:Cartesian:NotImplemented') + end + + % Return the names of all boundaries in this grid. + function bs = getBoundaryNames(obj) + switch obj.D() + case 1 + bs = {'l', 'r'}; + case 2 + bs = {'w', 'e', 's', 'n'}; + case 3 + bs = {'w', 'e', 's', 'n', 'd', 'u'}; + otherwise + error('not implemented'); + end + end + + % Return coordinates for the given boundary + function X = getBoundary(obj, name) + % In what dimension is the boundary? + switch name + case {'l', 'r', 'w', 'e'} + D = 1; + case {'s', 'n'} + D = 2; + case {'d', 'u'} + D = 3; + otherwise + error('not implemented'); + end + + % At what index is the boundary? + switch name + case {'l', 'w', 's', 'd'} + index = 1; + case {'r', 'e', 'n', 'u'} + index = obj.m(D); + otherwise + error('not implemented'); + end + + + + I = cell(1, obj.d); + for i = 1:obj.d + if i == D + I{i} = index; + else + I{i} = ':'; + end + end + + % Calculate size of result: + m = obj.m; + m(D) = []; + N = prod(m); + + X = zeros(N, obj.d); + + coordMat = obj.matrices(); + for i = 1:length(coordMat) + Xtemp = coordMat{i}(I{:}); + X(:,i) = reshapeRowMaj(Xtemp, [N,1]); + end + end + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/CartesianTest.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,318 @@ +function tests = CartesianTest() + tests = functiontests(localfunctions); +end + + +function testWarningEmptyGrid(testCase) + in = { + {[]}, + {[],[1]}, + {[1],[2], []}, + }; + + for i = 1:length(in) + testCase.verifyError(@()grid.Cartesian(in{i}{:}),'grid:Cartesian:EmptyGrid'); + end +end + +function testN(testCase) + in = { + {[1 2 3]}, + {[1 2 3],[1 2]}, + {[1 2 3],[1 2 3]}, + {[1 2 3],[1 2 3], [1]}, + {[1 2 3],[1 2 3], [1 3 4]}, + }; + + out = [3,6,9,9,27]; + + for i = 1:length(in) + g = grid.Cartesian(in{i}{:}); + testCase.verifyEqual(g.N(),out(i)); + end +end + + +function testD(testCase) + in = { + {[1 2 3]}, + {[1 2 3],[1 2]}, + {[1 2 3],[1 2 3]}, + {[1 2 3],[1 2 3], [1]}, + {[1 2 3],[1 2 3], [1 3 4]}, + }; + + out = [1,2,2,3,3]; + + for i = 1:length(in) + g = grid.Cartesian(in{i}{:}); + testCase.verifyEqual(g.D(),out(i)); + end +end + +function testSize(testCase) + in = { + {[1 2 3]}, + {[1 2 3],[1 2]}, + {[1 2 3],[1 2 3]}, + {[1 2 3],[1 2 3], [1]}, + {[1 2 3],[1 2 3], [1 3 4]}, + }; + + out = { + [3], + [3 2], + [3 3], + [3 3 1], + [3 3 3], + }; + + for i = 1:length(in) + g = grid.Cartesian(in{i}{:}); + testCase.verifyEqual(g.size(),out{i}); + end +end + +function testPoints(testCase) + in = { + {[1 2]}, + {[1 2],[3 4]}, + {[1 2],[3 4], [5 6]}, + }; + + out = { + [[1; 2]], + [[1; 1; 2; 2],[3; 4; 3; 4]], + [[1; 1; 1; 1; 2; 2; 2; 2],[3; 3; 4; 4; 3; 3; 4; 4],[ 5; 6; 5; 6; 5; 6; 5; 6]], + }; + + for i = 1:length(in) + g = grid.Cartesian(in{i}{:}); + testCase.verifyEqual(g.points(),out{i}); + end +end + +function testMatrices(testCase) + in = { + {[1 2]}, + {[1 2],[3 4]}, + {[1 2],[3 4], [5 6]}, + }; + + out{1}{1} = [1; 2]; + + out{2}{1} = [1, 1; 2, 2]; + out{2}{2} = [3, 4; 3, 4]; + + out{3}{1}(:,:,1) = [1, 1; 2, 2]; + out{3}{1}(:,:,2) = [1, 1; 2, 2]; + + out{3}{2}(:,:,1) = [3, 4; 3, 4]; + out{3}{2}(:,:,2) = [3, 4; 3, 4]; + + out{3}{3}(:,:,1) = [5, 5; 5, 5]; + out{3}{3}(:,:,2) = [6, 6; 6, 6]; + + for i = 1:length(in) + g = grid.Cartesian(in{i}{:}); + testCase.verifyEqual(g.matrices(),out{i}); + end +end + + +function testRestrictFuncInvalidInput(testCase) + inG1 = { + {[1 2 3 4 5]}, + {[1 2 3],[4 5 6 7 8]}, + {[1 2 3],[4 5 6 7 8]}, + {[1 2 3],[4 5 6 7 8]}, + }; + + inG2 = { + {[1 3 4 5]}, + {[1 3],[4 5 6 8]}, + {[1 3],[4 6 8]}, + {[1 3],[4 6 8]}, + }; + + inGf = { + [1; 2; 3; 4; 5], + [14; 15; 16; 17; 18; 24; 25; 26; 27; 28; 34; 35; 36; 37; 38]; + [14; 15; 16; 17; 18; 24; 25; 26; 27; 28; 34; 35; 36]; + [14; 15; 16; 17; 18; 24; 25; 26; 27; 28; 34; 35; 36; 37; 38; 39; 40]; + }; + + out = { + 'grid:Cartesian:restrictFunc:NonMatchingGrids', + 'grid:Cartesian:restrictFunc:NonMatchingGrids', + 'grid:Cartesian:restrictFunc:NonMatchingFunctionSize', + 'grid:Cartesian:restrictFunc:NonMatchingFunctionSize', + }; + + for i = 1:length(inG1) + g1 = grid.Cartesian(inG1{i}{:}); + g2 = grid.Cartesian(inG2{i}{:}); + testCase.verifyError(@()g1.restrictFunc(inGf{i},g2),out{i}); + end +end + +function testRestrictFunc(testCase) + inG1 = { + {[1 2 3 4 5]}, + {[1 2 3],[4 5 6 7 8]}, + }; + + inG2 = { + {[1 3 5]}, + {[1 3],[4 6 8]}, + }; + + inGf = { + [1; 2; 3; 4; 5], + [14; 15; 16; 17; 18; 24; 25; 26; 27; 28; 34; 35; 36; 37; 38]; + }; + + outGf = { + [1; 3; 5], + [14; 16; 18; 34; 36; 38]; + }; + + for i = 1:length(inG1) + g1 = grid.Cartesian(inG1{i}{:}); + g2 = grid.Cartesian(inG2{i}{:}); + testCase.verifyEqual(g1.restrictFunc(inGf{i}, g2), outGf{i}); + end +end + +function testScaling(testCase) + in = {[1 2 3], [1 2]}; + g = grid.Cartesian(in{:}); + + testCase.verifyError(@()g.scaling(),'grid:Cartesian:NoScalingSet'); + + g.h = [2 1]; + testCase.verifyEqual(g.scaling(),[2 1]); + +end + + +function testGetBoundaryNames(testCase) + in = { + {[1 2 3]}, + {[1 2 3], [4 5]}, + {[1 2 3], [4 5], [6 7 8]}, + }; + + out = { + {'l', 'r'}, + {'w', 'e', 's', 'n'}, + {'w', 'e', 's', 'n', 'd', 'u'}, + }; + + for i = 1:length(in) + g = grid.Cartesian(in{i}{:}); + testCase.verifyEqual(g.getBoundaryNames(), out{i}); + end +end + +function testGetBoundary(testCase) + grids = { + {[1 2 3]}, + {[1 2 3], [4 5]}, + {[1 2 3], [4 5], [6 7 8]}, + }; + + boundaries = { + {'l', 'r'}, + {'w', 'e', 's', 'n'}, + {'w', 'e', 's', 'n', 'd', 'u'}, + }; + + + % 1d + out{1,1} = 1; + out{1,2} = 3; + + % 2d + out{2,1} = [ + 1,4; + 1,5; + ]; + out{2,2} = [ + 3,4; + 3,5; + ]; + out{2,3} = [ + 1,4; + 2,4; + 3,4; + ]; + out{2,4} = [ + 1,5; + 2,5; + 3,5; + ]; + + % 3d + out{3,1} = [ + 1,4,6; + 1,4,7; + 1,4,8; + 1,5,6; + 1,5,7; + 1,5,8; + ]; + out{3,2} = [ + 3,4,6; + 3,4,7; + 3,4,8; + 3,5,6; + 3,5,7; + 3,5,8; + ]; + out{3,3} = [ + 1,4,6; + 1,4,7; + 1,4,8; + 2,4,6; + 2,4,7; + 2,4,8; + 3,4,6; + 3,4,7; + 3,4,8; + ]; + out{3,4} = [ + 1,5,6; + 1,5,7; + 1,5,8; + 2,5,6; + 2,5,7; + 2,5,8; + 3,5,6; + 3,5,7; + 3,5,8; + ]; + out{3,5} = [ + 1,4,6; + 1,5,6; + 2,4,6; + 2,5,6; + 3,4,6; + 3,5,6; + ]; + out{3,6} = [ + 1,4,8; + 1,5,8; + 2,4,8; + 2,5,8; + 3,4,8; + 3,5,8; + ]; + + for ig = 1:length(grids) + g = grid.Cartesian(grids{ig}{:}); + for ib = 1:length(boundaries{ig}) + testCase.verifyEqual(g.getBoundary(boundaries{ig}{ib}), out{ig,ib}); + end + end +end
--- a/+grid/Curve.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,388 +0,0 @@ -classdef Curve - properties - g - gp - transformation - end - - methods - %TODO: - % Concatenation of curves - % Subsections of curves - % Stretching of curve parameter - done for arc length. - % Curve to cell array of linesegments - - % Returns a curve object. - % g -- curve parametrization for parameter between 0 and 1 - % gp -- parametrization of curve derivative - function obj = Curve(g,gp,transformation) - default_arg('gp',[]); - default_arg('transformation',[]); - p_test = g(0); - assert(all(size(p_test) == [2,1]), 'A curve parametrization must return a 2x1 vector.'); - - if ~isempty(transformation) - transformation.base_g = g; - transformation.base_gp = gp; - [g,gp] = grid.Curve.transform_g(g,gp,transformation); - end - - obj.g = g; - obj.gp = gp; - obj.transformation = transformation; - - end - - function n = normal(obj,t) - assert(~isempty(obj.gp),'Curve has no derivative!'); - deriv = obj.gp(t); - normalization = sqrt(sum(deriv.^2,1)); - n = [-deriv(2,:)./normalization; deriv(1,:)./normalization]; - end - - - % Plots a curve g(t) for 0<t<1, using n points. Returns a handle h to the plotted curve. - % h = plot_curve(g,n) - function h = plot(obj,n) - default_arg('n',100); - - t = linspace(0,1,n); - p = obj.g(t); - h = line(p(1,:),p(2,:)); - end - - function h= plot_normals(obj,l,n,m) - default_arg('l',0.1); - default_arg('n',10); - default_arg('m',100); - t_n = linspace(0,1,n); - - normals = obj.normal(t_n)*l; - - n0 = obj.g(t_n); - n1 = n0 + normals; - - h = line([n0(1,:); n1(1,:)],[n0(2,:); n1(2,:)]); - set(h,'Color',Color.red); - obj.plot(m); - end - - function h= show(obj,name) - p = obj.g(1/2); - n = obj.normal(1/2); - p = p + n*0.1; - - % Add arrow - - h = text(p(1),p(2),name); - h.HorizontalAlignment = 'center'; - h.VerticalAlignment = 'middle'; - - obj.plot(); - end - % Shows curve with name and arrow for direction. - - % Length of arc from parameter t0 to t1 (which may be vectors). - % Computed using derivative. - function L = arcLength(obj,t0,t1) - assert(~isempty(obj.gp),'Curve has no derivative!'); - speed = @(t) sqrt(sum(obj.gp(t).^2,1)); - L = integral_vec(speed,t0,t1); - end - - % Creates the arc length parameterization of a curve. - % N -- number of points used to approximate the arclength function - function curve = arcLengthParametrization(obj,N) - default_arg('N',100); - assert(~isempty(obj.gp),'Curve has no derivative!'); - - % Construct arcLength function using splines - tvec = linspace(0,1,N); - arcVec = obj.arcLength(0,tvec); - tFunc = spline(arcVec,tvec); % t as a function of arcLength - L = obj.arcLength(0,1); - arcPar = @(s) tFunc(s*L); - - % New function and derivative - g_new = @(t)obj.g(arcPar(t)); - gp_new = @(t) normalize(obj.gp(arcPar(t))); - curve = grid.Curve(g_new,gp_new); - - end - - % how to make it work for methods without returns - function p = subsref(obj,S) - %Should i add error checking here? - %Maybe if you want performance you fetch obj.g and then use that - switch S(1).type - case '()' - p = obj.g(S.subs{1}); - % case '.' - - % p = obj.(S.subs); - otherwise - p = builtin('subsref',obj,S); - % error() - end - end - - - %% TRANSFORMATION OF A CURVE - function D = reverse(C) - % g = C.g; - % gp = C.gp; - % D = grid.Curve(@(t)g(1-t),@(t)-gp(1-t)); - D = C.transform([],[],-1); - end - - function D = transform(C,A,b,flip) - default_arg('A',[1 0; 0 1]); - default_arg('b',[0; 0]); - default_arg('flip',1); - if isempty(C.transformation) - g = C.g; - gp = C.gp; - transformation.A = A; - transformation.b = b; - transformation.flip = flip; - else - g = C.transformation.base_g; - gp = C.transformation.base_gp; - A_old = C.transformation.A; - b_old = C.transformation.b; - flip_old = C.transformation.flip; - - transformation.A = A*A_old; - transformation.b = A*b_old + b; - transformation.flip = flip*flip_old; - end - - D = grid.Curve(g,gp,transformation); - - end - - function D = translate(C,a) - g = C.g; - gp = C.gp; - - % function v = g_fun(t) - % x = g(t); - % v(1,:) = x(1,:)+a(1); - % v(2,:) = x(2,:)+a(2); - % end - - % D = grid.Curve(@g_fun,gp); - - D = C.transform([],a); - end - - function D = mirror(C, a, b) - assert_size(a,[2,1]); - assert_size(b,[2,1]); - - g = C.g; - gp = C.gp; - - l = b-a; - lx = l(1); - ly = l(2); - - - % fprintf('Singular?\n') - - A = [lx^2-ly^2 2*lx*ly; 2*lx*ly ly^2-lx^2]/(l'*l); - - % function v = g_fun(t) - % % v = a + A*(g(t)-a) - % x = g(t); - - % ax1 = x(1,:)-a(1); - % ax2 = x(2,:)-a(2); - % v(1,:) = a(1)+A(1,:)*[ax1;ax2]; - % v(2,:) = a(2)+A(2,:)*[ax1;ax2]; - % end - - % function v = gp_fun(t) - % v = A*gp(t); - % end - - % D = grid.Curve(@g_fun,@gp_fun); - - % g = A(g-a)+a = Ag - Aa + a; - b = - A*a + a; - D = C.transform(A,b); - - end - - function D = rotate(C,a,rad) - assert_size(a, [2,1]); - assert_size(rad, [1,1]); - g = C.g; - gp = C.gp; - - - A = [cos(rad) -sin(rad); sin(rad) cos(rad)]; - - % function v = g_fun(t) - % % v = a + A*(g(t)-a) - % x = g(t); - - % ax1 = x(1,:)-a(1); - % ax2 = x(2,:)-a(2); - % v(1,:) = a(1)+A(1,:)*[ax1;ax2]; - % v(2,:) = a(2)+A(2,:)*[ax1;ax2]; - % end - - % function v = gp_fun(t) - % v = A*gp(t); - % end - - % D = grid.Curve(@g_fun,@gp_fun); - - - % g = A(g-a)+a = Ag - Aa + a; - b = - A*a + a; - D = C.transform(A,b); - end - end - - methods (Static) - - % Computes the derivative of g: R -> R^2 using an operator D1 - function gp_out = numericalDerivative(g,D1) - m = length(D1); - t = linspace(0,1,m); - gVec = g(t)'; - gpVec = (D1*gVec)'; - - gp1_fun = spline(t,gpVec(1,:)); - gp2_fun = spline(t,gpVec(2,:)); - gp_out = @(t) [gp1_fun(t);gp2_fun(t)]; - end - - function obj = line(p1, p2) - - function v = g_fun(t) - v(1,:) = p1(1) + t.*(p2(1)-p1(1)); - v(2,:) = p1(2) + t.*(p2(2)-p1(2)); - end - g = @g_fun; - - obj = grid.Curve(g); - end - - function obj = circle(c,r,phi) - default_arg('phi',[0; 2*pi]) - default_arg('c',[0; 0]) - default_arg('r',1) - - function v = g_fun(t) - w = phi(1)+t*(phi(2)-phi(1)); - v(1,:) = c(1) + r*cos(w); - v(2,:) = c(2) + r*sin(w); - end - - function v = g_fun_deriv(t) - w = phi(1)+t*(phi(2)-phi(1)); - v(1,:) = -(phi(2)-phi(1))*r*sin(w); - v(2,:) = (phi(2)-phi(1))*r*cos(w); - end - - obj = grid.Curve(@g_fun,@g_fun_deriv); - end - - function obj = bezier(p0, p1, p2, p3) - function v = g_fun(t) - v(1,:) = (1-t).^3*p0(1) + 3*(1-t).^2.*t*p1(1) + 3*(1-t).*t.^2*p2(1) + t.^3*p3(1); - v(2,:) = (1-t).^3*p0(2) + 3*(1-t).^2.*t*p1(2) + 3*(1-t).*t.^2*p2(2) + t.^3*p3(2); - end - - function v = g_fun_deriv(t) - v(1,:) = 3*(1-t).^2*(p1(1)-p0(1)) + 6*(1-t).*t*(p2(1)-p1(1)) + 3*t.^2*(p3(1)-p2(1)); - v(2,:) = 3*(1-t).^2*(p1(2)-p0(2)) + 6*(1-t).*t*(p2(2)-p1(2)) + 3*t.^2*(p3(2)-p2(2)); - end - - obj = grid.Curve(@g_fun,@g_fun_deriv); - end - - - function [g_out,gp_out] = transform_g(g,gp,tr) - A = tr.A; - b = tr.b; - flip = tr.flip; - - function v = g_fun_noflip(t) - % v = A*g + b - x = g(t); - - v(1,:) = A(1,:)*x+b(1); - v(2,:) = A(2,:)*x+b(2); - end - - function v = g_fun_flip(t) - % v = A*g + b - x = g(1-t); - - v(1,:) = A(1,:)*x+b(1); - v(2,:) = A(2,:)*x+b(2); - end - - - switch flip - case 1 - g_out = @g_fun_noflip; - gp_out = @(t)A*gp(t); - case -1 - g_out = @g_fun_flip; - gp_out = @(t)-A*gp(1-t); - end - end - - end -end - - - -function g_norm = normalize(g0) - g1 = g0(1,:); - g2 = g0(2,:); - normalization = sqrt(sum(g0.^2,1)); - g_norm = [g1./normalization; g2./normalization]; -end - -function I = integral_vec(f,a,b) -% Wrapper around the built-in function integral that -% handles multiple limits. - - Na = length(a); - Nb = length(b); - assert(Na == 1 || Nb == 1 || Na==Nb,... - 'a and b must have same length, unless one is a scalar.'); - - if(Na>Nb); - I = zeros(size(a)); - for i = 1:Na - I(i) = integral(f,a(i),b); - end - elseif(Nb>Na) - I = zeros(size(b)); - for i = 1:Nb - I(i) = integral(f,a,b(i)); - end - else - I = zeros(size(b)); - for i = 1:Nb - I(i) = integral(f,a(i),b(i)); - end - end -end - -% Returns a function handle to the spline. -function f = spline(tval,fval,spline_order) - default_arg('spline_order',4); - [m,~] = size(tval); - assert(m==1,'Need row vectors.'); - - f_spline = spapi( optknt(tval,spline_order), tval, fval ); - f = @(t) fnval(f_spline,t); -end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/Curvilinear.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,170 @@ +classdef Curvilinear < grid.Structured & grid.Mapped + properties + logic % Grid of Logical domain + coords % N x D matrix with coordinates of each point in the physical domain + end + + methods + % Creates a curvilinear grid. + % Ex: grid.Curvilinear(mapping, xi, eta, ...) + % mapping -- either a function handle, a matrix or a cell array with physical coordinates. + % A function handle should be a vector valued function of the coordinate mapping. + % A matrix should be a grid function (N*D x 1 vector) or a N x D + % A cell array should be a 1 x D cell array with either N x 1 vectors + % or matrices of the same dimesions as the logical grid. + % xi, eta, ... -- are the coordinate positions of the cartesian logical grid. + function obj = Curvilinear(mapping, varargin) + xi = varargin; + obj.logic = grid.Cartesian(xi{:}); + + % If mapping is a function evaluate it + if isa(mapping, 'function_handle') + mapping = grid.evalOn(obj.logic, mapping); + end + + D = obj.logic.D(); + N = obj.logic.N(); + + obj.coords = zeros(N,D); + + if iscell(mapping) + obj.coords = cellMappingToCoords(mapping, N, D, obj.logic.m); + elseif isnumeric(mapping) + obj.coords = matrixMappingToCoords(mapping, N, D); + else + error('grid:Curvilinear:Curvilinear','mapping must be a matrix or a cell array.'); + end + end + + function m = size(obj) + m = obj.logic.size(); + end + + % logicalGrid returns the domain grid of the mapping. + function g = logicalGrid(obj) + g = obj.logic; + end + + % mapping returns the mapped coordinates as a grid.Function + function m = mapping(obj); + m = obj.coords; + end + + % n returns the number of points in the grid + function o = N(obj) + o = obj.logic.N(); + end + + % d returns the spatial dimension of the grid + function o = D(obj) + o = obj.logic.D(); + end + + % points returns a n x d matrix containing the coordinates for all points. + function X = points(obj) + X = obj.coords; + end + + % Restricts the grid function gf on obj to the subgrid g. + function gf = restrictFunc(obj, gf, g) + gf = obj.logic.restrictFunc(gf, g.baseGrid()); + end + + % Projects the grid function gf on obj to the grid g. + function gf = projectFunc(obj, gf, g) + gf = obj.logic.projectFunc(gf,g.baseGrid()); + end + + function h = scaling(obj) + if isempty(obj.logic.h) + error('grid:Curvilinear:NoScalingSet','No scaling set'); + end + h = obj.logic.h; + end + + % Return the names of all boundaries in this grid. + function bs = getBoundaryNames(obj) + bs = obj.logic.getBoundaryNames(); + end + + % Return coordinates for the given boundary + function X = getBoundary(obj, name) + % In what dimension is the boundary? + switch name + case {'l', 'r', 'w', 'e'} + D = 1; + case {'s', 'n'} + D = 2; + case {'d', 'u'} + D = 3; + otherwise + error('not implemented'); + end + + % At what index is the boundary? + switch name + case {'l', 'w', 's', 'd'} + index = 1; + case {'r', 'e', 'n', 'u'} + index = obj.logic.m(D); + otherwise + error('not implemented'); + end + + + + I = cell(1, obj.D); + for i = 1:obj.D + if i == D + I{i} = index; + else + I{i} = ':'; + end + end + + % Calculate size of result: + m = obj.logic.m; + m(D) = []; + N = prod(m); + + X = zeros(N, obj.D); + + p = obj.points; + for i = 1:obj.D() + coordMat{i} = reshapeRowMaj(p(:,i), obj.logic.m); + end + + for i = 1:length(coordMat) + Xtemp = coordMat{i}(I{:}); + X(:,i) = reshapeRowMaj(Xtemp, [N,1]); + end + end + end +end + + +function coords = cellMappingToCoords(mapping, N, D, m) + if ~isequal(size(mapping),[1 D]) + error('grid:Curvilinear:Curvilinear','The cell array must be a 1xD array.'); + end + + if isequal(size(mapping{1}),[N 1]) + coords = cell2mat(mapping); + elseif isequal(size(mapping{1}), m) + for i = 1:length(mapping) + coords(:,i) = reshapeRowMaj(mapping{i}, [N 1]); + end + else + error('grid:Curvilinear:Curvilinear','The matrix must have size [N 1] or the same dimension as the grid. Actual: %s', toString(m)); + end +end + +function coords = matrixMappingToCoords(mapping, N, D) + if isequal(size(mapping), [N, D]) + coords = mapping; + elseif isequal(size(mapping), [N*D, 1]) + coords = reshapeRowMaj(mapping,[N D]); + else + error('grid:Curvilinear:Curvilinear','A matrix mapping must be of size [N D] or [N*D 1].'); + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/CurvilinearTest.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,131 @@ +function tests = CurvilinearTest() + tests = functiontests(localfunctions); +end + +function testMappingInputGridFunction(testCase) + in = { + {{1:10}, @(x) exp(x)}, + {{1:10,1:6}, @(x,y) [exp(x+y); exp(x-y)]}, + {{1:10,1:5,1:7}, @(x,y,z)[exp(x+y+z); exp(x-y-z); 2+x+y-z]}, + }; + + out = { + [10, 1]; + [10*6, 2]; + [10*5*7, 3]; + }; + + + % How to test this? Just make sure it runs without errors. + + for i = 1:length(in) + g = grid.Curvilinear(in{i}{2},in{i}{1}{:}); + testCase.verifyEqual(size(g.coords),out{i}); + end +end + +function testMappingInputComponentMatrix(testCase) + in = { + {{1:3}, [1 2 3]'}, + {{1:2, 1:3}, [1 2 3 4 5 6; 7 8 9 10 11 12]'}, + }; + + for i = 1:length(in) + g = grid.Curvilinear(in{i}{2},in{i}{1}{:}); + testCase.verifyEqual(g.coords,in{i}{2}); + end +end + +function testMappingInputCellOfMatrix(testCase) + + in = { + {{1:3}, {[1 2 3]'}}, + {{1:2, 1:3}, {[1 2 3; 4 5 6], [7 8 9; 10 11 12]}}, + }; + + out = { + [1 2 3]', + [1 2 3 4 5 6; 7 8 9 10 11 12]', + }; + + for i = 1:length(in) + g = grid.Curvilinear(in{i}{2},in{i}{1}{:}); + testCase.verifyEqual(g.coords,out{i}); + end +end + +function testMappingInputCellOfVectors(testCase) + in = { + {{1:3}, {[1 2 3]'}}, + {{1:2, 1:3}, {[1 2 3 4 5 6]', [7 8 9 10 11 12]'}}, + }; + + out = { + [1 2 3]', + [1 2 3 4 5 6; 7 8 9 10 11 12]', + }; +end + +function testMappingInputError(testCase) + testCase.verifyFail(); +end + +function testScaling(testCase) + in = {{1:2, 1:3}, {[1 2 3 4 5 6]', [7 8 9 10 11 12]'}}; + g = grid.Curvilinear(in{2},in{1}{:}); + + testCase.verifyError(@()g.scaling(),'grid:Curvilinear:NoScalingSet'); + + g.logicalGrid.h = [2 1]; + testCase.verifyEqual(g.scaling(),[2 1]); +end + +function testGetBoundaryNames(testCase) + in = { + {{1:10}, @(x) exp(x)}, + {{1:10,1:6}, @(x,y) [exp(x+y); exp(x-y)]}, + {{1:10,1:5,1:7}, @(x,y,z)[exp(x+y+z); exp(x-y-z); 2+x+y-z]}, + }; + + out = { + {'l', 'r'}, + {'w', 'e', 's', 'n'}, + {'w', 'e', 's', 'n', 'd', 'u'}, + }; + + for i = 1:length(in) + g = grid.Curvilinear(in{i}{2},in{i}{1}{:}); + testCase.verifyEqual(g.getBoundaryNames(), out{i}); + end +end + +function testGetBoundary(testCase) + grids = { + {{1:10}, @(x) exp(x)}, + {{1:10,1:6}, @(x,y) [exp(x+y); exp(x-y)]}, + {{1:10,1:5,1:7}, @(x,y,z)[exp(x+y+z); exp(x-y-z); 2+x+y-z]}, + }; + + boundaries = { + {'l', 'r'}, + {'w', 'e', 's', 'n'}, + {'w', 'e', 's', 'n', 'd', 'u'}, + }; + + + for ig = 1:length(grids) + g = grid.Curvilinear(grids{ig}{2},grids{ig}{1}{:}); + + logicalGrid = grid.Cartesian(grids{ig}{1}{:}); + + for ib = 1:length(boundaries{ig}) + + logicalBoundary = logicalGrid.getBoundary(boundaries{ig}{ib}); + + x = num2cell(logicalBoundary',2); + expectedBoundary = grids{ig}{2}(x{:})'; + testCase.verifyEqual(g.getBoundary(boundaries{ig}{ib}), expectedBoundary); + end + end +end +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/Grid.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,25 @@ +classdef Grid < handle + % General d dimensional grid with n points + methods (Abstract) + % n returns the number of points in the grid + o = N(obj) + + % d returns the spatial dimension of the grid + o = D(obj) + + % points returns a n x d matrix containing the coordinates for all points. + X = points(obj) + + % Restricts the grid function gf on obj to the subgrid g. + gf = restrictFunc(obj, gf, g) + + % Projects the grid function gf on obj to the grid g. + gf = projectFunc(obj, gf, g) + + % Return the names of all boundaries in this grid. + bs = getBoundaryNames(obj) + + % Return coordinates for the given boundary + b = getBoundary(obj, name) + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/Mapped.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,10 @@ +classdef Mapped < grid.Grid + % General grid mapping + methods (Abstract) + % logicalGrid returns the domain grid of the mapping. + g = logicalGrid(obj); + + % mapping returns the mapped coordinates as a N x D component matrix + m = mapping(obj); + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/Structured.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,7 @@ +classdef Structured < grid.Grid + methods (Abstract) + % Returns the size of the grid in each dimension m = [mx my mz ...] + m = size(obj); % Is this a good idea? Isn't immersed a structured grid? + h = scaling(obj); + end +end
--- a/+grid/Ti.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,201 +0,0 @@ -classdef Ti - properties - gs % {4}Curve - S % FunctionHandle(u,v) - end - - methods - % TODO function to label boundary names. - % function to find largest and smallest delta h in the grid. Maybe shouldnt live here - function obj = Ti(C1,C2,C3,C4) - obj.gs = {C1,C2,C3,C4}; - - g1 = C1.g; - g2 = C2.g; - g3 = C3.g; - g4 = C4.g; - - A = g1(0); - B = g2(0); - C = g3(0); - D = g4(0); - - function o = S_fun(u,v) - x1 = g1(u); - x2 = g2(v); - x3 = g3(1-u); - x4 = g4(1-v); - o1 = (1-v).*x1(1,:) + u.*x2(1,:) + v.*x3(1,:) + (1-u).*x4(1,:) ... - -((1-u)*(1-v).*A(1,:) + u*(1-v).*B(1,:) + u*v.*C(1,:) + (1-u)*v.*D(1,:)); - o2 = (1-v).*x1(2,:) + u.*x2(2,:) + v.*x3(2,:) + (1-u).*x4(2,:) ... - -((1-u)*(1-v).*A(2,:) + u*(1-v).*B(2,:) + u*v.*C(2,:) + (1-u)*v.*D(2,:)); - - o = [o1;o2]; - end - - obj.S = @S_fun; - end - - function [X,Y] = map(obj,u,v) - default_arg('v',u); - - if isscalar(u) - u = linspace(0,1,u); - end - - if isscalar(v) - v = linspace(0,1,v); - end - - S = obj.S; - - nu = length(u); - nv = length(v); - - X = zeros(nv,nu); - Y = zeros(nv,nu); - - u = rowVector(u); - v = rowVector(v); - - for i = 1:nv - p = S(u,v(i)); - X(i,:) = p(1,:); - Y(i,:) = p(2,:); - end - end - - function h = plot(obj,nu,nv) - S = obj.S; - - default_arg('nv',nu) - - u = linspace(0,1,nu); - v = linspace(0,1,nv); - - m = 100; - - X = zeros(nu+nv,m); - Y = zeros(nu+nv,m); - - - t = linspace(0,1,m); - for i = 1:nu - p = S(u(i),t); - X(i,:) = p(1,:); - Y(i,:) = p(2,:); - end - - for i = 1:nv - p = S(t,v(i)); - X(i+nu,:) = p(1,:); - Y(i+nu,:) = p(2,:); - end - - h = line(X',Y'); - end - - - function h = show(obj,nu,nv) - default_arg('nv',nu) - S = obj.S; - - if(nu>2 || nv>2) - h_grid = obj.plot(nu,nv); - set(h_grid,'Color',[0 0.4470 0.7410]); - end - - h_bord = obj.plot(2,2); - set(h_bord,'Color',[0.8500 0.3250 0.0980]); - set(h_bord,'LineWidth',2); - end - - - % TRANSFORMATIONS - function ti = translate(obj,a) - gs = obj.gs; - - for i = 1:length(gs) - new_gs{i} = gs{i}.translate(a); - end - - ti = grid.Ti(new_gs{:}); - end - - % Mirrors the Ti so that the resulting Ti is still left handed. - % (Corrected by reversing curves and switching e and w) - function ti = mirror(obj, a, b) - gs = obj.gs; - - new_gs = cell(1,4); - - new_gs{1} = gs{1}.mirror(a,b).reverse(); - new_gs{3} = gs{3}.mirror(a,b).reverse(); - new_gs{2} = gs{4}.mirror(a,b).reverse(); - new_gs{4} = gs{2}.mirror(a,b).reverse(); - - ti = grid.Ti(new_gs{:}); - end - - function ti = rotate(obj,a,rad) - gs = obj.gs; - - for i = 1:length(gs) - new_gs{i} = gs{i}.rotate(a,rad); - end - - ti = grid.Ti(new_gs{:}); - end - - function ti = rotate_edges(obj,n); - new_gs = cell(1,4); - for i = 0:3 - new_i = mod(i - n,4); - new_gs{new_i+1} = obj.gs{i+1}; - end - ti = grid.Ti(new_gs{:}); - end - end - - methods(Static) - function obj = points(p1, p2, p3, p4) - g1 = grid.Curve.line(p1,p2); - g2 = grid.Curve.line(p2,p3); - g3 = grid.Curve.line(p3,p4); - g4 = grid.Curve.line(p4,p1); - - obj = grid.Ti(g1,g2,g3,g4); - end - - function label(varargin) - if nargin == 2 && ischar(varargin{2}) - label_impl(varargin{:}); - else - for i = 1:length(varargin) - label_impl(varargin{i},inputname(i)); - end - end - - - function label_impl(ti,str) - S = ti.S; - - pc = S(0.5,0.5); - - margin = 0.1; - pw = S( margin, 0.5); - pe = S(1-margin, 0.5); - ps = S( 0.5, margin); - pn = S( 0.5, 1-margin); - - - ti.show(2,2); - grid.place_label(pc,str); - grid.place_label(pw,'w'); - grid.place_label(pe,'e'); - grid.place_label(ps,'s'); - grid.place_label(pn,'n'); - end - end - end -end \ No newline at end of file
--- a/+grid/equal_step_size.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,4 +0,0 @@ -% Calculates M so that the stepsize m/M is as close to n/M as possible -function M = equal_step_size(n,N,m) - M = round((m*(N-1)+n)/n); -end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/equidistant.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,29 @@ +% Creates a cartesian grid of dimension length(m). +% over the doman xlim, ylim, ... +% Examples: +% g = grid.equidistant([mx, my], xlim, ylim) +% g = grid.equidistant([10, 15], {0,1}, {0,2}) +function g = equidistant(m, varargin) + if length(m) ~= length(varargin) + error('grid:equidistant:NonMatchingParameters','The number of provided dimensions do not match.') + end + + for i = 1:length(m) + if ~iscell(varargin{i}) || numel(varargin{i}) ~= 2 + error('grid:equidistant:InvalidLimits','The limits should be cell arrays with 2 elements.'); + end + + if varargin{i}{1} > varargin{i}{2} + error('grid:equidistant:InvalidLimits','The elements of the limit must be increasing.'); + end + end + + X = {}; + h = []; + for i = 1:length(m) + [X{i}, h(i)] = util.get_grid(varargin{i}{:},m(i)); + end + + g = grid.Cartesian(X{:}); + g.h = h; +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/equidistantCurvilinear.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,35 @@ +% Creates a curvilinear grid of dimension length(m). +% over the logical domain xi_lim, eta_lim, ... +% If all limits are ommited they are set to {0,1}. +% Examples: +% g = grid.equidistantCurvilinear(mapping, [m_xi, m_eta]) +% g = grid.equidistantCurvilinear(mapping, [m_xi, m_eta], xi_lim, eta_lim) +% g = grid.equidistantCurvilinear(mapping, [10, 15], {0,1}, {0,1}) +function g = equidistantCurvilinear(mapping, m, varargin) + if isempty(varargin) + varargin = repmat({{0,1}}, [1 length(m)]); + end + + if length(m) ~= length(varargin) + error('grid:equidistant:NonMatchingParameters','The number of provided dimensions do not match.') + end + + for i = 1:length(m) + if ~iscell(varargin{i}) || numel(varargin{i}) ~= 2 + error('grid:equidistant:InvalidLimits','The limits should be cell arrays with 2 elements.'); + end + + if varargin{i}{1} > varargin{i}{2} + error('grid:equidistant:InvalidLimits','The elements of the limit must be increasing.'); + end + end + + X = {}; + h = []; + for i = 1:length(m) + [X{i}, h(i)] = util.get_grid(varargin{i}{:},m(i)); + end + + g = grid.Curvilinear(mapping, X{:}); + g.logic.h = h; +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/equidistantCurvilinearTest.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,7 @@ +function tests = equdistantCurvilinearTest() + tests = functiontests(localfunctions); +end + +function testNoTests(testCase) + testCase.verifyFail(); +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/equidistantTest.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,50 @@ +function tests = equidistantTest() + tests = functiontests(localfunctions); +end + + +function testErrorInvalidLimits(testCase) + in = { + {10,{1}}, + {10,[0,1]}, + {[10,10],{0,1},{1}}, + {[10,10],{1},{1,0}}, + {10,{1,0}}, + {[10, 5],{1,0}, {0,-1}}, + }; + + for i = 1:length(in) + testCase.verifyError(@()grid.equidistant(in{i}{:}),'grid:equidistant:InvalidLimits',sprintf('in(%d) = %s',i,toString(in{i}))); + end +end + +function testErrorNonMatchingParam(testCase) + in = { + {[],{1}}, + {[],{1},{0,1}}, + {[5,5],{0,1},{0,1},{0,1}}, + {[5,5,4],{0,1},{0,1}}, + }; + + for i = 1:length(in) + testCase.verifyError(@()grid.equidistant(in{i}{:}),'grid:equidistant:NonMatchingParameters',sprintf('in(%d) = %s',i,toString(in{i}))); + end +end + + +function testCompiles(testCase) + in = { + {5, {0,1}}, + {[3 3],{0,1},{0,2}}, + }; + + out = { + [[0; 0.25; 0.5; 0.75; 1]], + [[0; 0; 0; 0.5; 0.5; 0.5; 1; 1; 1;],[0; 1; 2; 0; 1; 2; 0; 1; 2;]], + }; + + for i = 1:length(in) + g = grid.equidistant(in{i}{:}); + testCase.verifyEqual(g.points(),out{i}); + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/evalOn.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,45 @@ +% Takes a function and evaluates it on a grid to return a grid function in the +% form of a (n*k)x1 vector, where n is the number of grid points and k is the +% number of components of the function. +% g -- Grid to evaluate on. +% func -- Function to evaluate. May be a function handle or a constant. If +% it is a vector value it has to be provided as a column vector, +function gf = evalOn(g, func) + if ~isa(func, 'function_handle') + % We should have a constant. + if size(func,2) ~= 1 + error('grid:evalOn:VectorValuedWrongDim', 'A vector valued function must be given as a column vector') + end + + gf = repmat(func,[g.N, 1]); + return + end + % func should now be a function_handle + + if g.D ~= nargin(func) + g.D + nargin(func) + error('grid:evalOn:WrongNumberOfInputs', 'The number of inputs of the function must match the dimension of the domain.') + end + + + % Get coordinates and convert to cell array for easier use as a parameter + x = num2cell(g.points()); + + % Find the number of components + x0 = x(1,:); + f0 = func(x0{:}); + k = length(f0); + + if size(f0,2) ~= 1 + error('grid:evalOn:VectorValuedWrongDim', 'A vector valued function must be given as a column vector') + end + + gf = zeros(g.N*k, 1); + % keyboard + for i = 1:g.N + % (1 + (i-1)*k):(i*k) + % func(x{i,:}) + gf((1 + (i-1)*k):(i*k)) = func(x{i,:}); + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/evalOnScalar.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,41 @@ +% WHAT KIND OF A FUNCTION NAME IS THIS?! +% This functions send matrixa arguments to func unlike grid.evalOn() +% Takes a funciton and evaluates it on a grid to return a grid function in the +% form of a (n*k)x1 vector, where n is the number of grid points and k is the +% number of components of the function. +% g -- Grid to evaluate on. +% func -- Function to evaluate. May be a function handle or a constant. If +% it is a vector value it has to be provided as a column vector, +function gf = evalOn(g, func) + if ~isa(func, 'function_handle') + % We should have a constant. + if size(func,2) ~= 1 + error('grid:evalOnScalar:VectorValuedWrongDim', 'A vector valued function must be given as a column vector') + end + + gf = repmat(func,[g.N, 1]); + return + end + % func should now be a function_handle + + % Get coordinates and convert to cell array for easier use as a parameter + x = g.points(); + X = {}; + for i = 1:size(x, 2) + X{i} = x(:,i); + end + + % Find the number of components + x0 = num2cell(x(1,:)); + f0 = func(x0{:}); + k = length(f0); + + if size(f0,2) ~= 1 + error('grid:evalOnScalar:VectorValuedWrongDim', 'A vector valued function must be given as a column vector') + end + + gf = func(X{:}); + if k > 1 % Reorder so that componets sits together. + gf = reshape(reshape(gf, [g.N, k])', [g.N*k, 1]); + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/evalOnScalarTest.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,118 @@ +function tests = evalOnScalarTest() + tests = functiontests(localfunctions); +end + +function testInputConstant(testCase) + in = { + 0, + 47, + 1, + [1; 2], + }; + + out = { + [0; 0; 0], + [47; 47; 47], + [1; 1; 1], + [1; 2; 1; 2; 1; 2], + }; + + g = getTestGrid('1d'); + + for i = 1:length(in) + gf = grid.evalOnScalar(g,in{i}); + testCase.verifyEqual(gf, out{i}); + end +end + +function testInputScalarFunction1d(testCase) + in = { + @(x)1+x*0, + @(x)x, + @(x)x.*x, + }; + + out = { + [1; 1; 1], + [0; 1; 2], + [0; 1; 4], + }; + + g = getTestGrid('1d'); + + for i = 1:length(in) + gf = grid.evalOnScalar(g,in{i}); + testCase.verifyEqual(gf, out{i}); + end +end + +function testInputScalarFunction2d(testCase) + in = { + @(x,y)1+x*0, + @(x,y)x-y, + @(x,y)x./(1+y), + }; + + out = { + [1; 1; 1; 1; 1; 1; 1; 1; 1], + [0; -1; -2; 1; 0; -1; 2; 1; 0], + [0; 0; 0; 1; 1/2; 1/3; 2; 1; 2/3], + }; + + g = getTestGrid('2d'); + + for i = 1:length(in) + gf = grid.evalOnScalar(g, in{i}); + testCase.verifyEqual(gf, out{i}); + end +end + + +function testInputVectorFunction(testCase) + g = getTestGrid('1d'); + in = @(x)[x; -2*x]; + out = [0; 0; 1; -2; 2; -4]; + + gf = grid.evalOnScalar(g,in); + testCase.verifyEqual(gf, out); + + g = getTestGrid('2d'); + in = @(x,y)[x.^2; -2*y]; + out = [ + 0; 0; + 0; -2; + 0; -4; + 1; 0; + 1; -2; + 1; -4; + 4; 0; + 4; -2; + 4; -4; + ]; + + gf = grid.evalOnScalar(g,in); + testCase.verifyEqual(gf, out); +end + + +function testInputErrorVectorValued(testCase) + in = { + [1,2,3], + @(x,y)[x,-y]; + }; + + g = getTestGrid('2d'); + + for i = 1:length(in) + testCase.verifyError(@()grid.evalOnScalar(g, in{i}),'grid:evalOnScalar:VectorValuedWrongDim',sprintf('in(%d) = %s',i,toString(in{i}))); + end +end + +function g = getTestGrid(d) + switch d + case '1d' + g = grid.equidistant(3,{0,2}); + case '2d' + g = grid.equidistant([3,3],{0,2},{0,2}); + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/evalOnTest.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,133 @@ +function tests = evalOnTest() + tests = functiontests(localfunctions); +end + +function testInputConstant(testCase) + in = { + 0, + 47, + 1, + [1; 2], + }; + + out = { + [0; 0; 0], + [47; 47; 47], + [1; 1; 1], + [1; 2; 1; 2; 1; 2], + }; + + g = getTestGrid('1d'); + + for i = 1:length(in) + gf = grid.evalOn(g,in{i}); + testCase.verifyEqual(gf, out{i}); + end +end + +% evalOn should give and error if the number of inputs to func is not the same as +% the number of dimensions of the grid. +function testNumberOfInputs(testCase) + cases = { + {getTestGrid('1d'), @(x,y)x-y}, + {getTestGrid('2d'), @(x)x }, + } + + for i = 1:length(cases) + g = cases{i}{1}; + f = cases{i}{2}; + testCase.verifyError(@()grid.evalOn(g, f),'grid:evalOn:WrongNumberOfInputs',sprintf('in(%d) = %s',i,toString(f))); + end +end + +function testInputScalarFunction1d(testCase) + in = { + @(x)1+x*0, + @(x)x, + @(x)x.*x, + }; + + out = { + [1; 1; 1], + [0; 1; 2], + [0; 1; 4], + }; + + g = getTestGrid('1d'); + + for i = 1:length(in) + gf = grid.evalOn(g,in{i}); + testCase.verifyEqual(gf, out{i}); + end +end + +function testInputScalarFunction2d(testCase) + in = { + @(x,y)1+x*0, + @(x,y)x-y, + @(x,y)x./(1+y), + }; + + out = { + [1; 1; 1; 1; 1; 1; 1; 1; 1], + [0; -1; -2; 1; 0; -1; 2; 1; 0], + [0; 0; 0; 1; 1/2; 1/3; 2; 1; 2/3], + }; + + g = getTestGrid('2d'); + + for i = 1:length(in) + gf = grid.evalOn(g, in{i}); + testCase.verifyEqual(gf, out{i}); + end +end + + +function testInputVectorFunction(testCase) + g = getTestGrid('1d'); + in = @(x)[x; -2*x]; + out = [0; 0; 1; -2; 2; -4]; + + gf = grid.evalOn(g,in); + testCase.verifyEqual(gf, out); + + g = getTestGrid('2d'); + in = @(x,y)[x.^2; -2*y]; + out = [ + 0; 0; + 0; -2; + 0; -4; + 1; 0; + 1; -2; + 1; -4; + 4; 0; + 4; -2; + 4; -4; + ]; + + gf = grid.evalOn(g,in); + testCase.verifyEqual(gf, out); +end + + +function testInputErrorVectorValued(testCase) + in = { + [1,2,3], + @(x,y)[x,-y]; + }; + + g = getTestGrid('2d'); + + for i = 1:length(in) + testCase.verifyError(@()grid.evalOn(g, in{i}),'grid:evalOn:VectorValuedWrongDim',sprintf('in(%d) = %s',i,toString(in{i}))); + end +end + +function g = getTestGrid(d) + switch d + case '1d' + g = grid.equidistant(3,{0,2}); + case '2d' + g = grid.equidistant([3,3],{0,2},{0,2}); + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/funcToComponents.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,8 @@ +% funcToComponents converts a grid function to a N x k matrix, where +% k is the number of vector components of the gridfunction and N is the +% number of points in the grid. +% +% Takes a grid function and and a grid. +function F = funcToComponents(g, gf); + F = reshapeRowMaj(gf, [g.N, length(gf)/g.N]); +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/funcToComponentsTest.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,23 @@ +function tests = funcToComponentsTest() + tests = functiontests(localfunctions); +end + + +function testScalarGf(testCase) + g = getTestGrid(); + gf_in = [1; 2; 3]; + + testCase.verifyEqual(grid.funcToComponents(g, gf_in), gf_in); +end + +function testVectorGf(testCase) + g = getTestGrid(); + gf_in = [1; 2; 3; 4; 5; 6]; + out = [1 2; 3 4; 5 6]; + + testCase.verifyEqual(grid.funcToComponents(g, gf_in), out); +end + +function g = getTestGrid() + g = grid.equidistant(3,{0,2}); +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/funcToMatrix.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,5 @@ +% Converts a gridfunction to a matrix +% Takes a grid function and and a structured grid. +function F = funcToMatrix(g, gf) + F = reshapeRowMaj(gf, g.size()); +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/funcToPlotMatrix.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,5 @@ +% Converts a gridfunction to a plot matrix +% Takes a grid function and and a structured grid. +function F = funcToPlotMatrix(g, gf) + F = reshapeToPlotMatrix(gf, g.size()); +end \ No newline at end of file
--- a/+grid/old/concat_curve.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,12 +0,0 @@ -% Concatenate two curves g1 and g2 intop -% g = concat_curve(g1,g2) -function g = concat_curve(g1,g2) - function v = g_fun(t) - if t < 1/2 - v = g1(2*t); - else - v = g2(2*t-1); - end - end - g = @g_fun; -end \ No newline at end of file
--- a/+grid/old/curve_discretise.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,97 +0,0 @@ -% Discretises the curve g with the smallest number of points such that all segments -% are shorter than h. If do_plot is true the points of the discretisation and -% the normals of the curve in those points are plotted. -% -% [t,p,d] = curve_discretise(g,h,do_plot) -% -% t is a vector of input values to g. -% p is a cector of points. -% d are the length of the segments. -function [t,p,d] = curve_discretise(g,h,do_plot) - default_arg('do_plot',false) - - n = 10; - - [t,p,d] = curve_discretise_n(g,n); - - % ni = 0; - while any(d>h) - [t,p,d] = curve_discretise_n(g,n); - n = ceil(n*d(1)/h); - % ni = ni+1; - end - - % nj = 0; - while all(d<h) - [t,p,d] = curve_discretise_n(g,n); - n = n-1; - % nj = nj+1; - end - [t,p,d] = curve_discretise_n(g,n+1); - - % fprintf('ni = %d, nj = %d\n',ni,nj); - - if do_plot - fprintf('n:%d max: %f min: %f\n', n, max(d),min(d)); - p = grid.map_curve(g,t); - figure - show(g,t,h); - end - -end - -function [t,p,d] = curve_discretise_n(g,n) - t = linspace(0,1,n); - t = equalize_d(g,t); - d = D(g,t); - p = grid.map_curve(g,t); -end - -function d = D(g,t) - p = grid.map_curve(g,t); - - d = zeros(1,length(t)-1); - for i = 1:length(d) - d(i) = norm(p(:,i) - p(:,i+1)); - end -end - -function t = equalize_d(g,t) - d = D(g,t); - v = d-mean(d); - while any(abs(v)>0.01*mean(d)) - dt = t(2:end)-t(1:end-1); - t(2:end) = t(2:end) - cumsum(dt.*v./d); - - t = t/t(end); - d = D(g,t); - v = d-mean(d); - end -end - - -function show(g,t,hh) - p = grid.map_curve(g,t); - - - - h = grid.plot_curve(g); - h.LineWidth = 2; - axis equal - hold on - h = plot(p(1,:),p(2,:),'.'); - h.Color = [0.8500 0.3250 0.0980]; - h.MarkerSize = 24; - hold off - - n = grid.curve_normals(g,t); - hold on - for i = 1:length(t) - p0 = p(:,i); - p1 = p0 + hh*n(:,i); - l = [p0, p1]; - h = plot(l(1,:),l(2,:)); - h.Color = [0.8500 0.3250 0.0980]; - end - -end \ No newline at end of file
--- a/+grid/old/curve_interp.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,58 +0,0 @@ -% Create a cubic spline from the points (x,y) using periodic conditions. -% g = curve_interp(x,y) -function g = curve_interp(x,y) - default_arg('x',[0 2 2 1 1 0]) - default_arg('y',[0 0 2 2 1 1]) - % solve for xp and yp - - % x(t) = at^4 + bt^2+ct+d - - % a = xp1 -2x1 + 2x0 + xp0 - % b = 3x1 -xp1 - 3x0 + 2xp0 - % c = xp0 - % d = x0 - - assert(length(x) == length(y)) - n = length(x); - A = spdiags(ones(n,1)*[2, 8, 2],-1:1,n,n); - A(n,1) = 2; - A(1,n) = 2; - - bx = zeros(n,1); - for i = 2:n-1 - bx(i) = -6*x(i-1)+6*x(i+1); - end - bx(1) = -6*x(n)+6*x(2); - bx(n) = -6*x(n-1)+6*x(1); - - by = zeros(n,1); - for i = 2:n-1 - by(i) = -6*y(i-1)+6*y(i+1); - end - by(1) = -6*y(n)+6*y(2); - by(n) = -6*y(n-1)+6*y(1); - - - xp = A\bx; - yp = A\by; - - x(end+1) = x(1); - y(end+1) = y(1); - - xp(end+1) = xp(1); - yp(end+1) = yp(1); - - function v = g_fun(t) - t = mod(t,1); - i = mod(floor(t*n),n) + 1; - t = t * n -(i-1); - X = (2*x(i)-2*x(i+1)+xp(i)+xp(i+1))*t.^3 + (-3*x(i)+3*x(i+1)-2*xp(i)-xp(i+1))*t.^2 + (xp(i))*t + x(i); - Y = (2*y(i)-2*y(i+1)+yp(i)+yp(i+1))*t.^3 + (-3*y(i)+3*y(i+1)-2*yp(i)-yp(i+1))*t.^2 + (yp(i))*t + y(i); - v = [X;Y]; - end - - g = @g_fun; -end - - -
--- a/+grid/old/max_h.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,33 +0,0 @@ -function [d_max, i1_max, j1_max, i2_max, j2_max] = max_h(X,Y) - ni = size(X,1); - nj = size(X,2); - d_max = 0; - - i1_max = 0; - j1_max = 0; - i2_max = 0; - j2_max = 0; - - D = {[0,-1],[1,0],[0,1],[-1,0]}; - - for i = 1:ni - for j = 1:nj - p1 = [X(i,j); Y(i,j)]; - for k = 1:length(D) - i2 = i+D{k}(1); - j2 = j+D{k}(2); - if i2 >= 1 && i2 <= ni && j2 >= 1 && j2 <= nj - p2 = [X(i2,j2); Y(i2,j2)]; - d = norm(p2-p1); - if d > d_max; - d_max = d; - i1_max = i; - j1_max = j; - i2_max = i2; - j2_max = j2; - end - end - end - end - end -end
--- a/+grid/old/min_h.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,34 +0,0 @@ -function [d_min, i1_min, j1_min, i2_min, j2_min] = min_h(X,Y) - ni = size(X,1); - nj = size(X,2); - d_min = norm([X(1,1);Y(1,1)] - [X(ni,nj);Y(ni,nj)]); - - i1_min = 0; - j1_min = 0; - i2_min = 0; - j2_min = 0; - - D = {[-1,-1],[0,-1],[1,-1],[1,0],[1,1],[0,1],[-1,1],[-1,0]}; - % D = {[0,-1],[1,0],[0,1],[-1,0]}; - - for i = 1:ni - for j = 1:nj - p1 = [X(i,j); Y(i,j)]; - for k = 1:length(D) - i2 = i+D{k}(1); - j2 = j+D{k}(2); - if i2 >= 1 && i2 <= ni && j2 >= 1 && j2 <= nj - p2 = [X(i2,j2); Y(i2,j2)]; - d = norm(p2-p1); - if d < d_min; - d_min = d; - i1_min = i; - j1_min = j; - i2_min = i2; - j2_min = j2; - end - end - end - end - end -end
--- a/+grid/old/plot_shape.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,12 +0,0 @@ -% Plot a shape using n points. Returns cell array of plot handles. -% hs = plot_shape(s,n) -function hs = plot_shape(s,n) - default_arg('n',100); - - hs = {}; - hold on - for i = 1:length(s) - hs{end+1} = grid.plot_curve(s{i},n); - end - hold off -end \ No newline at end of file
--- a/+grid/old/shape.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,4 +0,0 @@ -% Creates a shape from a number of curves. A shape is a cell array of curves. -function s = shape(varargin); - s = varargin; -end \ No newline at end of file
--- a/+grid/old/shape_discretise.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,8 +0,0 @@ -% Discretises a shape such that points on the curves are no further than h appart. -function p = shape_discretise(s,h) - p = []; - for i = 1:length(s) - [~,pt] = grid.curve_discretise(s{i},h); - p = [p, pt]; - end -end \ No newline at end of file
--- a/+grid/old/shape_linesegments.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,9 +0,0 @@ -% Converts a shape into a cell array of linesegments shorter than h. -function l = shape_linesegments(s,h) - l = {}; - - for i = 1:length(s) - t = grid.curve_discretise(s{i},h); - l = [l, grid.curve_linesegments(s{i},t)]; - end -end \ No newline at end of file
--- a/+grid/old/triang_interp.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,132 +0,0 @@ -classdef triang_interp - properties - g1, g2 ,g3 % Curves encirling the tirangle in the positive direction. - A,B,C % The corners of the triangle - Sa, Sb, Sc % Mappings from square with different sides collapsed - end - - methods - function o = triang_interp(g1,g2,g3) - o.g1 = g1; - o.g2 = g2; - o.g3 = g3; - o.A = g1(0); - o.B = g2(0); - o.C = g3(0); - o.Sa = grid.triang_interp.square_to_triangle_interp(g2,g3,g1); - o.Sb = grid.triang_interp.square_to_triangle_interp(g3,g1,g2); - o.Sc = grid.triang_interp.square_to_triangle_interp(g1,g2,g3); - end - - - function show(o,N) - % Show the mapped meridians of the triangle. - % Might be used for the barycentric coordinates. - ma = @(t)o.Sa(1/2,1-t); - mb = @(t)o.Sb(1/2,1-t); - mc = @(t)o.Sc(1/2,1-t); - - na = @(t)o.Sa(t,1/2); - - ka = @(t)(o.g1(1-t)+o.g2(t))/2; - - h = grid.plot_curve(ma); - h.Color = Color.blue; - h = grid.plot_curve(mb); - h.Color = Color.blue; - h = grid.plot_curve(mc); - h.Color = Color.blue; - - h = grid.plot_curve(na); - h.Color = Color.red; - - h = grid.plot_curve(ka); - h.Color = Color.red; - - [a(1),a(2)] = ma(1/3); - [b(1),b(2)] = mb(1/3); - [c(1),c(2)] = mc(1/3); - - d = ka(1-1/3); - - - grid.label_pt(a,b,c,d); - - - % t = linspace(0,1,N); - % for i = 1:N - % sa = @(s)o.Sa(s,t(i)); - % sb = @(s)o.Sb(s,t(i)); - % sc = @(s)o.Sc(s,t(i)); - - % h = grid.plot_curve(sa); - % h.Color = Color.blue; - % h = grid.plot_curve(sb); - % h.Color = Color.blue; - % h = grid.plot_curve(sc); - % h.Color = Color.blue; - % end - - h = grid.plot_curve(o.g1); - h.LineWidth = 2; - h.Color = Color.red; - - h = grid.plot_curve(o.g2); - h.LineWidth = 2; - h.Color = Color.red; - - h = grid.plot_curve(o.g3); - h.LineWidth = 2; - h.Color = Color.red; - - end - - - end - - methods(Static) - % Makes a mapping from the unit square to a triangle by collapsing - % one of the sides of the squares to a corner on the triangle - % The collapsed side is mapped to the corner oposite to g1. - % This is done such that for S(s,t), S(s,1) = g1(s) - function S = square_to_triangle_interp(g1,g2,g3) - corner = grid.line_segment(g3(0),g3(0)); - S = grid.transfinite_interp(corner,g3,f(g1),f(g2)) - - % Function to flip a curve - function h = f(g) - h = @(t)g(1-t); - end - end - end - -end - -% % Return a mapping from u.v to x,y of the domain encircled by g1 g2 g3 in the the positive direction. created be using transfinite interpolation. -% function S = triang_interp(g1,g2,g3) -% A = g1(0) -% B = g2(0) -% C = g3(0) - -% function [x,y] = S_fun(u,v) -% w = sqrt((u-1)^2+v^2)/sqrt(2); % Parameter for g3 -% v = v*(1-u-v)*g1(u) + u*(1-u-v)*g2(v) + u*v*g3(w) ... -% +(1-u)*(1-v)*A+u*(1-v)*B + (1-u)*v*C; -% x = v(1); -% y = v(2); -% end -% S = @S_fun; -% end - - - -% function subsref(obj,S) -% if ~all(isnumeric(S.subs{:})) -% error('Only supports calling object with number') -% end -% if numel(S.subs{:}) > 1 -% disp('You''ve called the object with more than one argument'); -% else -% disp(['You called the object with argument = ',num2str(S.subs{:})]); -% end -% end \ No newline at end of file
--- a/+grid/old/triang_interp_pts.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,12 +0,0 @@ -% Creates a transfinite interpolation from connecting the four points wiht straight lines. -function [S, g1, g2, g3] = triang_interp_pts(p1,p2,p3) - if size(p1) ~= [2 1] - error('p1 is strange!'); - end - - g1 = @(t)(p1 + t*(p2-p1)); - g2 = @(t)(p2 + t*(p3-p2)); - g3 = @(t)(p3 + t*(p1-p3)); - - S = grid.triang_interp(g1,g2,g3); -end
--- a/+grid/old/triang_map.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,29 +0,0 @@ -% Creates a grid [X,Y] from the mapping function S at points in vectors u,v -function [X, Y] = traing_map(S,u,v) - error('not done') - if nargin == 2 - v = u; - end - - if isscalar(u) - u = linspace(0,1,u); - end - - if isscalar(v) - v = linspace(0,1,v); - end - - nu = length(u); - nv = length(v); - - X = zeros(nu,nv); - Y = zeros(nu,nv); - - for i = 1:nu - for j = 1:nv - [x,y] = S(u(i),v(j)); - X(i,j) = x; - Y(i,j) = y; - end - end -end
--- a/+grid/old/triang_plot_interp.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,114 +0,0 @@ -% Plots a transfinite interpolation in x,y space using nu and nv curves along u and v axes. - - - - - - -% Plots a interp of a triangle where one the interpolation is from a square -% with one side collapsed to -function h = triang_plot_interp_kindaworking(S,n) - u = linspace(0,1,n); - v = linspace(0,1,n); - - m = 100; - m = 20; - - Xl_curves = cell(n,1); - Xr_curves = cell(n,1); - Y_curves = cell(n,1); - - - function u = wierdness(v,d,N) - if N == 0 - u = 0; - else - u = N*d./(1-v); - end - end - - - %Y curves - t = linspace(0,1,m); - for i = 1:n - x = []; y = []; - for j = 1:length(t) - [x(j),y(j)] = S(t(j),v(i)); - end - Y_curves{i} = [x', y']; - end - - - % Right and left X curves - t = linspace(0,1,m); - d = u(2); - for i = 1:n - xl = []; yl = []; - xr = []; yr = []; - N = i-1; - t = linspace(0,1-N*d,m); - for j = 1:length(t) - w = wierdness(t(j),d,N); - [xr(j),yr(j)] = S(w,t(j)); - [xl(j),yl(j)] = S(1-w,t(j)); - end - Xl_curves{i} = [xl', yl']; - Xr_curves{i} = [xr', yr']; - end - - for i = 1:n-1 - line(Xl_curves{i}(:,1),Xl_curves{i}(:,2)) - line(Xr_curves{i}(:,1),Xr_curves{i}(:,2)) - line(Y_curves{i}(:,1),Y_curves{i}(:,2)) - end -end - - - - -function h = triang_plot_interp_nonworking(S,n) - - u = linspace(0,1,n); - v = linspace(0,1,n); - - m = 100; - - X_curves = cell(n-1,1); - Y_curves = cell(n-1,1); - K_curves = cell(n-1,1); - - - t = linspace(0,1,m); - for i = 1:n-1 - x = []; y = []; - for j = find(t+u(i) <= 1) - [x(j),y(j)] = S(u(i),t(j)); - end - X_curves{i} = [x', y']; - end - - for i = 1:n-1 - x = []; y = []; - for j = find(t+v(i) <= 1) - [x(j),y(j)] = S(t(j),v(i)); - end - Y_curves{i} = [x', y']; - end - - for i = 2:n - x = []; y = []; - for j = find(t<u(i)) - [x(j),y(j)] = S(t(j), u(i)-t(j)); - end - K_curves{i-1} = [x', y']; - end - - for i = 1:n-1 - line(X_curves{i}(:,1),X_curves{i}(:,2)) - line(Y_curves{i}(:,1),Y_curves{i}(:,2)) - line(K_curves{i}(:,1),K_curves{i}(:,2)) - end - - h = -1; - % h = plot(X_curves{:},Y_curves{:},K_curves{:}); -end
--- a/+grid/old/triang_show.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,23 +0,0 @@ -% Show a grid as a matlab figure. -function triang_show(S,n) - - ih = ishold(); - - hold on - h_grid = grid.triang_plot_interp(S,n); - h_bord = grid.triang_plot_interp(S,2); - - set(h_grid,'Color',[0 0.4470 0.7410]); - set(h_bord,'Color',[0.8500 0.3250 0.0980]); - set(h_bord,'LineWidth',2); - - % axis auto - % axis equal - % axis square - - if ih - hold on - else - hold off - end -end
--- a/+grid/place_label.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,10 +0,0 @@ -function place_label(pt,str,horzAl,vertAl) - default_arg('horzAl','center'); - default_arg('vertAl', 'middle'); - - x = pt(1); - y = pt(2); - h = text(x,y,str); - h.HorizontalAlignment = horzAl; - h.VerticalAlignment = vertAl; -end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+multiblock/BoundaryGroup.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,51 @@ +% BoundaryGroup defines a boundary grouping in a multiblock grid. +classdef BoundaryGroup + properties + blockIDs + names + end + + methods + function obj = BoundaryGroup(varargin) + % Input arguemnts are arbitrary number or 1x2 cell arrays + % representing each boundary in the group. + % The 1st element of the cell array is an integer defining which grid it belongs to. + % The 2nd element of the cell array is the name of the boundary within the block. + % + % Ex: + % bg = multiblock.BoundaryGroup({1,'n'},{1,'s'},{2,'s'}) + + + obj.blockIDs = []; + obj.names = {}; + for i = 1:length(varargin) + if ~iscell(varargin{i}) || ~all(size(varargin{i}) == [1 2]) + error('multiblock:BoundaryGroup:BoundaryGroup:InvalidInput', 'Inputs must be 1x2 cell arrays'); + end + obj.blockIDs(i) = varargin{i}{1}; + obj.names{i} = varargin{i}{2}; + end + end + + function display(obj, name) + + disp(' ') + disp([name, ' =']) + disp(' ') + + if length(obj.names) == 1 + fprintf(' {}\n\n') + return + end + + fprintf(' {') + + fprintf('%d:%s', obj.blockIDs(1), obj.names{1}) + for i = 2:length(obj.names) + fprintf(', %d:%s', obj.blockIDs(i), obj.names{i}); + end + + fprintf('}\n\n') + end + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+multiblock/BoundaryGroupTest.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,30 @@ +function tests = BoundaryGroupTest() + tests = functiontests(localfunctions); +end + +function testCreation(testCase) + in = {{3,'n'},{2,'hoho'},{1,'s'}}; + + blockIDs = [3 2 1]; + names = {'n', 'hoho', 's'}; + + bg = multiblock.BoundaryGroup(in{:}); + testCase.verifyEqual(bg.blockIDs, blockIDs); + testCase.verifyEqual(bg.names, names); +end + +function testInputError(testCase) + in = { + {'n', 's'}, + {{3,'n'},{2,2,'hoho'},{1,'s'}}, + }; + + out = { + 'multiblock:BoundaryGroup:BoundaryGroup:InvalidInput', + 'multiblock:BoundaryGroup:BoundaryGroup:InvalidInput', + }; + + for i = 1:length(in) + testCase.verifyError(@()multiblock.BoundaryGroup(in{i}{:}), out{i}); + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+multiblock/DiffOp.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,151 @@ +classdef DiffOp < scheme.Scheme + properties + grid + order + diffOps + D + H + + blockmatrixDiv + end + + methods + function obj = DiffOp(doHand, grid, order, doParam) + % doHand -- may either be a function handle or a cell array of + % function handles for each grid. The function handle(s) + % should be on the form do = doHand(grid, order, ...) + % Additional parameters for each doHand may be provided in + % the doParam input. + % grid -- a multiblock grid + % order -- integer specifying the order of accuracy + % doParam -- may either be a cell array or a cell array of cell arrays + % for each block. If it is a cell array with length equal + % to the number of blocks then each element is sent to the + % corresponding function handle as extra parameters: + % doHand(..., doParam{i}{:}) Otherwise doParam is sent as + % extra parameters to all doHand: doHand(..., doParam{:}) + default_arg('doParam', []) + + [getHand, getParam] = parseInput(doHand, grid, doParam); + + nBlocks = grid.nBlocks(); + + obj.order = order; + + % Create the diffOps for each block + obj.diffOps = cell(1, nBlocks); + for i = 1:nBlocks + h = getHand(i); + p = getParam(i); + obj.diffOps{i} = h(grid.grids{i}, order, p{:}); + end + + + % Build the norm matrix + H = cell(nBlocks, nBlocks); + for i = 1:nBlocks + H{i,i} = obj.diffOps{i}.H; + end + obj.H = blockmatrix.toMatrix(H); + + + % Build the differentiation matrix + obj.blockmatrixDiv = {grid.Ns, grid.Ns}; + D = blockmatrix.zero(obj.blockmatrixDiv); + for i = 1:nBlocks + D{i,i} = obj.diffOps{i}.D; + end + + for i = 1:nBlocks + for j = 1:nBlocks + intf = grid.connections{i,j}; + if isempty(intf) + continue + end + + + [ii, ij] = obj.diffOps{i}.interface(intf{1}, obj.diffOps{j}, intf{2}); + D{i,i} = D{i,i} + ii; + D{i,j} = D{i,j} + ij; + + [jj, ji] = obj.diffOps{j}.interface(intf{2}, obj.diffOps{i}, intf{1}); + D{j,j} = D{j,j} + jj; + D{j,i} = D{j,i} + ji; + end + end + obj.D = blockmatrix.toMatrix(D); + + + function [getHand, getParam] = parseInput(doHand, grid, doParam) + if ~isa(grid, 'multiblock.Grid') + error('multiblock:DiffOp:DiffOp:InvalidGrid', 'Requires a multiblock grid.'); + end + + if iscell(doHand) && length(doHand) == grid.nBlocks() + getHand = @(i)doHand{i}; + elseif isa(doHand, 'function_handle') + getHand = @(i)doHand; + else + error('multiblock:DiffOp:DiffOp:InvalidGridDoHand', 'doHand must be a function handle or a cell array of length grid.nBlocks'); + end + + if isempty(doParam) + getParam = @(i){}; + elseif iscell(doParam) && length(doParam) == grid.nBlocks() + getParam = @(i)doParam{i}; + else + getParam = @(i)doParam; + end + end + end + + function ops = splitOp(obj, op) + % Splits a matrix operator into a cell-matrix of matrix operators for + % each grid. + ops = sparse2cell(op, obj.NNN); + end + + % Creates the closere and penalty matrix for a given boundary condition, + % boundary -- the name of the boundary on the form [id][name] where + % id is the number of a block and name is the name of a + % boundary of that block example: 1s or 3w + function [closure, penalty] = boundary_condition(obj,boundary,type) + I = boundary{1}; + name = boundary{2}; + + % Get the closure and penaly matrices + [blockClosure, blockPenalty] = obj.diffOps{I}.boundary_condition(name, type); + + % Expand to matrix for full domain. + div = obj.blockmatrixDiv; + closure = blockmatrix.zero(div); + closure{I,I} = blockClosure; + closure = blockmatrix.toMatrix(closure); + + div{2} = 1; % Penalty is a column vector + if ~iscell(blockPenalty) + p = blockmatrix.zero(div); + p{I} = blockPenalty; + penalty = blockmatrix.toMatrix(p); + else + for i = 1:length(blockPenalty) + p = blockmatrix.zero(div); + p{I} = blockPenalty{i}; + penalty{i} = blockmatrix.toMatrix(p); + end + end + end + + function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary) + + end + + % Size returns the number of degrees of freedom + function N = size(obj) + N = 0; + for i = 1:length(obj.diffOps) + N = N + obj.diffOps{i}.size(); + end + end + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+multiblock/DiffOpTest.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,59 @@ +function tests = DiffOpTest() + tests = functiontests(localfunctions); +end + +function testCreation(testCase) + do = newMultiblockOp(); +end + +function testSplitOp(testCase) + testCase.verifyFail(); +end + +function testBoundary_condition(testCase) + testCase.verifyFail(); +end + +function testInterface(testCase) + testCase.verifyFail(); +end + +function testSize(testCase) + mbDo = newMultiblockOp(); + testCase.verifyEqual(mbDo.size(), 15) +end + + +function do = mockDiffOp(size, bc, interface) + do.H = 1; + do.D = 1; + + do.size = size; + do.boundary_condition = bc; + do.interface = interface; +end + + +function do = newMultiblockOp() + grids = { + grid.Cartesian([0 1 2], [3 4 5]); + grid.Cartesian([1 2 3], [10 20]); + }; + + conn = cell(2,2); + conn{1, 2} = {'s','n'}; + + mbGrid = multiblock.Grid(grids, conn); + + function [c, p] = boundary_condition(~,~,~,~) + c = 1; p = 1; + end + + function [c, p] = interface(~,~,~,~) + c = 1; p = 1; + end + + doHand = @(grid,~)mockDiffOp(@(~)prod(grid.size()), @boundary_condition, @interface); + + do = multiblock.DiffOp(doHand, mbGrid, 0); +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+multiblock/Grid.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,120 @@ +classdef Grid < grid.Grid + properties + grids + connections + boundaryGroups + + nPoints + end + + % General multiblock grid + methods + + % grids -- cell array of N grids + % connections -- NxN upper triangular cell matrix. connections{i,j} + % specifies the connection between block i and j. If + % it's empty there is no connection otherwise it's a 2 + % -cell-vector with strings naming the boundaries to be + % connected. (inverted coupling?) + %% Should we have boundary groups at all? maybe it can be handled in a + %% cleaner way outside of the class. + function obj = Grid(grids, connections, boundaryGroups) + obj.grids = grids; + obj.connections = connections; + + obj.nPoints = 0; + for i = 1:length(grids) + obj.nPoints = obj.nPoints + grids{i}.N(); + end + + % if iscell(boundaryGroups) + end + + function n = size(obj) + n = length(obj.grids); + end + + % N returns the number of points in the grid + function o = N(obj) + o = obj.nPoints; + end + + % Ns returns the number of points in each sub grid as a vector + function o = Ns(obj) + ns = zeros(1,obj.nBlocks); + for i = 1:obj.nBlocks; + ns(i) = obj.grids{i}.N(); + end + o = ns; + end + + function n = nBlocks(obj) + n = length(obj.grids); + end + + % d returns the spatial dimension of the grid + function o = D(obj) + o = obj.grids{1}.D(); + end + + % points returns a n x d matrix containing the coordinates for all points. + function X = points(obj) + X = []; + for i = 1:length(obj.grids) + X = [X; obj.grids{i}.points]; + end + end + + % Split a grid function on obj to a cell array of grid function on each block + function gfs = splitFunc(obj, gf) + nComponents = length(gf)/obj.nPoints; + nBlocks = length(obj.grids); + + % Collect number of points in each block + N = zeros(1,nBlocks); + for i = 1:nBlocks + N(i) = obj.grids{i}.N(); + end + + gfs = mat2cell(gf, N, 1); + end + + % Restricts the grid function gf on obj to the subgrid g. + function gf = restrictFunc(obj, gf, g) + gfs = obj.splitFunc(gf); + + for i = 1:length(obj.grids) + gfs{i} = obj.grids{i}.restrictFunc(gfs{i}, g.grids{i}); + end + + gf = cell2mat(gfs); + end + + % Projects the grid function gf on obj to the grid g. + function o = projectFunc(obj, gf, g) + error('not implemented') + + p = g.points(); + o = zeros(length(p),1); + for i = 1:length(p) + I = whatGrid(p(i)); + o(i) = obj.grids{I}.projectFunc(gf, p(i)); + end + + + function I = whatGrid(p) + % Find what grid a point lies on + end + + end + + function bs = getBoundaryNames(obj) + bs = []; + end + + % Return coordinates for the given boundary + function b = getBoundary(obj, name) + b = []; + end + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+multiblock/GridTest.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,37 @@ +function tests = GridTest() + tests = functiontests(localfunctions); +end + +function testCreation(testCase) + g = multiblock.Grid({},{}); +end + +function testMissing(testCase) + testCase.verifyFail(); +end + +function testGetBoundaryNames(testCase) + [grids, conn] = prepareAdjecentBlocks(); + + mbg = multiblock.Grid(grids, conn, multiblock.BoundaryGroup({1,'w'},{2,'w'}) ); + + testCase.verifyFail(); +end + +function testGetBoundary(testCase) + [grids, conn] = prepareAdjecentBlocks(); + + mbg = multiblock.Grid(grids, conn, multiblock.BoundaryGroup({1,'w'},{2,'w'}) ); + testCase.verifyFail(); +end + + +function [grids, conn] = prepareAdjecentBlocks() + grids = { + grid.Cartesian([0 1 2], [3 4 5]); + grid.Cartesian([1 2], [10 20]); + }; + + conn = cell(2,2); + conn{1, 2} = {'s','n'}; +end \ No newline at end of file
--- a/+multiblock/gridVector1d.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,8 +0,0 @@ -function x = gridVector1d(schms) - n = length(schms); - x = cell(n,1); - - for i = 1:n - x{i} = schms{i}.x; - end -end \ No newline at end of file
--- a/+multiblock/gridVector2d.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,10 +0,0 @@ -function [x,y] = gridVector2d(schms) - n = length(schms); - x = cell(n,1); - y = cell(n,1); - - for i = 1:n - x{i} = schms{i}.x; - y{i} = schms{i}.y; - end -end \ No newline at end of file
--- a/+multiblock/solutionVector2cell.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,15 +0,0 @@ -function v_cell = solutionVector2cell(schms,v,components) - if nargin == 2 - components = 1; - end - - N = length(schms); - v_cell = cell(N,1); - - i_start = 1; - for i =1:N - i_end = i_start + schms{i}.size()*components - 1; - v_cell{i} = v(i_start:i_end); - i_start = i_end+1; - end -end \ No newline at end of file
--- a/+multiblock/stitchSchemes.m Tue Sep 06 10:36:33 2016 +0200 +++ b/+multiblock/stitchSchemes.m Thu Sep 08 08:47:16 2016 +0200 @@ -11,24 +11,24 @@ % Output parameters are cell arrays and cell matrices. % % Ex: [schms, D, H] = stitchSchemes(schmHand, order, schmParam, blocks, ms, conn, bound) -function [schms, D, H] = stitchSchemes(schmHand, order, schmParam, blocks, ms, conn, bound) +function [schms, D, H] = stitchSchemes(schmHand, order, schmParam, grids, conn, bound) default_arg('schmParam',[]); - n_blocks = numel(blocks); + n_blocks = numel(grids); % Creating Schemes for i = 1:n_blocks if isempty(schmParam); - schms{i} = schmHand(ms{i},blocks{i},order,[]); + schms{i} = schmHand(grids{i},order,[]); elseif ~iscell(schmParam) param = schmParam(i); - schms{i} = schmHand(ms{i},blocks{i},order,param); + schms{i} = schmHand(grids{i},order,param); else param = schmParam{i}; if iscell(param) - schms{i} = schmHand(ms{i},blocks{i},order,param{:}); + schms{i} = schmHand(grids{i},order,param{:}); else - schms{i} = schmHand(ms{i},blocks{i},order,param); + schms{i} = schmHand(grids{i},order,param); end end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+noname/calcSol.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,10 @@ +% Calculates the solution of a discr at a given time using aligned timesteps. +% Returns the solution as a grid function as defined in +grid +function gf = calculateSolution(discr, T, tsOpt) + k_max = discr.getTimestep(tsOpt); + [k,N] = alignedTimestep(k_max,T); + tsOpt.k = k; + ts = discr.getTimestepper(tsOpt); + + gf = ts.stepN(N-ts.n); +end \ No newline at end of file
--- a/+noname/calculateSolution.m Tue Sep 06 10:36:33 2016 +0200 +++ b/+noname/calculateSolution.m Thu Sep 08 08:47:16 2016 +0200 @@ -13,7 +13,7 @@ fprintf('File ''%s'' already exist.',filename); do_append = yesnoQuestion('Do you want to append to it?'); if ~do_append - fprintf('Exiting...\n'); + fprintf('Exiting. No Solutions calculated.\n'); return end end
--- a/+noname/convergence.m Tue Sep 06 10:36:33 2016 +0200 +++ b/+noname/convergence.m Thu Sep 08 08:47:16 2016 +0200 @@ -1,5 +1,5 @@ % Reference is either a key or a function handle -function [q, e, h, runtime] = convergence(filename, errorFunc, reference, method, order, m, T) +function [q, e, h, runtime] = convergence(filename, errorFunc, reference, name, order, m, T) default_arg('errorFunc', @scheme.error1d); sf = SolutionFile(filename); @@ -7,7 +7,7 @@ % Generate convergence, error, and efficiency plots for each search key with more than one entry. for i = 1:length(m) - key.method = method; + key.name = name; key.order = order; key.m = m(i); key.T = T; @@ -30,18 +30,11 @@ % Get the reference solution vector if isa(reference,'function_handle'); - x = v_repr.x; + x = v_repr.grid.points(); v_ref = reference(x,T); else % Downsample the reference solution - x = v_repr.x; - x_ref = reference.x; - - [~,I] = ismember(x,x_ref,'rows'); - if any(I == 0) - error('Solution and reference solution seem to be on different grids.'); - end - v_ref = reference.v(I); + v_ref = reference.grid.restrictFunc(reference.v, v_repr.grid); end e = errorFunc(discr,v, v_ref);
--- a/+noname/plotSolutions.m Tue Sep 06 10:36:33 2016 +0200 +++ b/+noname/plotSolutions.m Thu Sep 08 08:47:16 2016 +0200 @@ -10,7 +10,7 @@ key = sf.keys{i}; entry = sf.get(key); - method = key.method; + name = key.name; order = key.order; m = key.m; T = key.T; @@ -23,11 +23,11 @@ update(repr); if save_figures - figname = sprintf('%s_%s_o%d_m%d_T%d',figename_prefix,method,order,m,i); + figname = sprintf('%s_%s_o%d_m%d_T%d',figename_prefix,name,order,m,i); fprintf('Saving figure to ''%s''\n',figname); saveeps(hand,figname); end end -end \ No newline at end of file +end
--- a/+noname/testCfl.m Tue Sep 06 10:36:33 2016 +0200 +++ b/+noname/testCfl.m Thu Sep 08 08:47:16 2016 +0200 @@ -27,29 +27,37 @@ end end + if silentFlag + rsInterval = util.ReplaceableString(''); + end + % Use bisection to find sharp estimate while( (alpha0(2)-alpha0(1))/alpha0(1) > tol) alpha = mean(alpha0); if ~silentFlag fprintf('[%.3e,%.3e]: ', alpha0(1), alpha0(2)); + else + rsInterval.update('[%.3e,%.3e]: ', alpha0(1), alpha0(2)); end [ok, n_step, maxVal] = testAlpha(alpha); if ok alpha0(1) = alpha; + stability = 'STABLE'; else alpha0(2) = alpha; + stability = 'UNSTABLE'; end if ~silentFlag - fprintf('a = %.3e, n_step=%d max = %.2e\n', alpha, n_step, maxVal); + fprintf('a = %.3e, n_step=%d %8s max = %.2e\n', alpha, n_step, stability, maxVal); end end if silentFlag - fprintf('Last calculated: a = %.3e, n_step=%d max = %.2e\n', alpha, n_step, maxVal); + rsInterval = util.ReplaceableString(''); end fprintf('T = %-3d dof = %-4d order = %d: clf = %.4e\n',T, discr.size(), discr.order, alpha0(1));
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+parametrization/Curve.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,388 @@ +classdef Curve + properties + g + gp + transformation + end + + methods + %TODO: + % Concatenation of curves + % Subsections of curves + % Stretching of curve parameter - done for arc length. + % Curve to cell array of linesegments + + % Returns a curve object. + % g -- curve parametrization for parameter between 0 and 1 + % gp -- parametrization of curve derivative + function obj = Curve(g,gp,transformation) + default_arg('gp',[]); + default_arg('transformation',[]); + p_test = g(0); + assert(all(size(p_test) == [2,1]), 'A curve parametrization must return a 2x1 vector.'); + + if ~isempty(transformation) + transformation.base_g = g; + transformation.base_gp = gp; + [g,gp] = parametrization.Curve.transform_g(g,gp,transformation); + end + + obj.g = g; + obj.gp = gp; + obj.transformation = transformation; + + end + + function n = normal(obj,t) + assert(~isempty(obj.gp),'Curve has no derivative!'); + deriv = obj.gp(t); + normalization = sqrt(sum(deriv.^2,1)); + n = [-deriv(2,:)./normalization; deriv(1,:)./normalization]; + end + + + % Plots a curve g(t) for 0<t<1, using n points. Returns a handle h to the plotted curve. + % h = plot_curve(g,n) + function h = plot(obj,n) + default_arg('n',100); + + t = linspace(0,1,n); + p = obj.g(t); + h = line(p(1,:),p(2,:)); + end + + function h= plot_normals(obj,l,n,m) + default_arg('l',0.1); + default_arg('n',10); + default_arg('m',100); + t_n = linspace(0,1,n); + + normals = obj.normal(t_n)*l; + + n0 = obj.g(t_n); + n1 = n0 + normals; + + h = line([n0(1,:); n1(1,:)],[n0(2,:); n1(2,:)]); + set(h,'Color',Color.red); + obj.plot(m); + end + + function h= show(obj,name) + p = obj.g(1/2); + n = obj.normal(1/2); + p = p + n*0.1; + + % Add arrow + + h = text(p(1),p(2),name); + h.HorizontalAlignment = 'center'; + h.VerticalAlignment = 'middle'; + + obj.plot(); + end + % Shows curve with name and arrow for direction. + + % Length of arc from parameter t0 to t1 (which may be vectors). + % Computed using derivative. + function L = arcLength(obj,t0,t1) + assert(~isempty(obj.gp),'Curve has no derivative!'); + speed = @(t) sqrt(sum(obj.gp(t).^2,1)); + L = integral_vec(speed,t0,t1); + end + + % Creates the arc length parameterization of a curve. + % N -- number of points used to approximate the arclength function + function curve = arcLengthParametrization(obj,N) + default_arg('N',100); + assert(~isempty(obj.gp),'Curve has no derivative!'); + + % Construct arcLength function using splines + tvec = linspace(0,1,N); + arcVec = obj.arcLength(0,tvec); + tFunc = spline(arcVec,tvec); % t as a function of arcLength + L = obj.arcLength(0,1); + arcPar = @(s) tFunc(s*L); + + % New function and derivative + g_new = @(t)obj.g(arcPar(t)); + gp_new = @(t) normalize(obj.gp(arcPar(t))); + curve = parametrization.Curve(g_new,gp_new); + + end + + % how to make it work for methods without returns + function p = subsref(obj,S) + %Should i add error checking here? + %Maybe if you want performance you fetch obj.g and then use that + switch S(1).type + case '()' + p = obj.g(S.subs{1}); + % case '.' + + % p = obj.(S.subs); + otherwise + p = builtin('subsref',obj,S); + % error() + end + end + + + %% TRANSFORMATION OF A CURVE + function D = reverse(C) + % g = C.g; + % gp = C.gp; + % D = parametrization.Curve(@(t)g(1-t),@(t)-gp(1-t)); + D = C.transform([],[],-1); + end + + function D = transform(C,A,b,flip) + default_arg('A',[1 0; 0 1]); + default_arg('b',[0; 0]); + default_arg('flip',1); + if isempty(C.transformation) + g = C.g; + gp = C.gp; + transformation.A = A; + transformation.b = b; + transformation.flip = flip; + else + g = C.transformation.base_g; + gp = C.transformation.base_gp; + A_old = C.transformation.A; + b_old = C.transformation.b; + flip_old = C.transformation.flip; + + transformation.A = A*A_old; + transformation.b = A*b_old + b; + transformation.flip = flip*flip_old; + end + + D = parametrization.Curve(g,gp,transformation); + + end + + function D = translate(C,a) + g = C.g; + gp = C.gp; + + % function v = g_fun(t) + % x = g(t); + % v(1,:) = x(1,:)+a(1); + % v(2,:) = x(2,:)+a(2); + % end + + % D = parametrization.Curve(@g_fun,gp); + + D = C.transform([],a); + end + + function D = mirror(C, a, b) + assert_size(a,[2,1]); + assert_size(b,[2,1]); + + g = C.g; + gp = C.gp; + + l = b-a; + lx = l(1); + ly = l(2); + + + % fprintf('Singular?\n') + + A = [lx^2-ly^2 2*lx*ly; 2*lx*ly ly^2-lx^2]/(l'*l); + + % function v = g_fun(t) + % % v = a + A*(g(t)-a) + % x = g(t); + + % ax1 = x(1,:)-a(1); + % ax2 = x(2,:)-a(2); + % v(1,:) = a(1)+A(1,:)*[ax1;ax2]; + % v(2,:) = a(2)+A(2,:)*[ax1;ax2]; + % end + + % function v = gp_fun(t) + % v = A*gp(t); + % end + + % D = parametrization.Curve(@g_fun,@gp_fun); + + % g = A(g-a)+a = Ag - Aa + a; + b = - A*a + a; + D = C.transform(A,b); + + end + + function D = rotate(C,a,rad) + assert_size(a, [2,1]); + assert_size(rad, [1,1]); + g = C.g; + gp = C.gp; + + + A = [cos(rad) -sin(rad); sin(rad) cos(rad)]; + + % function v = g_fun(t) + % % v = a + A*(g(t)-a) + % x = g(t); + + % ax1 = x(1,:)-a(1); + % ax2 = x(2,:)-a(2); + % v(1,:) = a(1)+A(1,:)*[ax1;ax2]; + % v(2,:) = a(2)+A(2,:)*[ax1;ax2]; + % end + + % function v = gp_fun(t) + % v = A*gp(t); + % end + + % D = parametrization.Curve(@g_fun,@gp_fun); + + + % g = A(g-a)+a = Ag - Aa + a; + b = - A*a + a; + D = C.transform(A,b); + end + end + + methods (Static) + + % Computes the derivative of g: R -> R^2 using an operator D1 + function gp_out = numericalDerivative(g,D1) + m = length(D1); + t = linspace(0,1,m); + gVec = g(t)'; + gpVec = (D1*gVec)'; + + gp1_fun = spline(t,gpVec(1,:)); + gp2_fun = spline(t,gpVec(2,:)); + gp_out = @(t) [gp1_fun(t);gp2_fun(t)]; + end + + function obj = line(p1, p2) + + function v = g_fun(t) + v(1,:) = p1(1) + t.*(p2(1)-p1(1)); + v(2,:) = p1(2) + t.*(p2(2)-p1(2)); + end + g = @g_fun; + + obj = parametrization.Curve(g); + end + + function obj = circle(c,r,phi) + default_arg('phi',[0; 2*pi]) + default_arg('c',[0; 0]) + default_arg('r',1) + + function v = g_fun(t) + w = phi(1)+t*(phi(2)-phi(1)); + v(1,:) = c(1) + r*cos(w); + v(2,:) = c(2) + r*sin(w); + end + + function v = g_fun_deriv(t) + w = phi(1)+t*(phi(2)-phi(1)); + v(1,:) = -(phi(2)-phi(1))*r*sin(w); + v(2,:) = (phi(2)-phi(1))*r*cos(w); + end + + obj = parametrization.Curve(@g_fun,@g_fun_deriv); + end + + function obj = bezier(p0, p1, p2, p3) + function v = g_fun(t) + v(1,:) = (1-t).^3*p0(1) + 3*(1-t).^2.*t*p1(1) + 3*(1-t).*t.^2*p2(1) + t.^3*p3(1); + v(2,:) = (1-t).^3*p0(2) + 3*(1-t).^2.*t*p1(2) + 3*(1-t).*t.^2*p2(2) + t.^3*p3(2); + end + + function v = g_fun_deriv(t) + v(1,:) = 3*(1-t).^2*(p1(1)-p0(1)) + 6*(1-t).*t*(p2(1)-p1(1)) + 3*t.^2*(p3(1)-p2(1)); + v(2,:) = 3*(1-t).^2*(p1(2)-p0(2)) + 6*(1-t).*t*(p2(2)-p1(2)) + 3*t.^2*(p3(2)-p2(2)); + end + + obj = parametrization.Curve(@g_fun,@g_fun_deriv); + end + + + function [g_out,gp_out] = transform_g(g,gp,tr) + A = tr.A; + b = tr.b; + flip = tr.flip; + + function v = g_fun_noflip(t) + % v = A*g + b + x = g(t); + + v(1,:) = A(1,:)*x+b(1); + v(2,:) = A(2,:)*x+b(2); + end + + function v = g_fun_flip(t) + % v = A*g + b + x = g(1-t); + + v(1,:) = A(1,:)*x+b(1); + v(2,:) = A(2,:)*x+b(2); + end + + + switch flip + case 1 + g_out = @g_fun_noflip; + gp_out = @(t)A*gp(t); + case -1 + g_out = @g_fun_flip; + gp_out = @(t)-A*gp(1-t); + end + end + + end +end + + + +function g_norm = normalize(g0) + g1 = g0(1,:); + g2 = g0(2,:); + normalization = sqrt(sum(g0.^2,1)); + g_norm = [g1./normalization; g2./normalization]; +end + +function I = integral_vec(f,a,b) +% Wrapper around the built-in function integral that +% handles multiple limits. + + Na = length(a); + Nb = length(b); + assert(Na == 1 || Nb == 1 || Na==Nb,... + 'a and b must have same length, unless one is a scalar.'); + + if(Na>Nb); + I = zeros(size(a)); + for i = 1:Na + I(i) = integral(f,a(i),b); + end + elseif(Nb>Na) + I = zeros(size(b)); + for i = 1:Nb + I(i) = integral(f,a,b(i)); + end + else + I = zeros(size(b)); + for i = 1:Nb + I(i) = integral(f,a(i),b(i)); + end + end +end + +% Returns a function handle to the spline. +function f = spline(tval,fval,spline_order) + default_arg('spline_order',4); + [m,~] = size(tval); + assert(m==1,'Need row vectors.'); + + f_spline = spapi( optknt(tval,spline_order), tval, fval ); + f = @(t) fnval(f_spline,t); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+parametrization/Ti.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,216 @@ +classdef Ti + properties + gs % {4}Curve + S % FunctionHandle(u,v) + end + + methods + % TODO function to label boundary names. + % function to find largest and smallest delta h in the grid. Maybe shouldnt live here + function obj = Ti(C1,C2,C3,C4) + obj.gs = {C1,C2,C3,C4}; + + g1 = C1.g; + g2 = C2.g; + g3 = C3.g; + g4 = C4.g; + + A = g1(0); + B = g2(0); + C = g3(0); + D = g4(0); + + function o = S_fun(u,v) + x1 = g1(u); + x2 = g2(v); + x3 = g3(1-u); + x4 = g4(1-v); + o1 = (1-v).*x1(1,:) + u.*x2(1,:) + v.*x3(1,:) + (1-u).*x4(1,:) ... + -((1-u)*(1-v).*A(1,:) + u*(1-v).*B(1,:) + u*v.*C(1,:) + (1-u)*v.*D(1,:)); + o2 = (1-v).*x1(2,:) + u.*x2(2,:) + v.*x3(2,:) + (1-u).*x4(2,:) ... + -((1-u)*(1-v).*A(2,:) + u*(1-v).*B(2,:) + u*v.*C(2,:) + (1-u)*v.*D(2,:)); + + o = [o1;o2]; + end + + obj.S = @S_fun; + end + + % Does this funciton make sense? + % Should it always be eval? + function [X,Y] = map(obj,u,v) + default_arg('v',u); + + if isscalar(u) + u = linspace(0,1,u); + end + + if isscalar(v) + v = linspace(0,1,v); + end + + S = obj.S; + + nu = length(u); + nv = length(v); + + X = zeros(nv,nu); + Y = zeros(nv,nu); + + u = rowVector(u); + v = rowVector(v); + + for i = 1:nv + p = S(u,v(i)); + X(i,:) = p(1,:); + Y(i,:) = p(2,:); + end + end + + % Evaluate S for each pair of u and v, + % Return same shape as u + function [x, y] = eval(obj, u, v) + x = zeros(size(u)); + y = zeros(size(u)); + + for i = 1:numel(u) + p = obj.S(u(i), v(i)); + x(i) = p(1,:); + y(i) = p(2,:); + end + end + + function h = plot(obj,nu,nv) + S = obj.S; + + default_arg('nv',nu) + + u = linspace(0,1,nu); + v = linspace(0,1,nv); + + m = 100; + + X = zeros(nu+nv,m); + Y = zeros(nu+nv,m); + + + t = linspace(0,1,m); + for i = 1:nu + p = S(u(i),t); + X(i,:) = p(1,:); + Y(i,:) = p(2,:); + end + + for i = 1:nv + p = S(t,v(i)); + X(i+nu,:) = p(1,:); + Y(i+nu,:) = p(2,:); + end + + h = line(X',Y'); + end + + + function h = show(obj,nu,nv) + default_arg('nv',nu) + S = obj.S; + + if(nu>2 || nv>2) + h_grid = obj.plot(nu,nv); + set(h_grid,'Color',[0 0.4470 0.7410]); + end + + h_bord = obj.plot(2,2); + set(h_bord,'Color',[0.8500 0.3250 0.0980]); + set(h_bord,'LineWidth',2); + end + + + % TRANSFORMATIONS + function ti = translate(obj,a) + gs = obj.gs; + + for i = 1:length(gs) + new_gs{i} = gs{i}.translate(a); + end + + ti = parametrization.Ti(new_gs{:}); + end + + % Mirrors the Ti so that the resulting Ti is still left handed. + % (Corrected by reversing curves and switching e and w) + function ti = mirror(obj, a, b) + gs = obj.gs; + + new_gs = cell(1,4); + + new_gs{1} = gs{1}.mirror(a,b).reverse(); + new_gs{3} = gs{3}.mirror(a,b).reverse(); + new_gs{2} = gs{4}.mirror(a,b).reverse(); + new_gs{4} = gs{2}.mirror(a,b).reverse(); + + ti = parametrization.Ti(new_gs{:}); + end + + function ti = rotate(obj,a,rad) + gs = obj.gs; + + for i = 1:length(gs) + new_gs{i} = gs{i}.rotate(a,rad); + end + + ti = parametrization.Ti(new_gs{:}); + end + + function ti = rotate_edges(obj,n); + new_gs = cell(1,4); + for i = 0:3 + new_i = mod(i - n,4); + new_gs{new_i+1} = obj.gs{i+1}; + end + ti = parametrization.Ti(new_gs{:}); + end + end + + methods(Static) + function obj = points(p1, p2, p3, p4) + g1 = parametrization.Curve.line(p1,p2); + g2 = parametrization.Curve.line(p2,p3); + g3 = parametrization.Curve.line(p3,p4); + g4 = parametrization.Curve.line(p4,p1); + + obj = parametrization.Ti(g1,g2,g3,g4); + end + + function label(varargin) + if nargin == 2 && ischar(varargin{2}) + label_impl(varargin{:}); + else + for i = 1:length(varargin) + label_impl(varargin{i},inputname(i)); + end + end + + + function label_impl(ti,str) + S = ti.S; + + pc = S(0.5,0.5); + + margin = 0.1; + pw = S( margin, 0.5); + pe = S(1-margin, 0.5); + ps = S( 0.5, margin); + pn = S( 0.5, 1-margin); + + + ti.show(2,2); + parametrization.place_label(pc,str); + parametrization.place_label(pw,'w'); + parametrization.place_label(pe,'e'); + parametrization.place_label(ps,'s'); + parametrization.place_label(pn,'n'); + end + end + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+parametrization/equal_step_size.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,4 @@ +% Calculates M so that the stepsize m/M is as close to n/M as possible +function M = equal_step_size(n,N,m) + M = round((m*(N-1)+n)/n); +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+parametrization/old/concat_curve.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,12 @@ +% Concatenate two curves g1 and g2 intop +% g = concat_curve(g1,g2) +function g = concat_curve(g1,g2) + function v = g_fun(t) + if t < 1/2 + v = g1(2*t); + else + v = g2(2*t-1); + end + end + g = @g_fun; +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+parametrization/old/curve_discretise.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,97 @@ +% Discretises the curve g with the smallest number of points such that all segments +% are shorter than h. If do_plot is true the points of the discretisation and +% the normals of the curve in those points are plotted. +% +% [t,p,d] = curve_discretise(g,h,do_plot) +% +% t is a vector of input values to g. +% p is a cector of points. +% d are the length of the segments. +function [t,p,d] = curve_discretise(g,h,do_plot) + default_arg('do_plot',false) + + n = 10; + + [t,p,d] = curve_discretise_n(g,n); + + % ni = 0; + while any(d>h) + [t,p,d] = curve_discretise_n(g,n); + n = ceil(n*d(1)/h); + % ni = ni+1; + end + + % nj = 0; + while all(d<h) + [t,p,d] = curve_discretise_n(g,n); + n = n-1; + % nj = nj+1; + end + [t,p,d] = curve_discretise_n(g,n+1); + + % fprintf('ni = %d, nj = %d\n',ni,nj); + + if do_plot + fprintf('n:%d max: %f min: %f\n', n, max(d),min(d)); + p = parametrization.map_curve(g,t); + figure + show(g,t,h); + end + +end + +function [t,p,d] = curve_discretise_n(g,n) + t = linspace(0,1,n); + t = equalize_d(g,t); + d = D(g,t); + p = parametrization.map_curve(g,t); +end + +function d = D(g,t) + p = parametrization.map_curve(g,t); + + d = zeros(1,length(t)-1); + for i = 1:length(d) + d(i) = norm(p(:,i) - p(:,i+1)); + end +end + +function t = equalize_d(g,t) + d = D(g,t); + v = d-mean(d); + while any(abs(v)>0.01*mean(d)) + dt = t(2:end)-t(1:end-1); + t(2:end) = t(2:end) - cumsum(dt.*v./d); + + t = t/t(end); + d = D(g,t); + v = d-mean(d); + end +end + + +function show(g,t,hh) + p = parametrization.map_curve(g,t); + + + + h = parametrization.plot_curve(g); + h.LineWidth = 2; + axis equal + hold on + h = plot(p(1,:),p(2,:),'.'); + h.Color = [0.8500 0.3250 0.0980]; + h.MarkerSize = 24; + hold off + + n = parametrization.curve_normals(g,t); + hold on + for i = 1:length(t) + p0 = p(:,i); + p1 = p0 + hh*n(:,i); + l = [p0, p1]; + h = plot(l(1,:),l(2,:)); + h.Color = [0.8500 0.3250 0.0980]; + end + +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+parametrization/old/curve_interp.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,58 @@ +% Create a cubic spline from the points (x,y) using periodic conditions. +% g = curve_interp(x,y) +function g = curve_interp(x,y) + default_arg('x',[0 2 2 1 1 0]) + default_arg('y',[0 0 2 2 1 1]) + % solve for xp and yp + + % x(t) = at^4 + bt^2+ct+d + + % a = xp1 -2x1 + 2x0 + xp0 + % b = 3x1 -xp1 - 3x0 + 2xp0 + % c = xp0 + % d = x0 + + assert(length(x) == length(y)) + n = length(x); + A = spdiags(ones(n,1)*[2, 8, 2],-1:1,n,n); + A(n,1) = 2; + A(1,n) = 2; + + bx = zeros(n,1); + for i = 2:n-1 + bx(i) = -6*x(i-1)+6*x(i+1); + end + bx(1) = -6*x(n)+6*x(2); + bx(n) = -6*x(n-1)+6*x(1); + + by = zeros(n,1); + for i = 2:n-1 + by(i) = -6*y(i-1)+6*y(i+1); + end + by(1) = -6*y(n)+6*y(2); + by(n) = -6*y(n-1)+6*y(1); + + + xp = A\bx; + yp = A\by; + + x(end+1) = x(1); + y(end+1) = y(1); + + xp(end+1) = xp(1); + yp(end+1) = yp(1); + + function v = g_fun(t) + t = mod(t,1); + i = mod(floor(t*n),n) + 1; + t = t * n -(i-1); + X = (2*x(i)-2*x(i+1)+xp(i)+xp(i+1))*t.^3 + (-3*x(i)+3*x(i+1)-2*xp(i)-xp(i+1))*t.^2 + (xp(i))*t + x(i); + Y = (2*y(i)-2*y(i+1)+yp(i)+yp(i+1))*t.^3 + (-3*y(i)+3*y(i+1)-2*yp(i)-yp(i+1))*t.^2 + (yp(i))*t + y(i); + v = [X;Y]; + end + + g = @g_fun; +end + + +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+parametrization/old/max_h.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,33 @@ +function [d_max, i1_max, j1_max, i2_max, j2_max] = max_h(X,Y) + ni = size(X,1); + nj = size(X,2); + d_max = 0; + + i1_max = 0; + j1_max = 0; + i2_max = 0; + j2_max = 0; + + D = {[0,-1],[1,0],[0,1],[-1,0]}; + + for i = 1:ni + for j = 1:nj + p1 = [X(i,j); Y(i,j)]; + for k = 1:length(D) + i2 = i+D{k}(1); + j2 = j+D{k}(2); + if i2 >= 1 && i2 <= ni && j2 >= 1 && j2 <= nj + p2 = [X(i2,j2); Y(i2,j2)]; + d = norm(p2-p1); + if d > d_max; + d_max = d; + i1_max = i; + j1_max = j; + i2_max = i2; + j2_max = j2; + end + end + end + end + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+parametrization/old/min_h.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,34 @@ +function [d_min, i1_min, j1_min, i2_min, j2_min] = min_h(X,Y) + ni = size(X,1); + nj = size(X,2); + d_min = norm([X(1,1);Y(1,1)] - [X(ni,nj);Y(ni,nj)]); + + i1_min = 0; + j1_min = 0; + i2_min = 0; + j2_min = 0; + + D = {[-1,-1],[0,-1],[1,-1],[1,0],[1,1],[0,1],[-1,1],[-1,0]}; + % D = {[0,-1],[1,0],[0,1],[-1,0]}; + + for i = 1:ni + for j = 1:nj + p1 = [X(i,j); Y(i,j)]; + for k = 1:length(D) + i2 = i+D{k}(1); + j2 = j+D{k}(2); + if i2 >= 1 && i2 <= ni && j2 >= 1 && j2 <= nj + p2 = [X(i2,j2); Y(i2,j2)]; + d = norm(p2-p1); + if d < d_min; + d_min = d; + i1_min = i; + j1_min = j; + i2_min = i2; + j2_min = j2; + end + end + end + end + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+parametrization/old/plot_shape.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,12 @@ +% Plot a shape using n points. Returns cell array of plot handles. +% hs = plot_shape(s,n) +function hs = plot_shape(s,n) + default_arg('n',100); + + hs = {}; + hold on + for i = 1:length(s) + hs{end+1} = parametrization.plot_curve(s{i},n); + end + hold off +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+parametrization/old/shape.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,4 @@ +% Creates a shape from a number of curves. A shape is a cell array of curves. +function s = shape(varargin); + s = varargin; +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+parametrization/old/shape_discretise.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,8 @@ +% Discretises a shape such that points on the curves are no further than h appart. +function p = shape_discretise(s,h) + p = []; + for i = 1:length(s) + [~,pt] = parametrization.curve_discretise(s{i},h); + p = [p, pt]; + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+parametrization/old/shape_linesegments.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,9 @@ +% Converts a shape into a cell array of linesegments shorter than h. +function l = shape_linesegments(s,h) + l = {}; + + for i = 1:length(s) + t = parametrization.curve_discretise(s{i},h); + l = [l, parametrization.curve_linesegments(s{i},t)]; + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+parametrization/old/triang_interp.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,132 @@ +classdef triang_interp + properties + g1, g2 ,g3 % Curves encirling the tirangle in the positive direction. + A,B,C % The corners of the triangle + Sa, Sb, Sc % Mappings from square with different sides collapsed + end + + methods + function o = triang_interp(g1,g2,g3) + o.g1 = g1; + o.g2 = g2; + o.g3 = g3; + o.A = g1(0); + o.B = g2(0); + o.C = g3(0); + o.Sa = parametrization.triang_interp.square_to_triangle_interp(g2,g3,g1); + o.Sb = parametrization.triang_interp.square_to_triangle_interp(g3,g1,g2); + o.Sc = parametrization.triang_interp.square_to_triangle_interp(g1,g2,g3); + end + + + function show(o,N) + % Show the mapped meridians of the triangle. + % Might be used for the barycentric coordinates. + ma = @(t)o.Sa(1/2,1-t); + mb = @(t)o.Sb(1/2,1-t); + mc = @(t)o.Sc(1/2,1-t); + + na = @(t)o.Sa(t,1/2); + + ka = @(t)(o.g1(1-t)+o.g2(t))/2; + + h = parametrization.plot_curve(ma); + h.Color = Color.blue; + h = parametrization.plot_curve(mb); + h.Color = Color.blue; + h = parametrization.plot_curve(mc); + h.Color = Color.blue; + + h = parametrization.plot_curve(na); + h.Color = Color.red; + + h = parametrization.plot_curve(ka); + h.Color = Color.red; + + [a(1),a(2)] = ma(1/3); + [b(1),b(2)] = mb(1/3); + [c(1),c(2)] = mc(1/3); + + d = ka(1-1/3); + + + parametrization.label_pt(a,b,c,d); + + + % t = linspace(0,1,N); + % for i = 1:N + % sa = @(s)o.Sa(s,t(i)); + % sb = @(s)o.Sb(s,t(i)); + % sc = @(s)o.Sc(s,t(i)); + + % h = parametrization.plot_curve(sa); + % h.Color = Color.blue; + % h = parametrization.plot_curve(sb); + % h.Color = Color.blue; + % h = parametrization.plot_curve(sc); + % h.Color = Color.blue; + % end + + h = parametrization.plot_curve(o.g1); + h.LineWidth = 2; + h.Color = Color.red; + + h = parametrization.plot_curve(o.g2); + h.LineWidth = 2; + h.Color = Color.red; + + h = parametrization.plot_curve(o.g3); + h.LineWidth = 2; + h.Color = Color.red; + + end + + + end + + methods(Static) + % Makes a mapping from the unit square to a triangle by collapsing + % one of the sides of the squares to a corner on the triangle + % The collapsed side is mapped to the corner oposite to g1. + % This is done such that for S(s,t), S(s,1) = g1(s) + function S = square_to_triangle_interp(g1,g2,g3) + corner = parametrization.line_segment(g3(0),g3(0)); + S = parametrization.transfinite_interp(corner,g3,f(g1),f(g2)) + + % Function to flip a curve + function h = f(g) + h = @(t)g(1-t); + end + end + end + +end + +% % Return a mapping from u.v to x,y of the domain encircled by g1 g2 g3 in the the positive direction. created be using transfinite interpolation. +% function S = triang_interp(g1,g2,g3) +% A = g1(0) +% B = g2(0) +% C = g3(0) + +% function [x,y] = S_fun(u,v) +% w = sqrt((u-1)^2+v^2)/sqrt(2); % Parameter for g3 +% v = v*(1-u-v)*g1(u) + u*(1-u-v)*g2(v) + u*v*g3(w) ... +% +(1-u)*(1-v)*A+u*(1-v)*B + (1-u)*v*C; +% x = v(1); +% y = v(2); +% end +% S = @S_fun; +% end + + + +% function subsref(obj,S) +% if ~all(isnumeric(S.subs{:})) +% error('Only supports calling object with number') +% end +% if numel(S.subs{:}) > 1 +% disp('You''ve called the object with more than one argument'); +% else +% disp(['You called the object with argument = ',num2str(S.subs{:})]); +% end +% end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+parametrization/old/triang_interp_pts.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,12 @@ +% Creates a transfinite interpolation from connecting the four points wiht straight lines. +function [S, g1, g2, g3] = triang_interp_pts(p1,p2,p3) + if size(p1) ~= [2 1] + error('p1 is strange!'); + end + + g1 = @(t)(p1 + t*(p2-p1)); + g2 = @(t)(p2 + t*(p3-p2)); + g3 = @(t)(p3 + t*(p1-p3)); + + S = parametrization.triang_interp(g1,g2,g3); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+parametrization/old/triang_map.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,29 @@ +% Creates a grid [X,Y] from the mapping function S at points in vectors u,v +function [X, Y] = traing_map(S,u,v) + error('not done') + if nargin == 2 + v = u; + end + + if isscalar(u) + u = linspace(0,1,u); + end + + if isscalar(v) + v = linspace(0,1,v); + end + + nu = length(u); + nv = length(v); + + X = zeros(nu,nv); + Y = zeros(nu,nv); + + for i = 1:nu + for j = 1:nv + [x,y] = S(u(i),v(j)); + X(i,j) = x; + Y(i,j) = y; + end + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+parametrization/old/triang_plot_interp.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,114 @@ +% Plots a transfinite interpolation in x,y space using nu and nv curves along u and v axes. + + + + + + +% Plots a interp of a triangle where one the interpolation is from a square +% with one side collapsed to +function h = triang_plot_interp_kindaworking(S,n) + u = linspace(0,1,n); + v = linspace(0,1,n); + + m = 100; + m = 20; + + Xl_curves = cell(n,1); + Xr_curves = cell(n,1); + Y_curves = cell(n,1); + + + function u = wierdness(v,d,N) + if N == 0 + u = 0; + else + u = N*d./(1-v); + end + end + + + %Y curves + t = linspace(0,1,m); + for i = 1:n + x = []; y = []; + for j = 1:length(t) + [x(j),y(j)] = S(t(j),v(i)); + end + Y_curves{i} = [x', y']; + end + + + % Right and left X curves + t = linspace(0,1,m); + d = u(2); + for i = 1:n + xl = []; yl = []; + xr = []; yr = []; + N = i-1; + t = linspace(0,1-N*d,m); + for j = 1:length(t) + w = wierdness(t(j),d,N); + [xr(j),yr(j)] = S(w,t(j)); + [xl(j),yl(j)] = S(1-w,t(j)); + end + Xl_curves{i} = [xl', yl']; + Xr_curves{i} = [xr', yr']; + end + + for i = 1:n-1 + line(Xl_curves{i}(:,1),Xl_curves{i}(:,2)) + line(Xr_curves{i}(:,1),Xr_curves{i}(:,2)) + line(Y_curves{i}(:,1),Y_curves{i}(:,2)) + end +end + + + + +function h = triang_plot_interp_nonworking(S,n) + + u = linspace(0,1,n); + v = linspace(0,1,n); + + m = 100; + + X_curves = cell(n-1,1); + Y_curves = cell(n-1,1); + K_curves = cell(n-1,1); + + + t = linspace(0,1,m); + for i = 1:n-1 + x = []; y = []; + for j = find(t+u(i) <= 1) + [x(j),y(j)] = S(u(i),t(j)); + end + X_curves{i} = [x', y']; + end + + for i = 1:n-1 + x = []; y = []; + for j = find(t+v(i) <= 1) + [x(j),y(j)] = S(t(j),v(i)); + end + Y_curves{i} = [x', y']; + end + + for i = 2:n + x = []; y = []; + for j = find(t<u(i)) + [x(j),y(j)] = S(t(j), u(i)-t(j)); + end + K_curves{i-1} = [x', y']; + end + + for i = 1:n-1 + line(X_curves{i}(:,1),X_curves{i}(:,2)) + line(Y_curves{i}(:,1),Y_curves{i}(:,2)) + line(K_curves{i}(:,1),K_curves{i}(:,2)) + end + + h = -1; + % h = plot(X_curves{:},Y_curves{:},K_curves{:}); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+parametrization/old/triang_show.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,23 @@ +% Show a grid as a matlab figure. +function triang_show(S,n) + + ih = ishold(); + + hold on + h_grid = parametrization.triang_plot_interp(S,n); + h_bord = parametrization.triang_plot_interp(S,2); + + set(h_grid,'Color',[0 0.4470 0.7410]); + set(h_bord,'Color',[0.8500 0.3250 0.0980]); + set(h_bord,'LineWidth',2); + + % axis auto + % axis equal + % axis square + + if ih + hold on + else + hold off + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+parametrization/place_label.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,10 @@ +function place_label(pt,str,horzAl,vertAl) + default_arg('horzAl','center'); + default_arg('vertAl', 'middle'); + + x = pt(1); + y = pt(2); + h = text(x,y,str); + h.HorizontalAlignment = horzAl; + h.VerticalAlignment = vertAl; +end \ No newline at end of file
--- a/+sbp/HigherCompatibleVariable.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,63 +0,0 @@ -classdef HigherCompatibleVariable < sbp.OpSet - properties - norms % Struct containing norm matrices such as H,Q, M - boundary % Struct contanging vectors for boundry point approximations - derivatives % Struct containging differentiation operators - borrowing % Struct with borrowing limits for different norm matrices - m % Number of grid points. - h % Step size - end - - - - methods - function obj = HigherCompatibleVariable(m,h,order) - - if order == 2 - [H, HI, D1, D2, D3, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher2_compatible_halfvariable(m,h); - obj.borrowing.N.S2 = 1.2500; - obj.borrowing.N.S3 = 0.4000; - elseif order == 4 - [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher4_compatible_halfvariable(m,h); - obj.borrowing.N.S2 = 0.5055; - obj.borrowing.N.S3 = 0.9290; - elseif order == 6 - [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher6_compatible_halfvariable(m,h); - obj.borrowing.N.S2 = 0.3259; - obj.borrowing.N.S3 = 0.1580; - else - error('Invalid operator order.'); - end - - obj.h = h; - obj.m = m; - - obj.norms.H = H; - obj.norms.HI = HI; - obj.norms.N = M4; - - obj.boundary.e_1 = e_1; - obj.boundary.S_1 = S_1; - obj.boundary.S2_1 = S2_1; - obj.boundary.S3_1 = S3_1; - - obj.boundary.e_m = e_m; - obj.boundary.S_m = S_m; - obj.boundary.S2_m = S2_m; - obj.boundary.S3_m = S3_m; - - obj.derivatives.D2 = D2; - obj.derivatives.D4 = D4; - - end - end - - methods (Static) - function lambda = smallestGrid(obj) - error('Not implmented') - end - end - - - -end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/HigherVariable.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,98 @@ +classdef HigherVariable < sbp.OpSet + properties + norms % Struct containing norm matrices such as H,Q, M + boundary % Struct contanging vectors for boundry point approximations + derivatives % Struct containging differentiation operators + borrowing % Struct with borrowing limits for different norm matrices + m % Number of grid points. + h % Step size + end + + + + methods + function obj = HigherVariable(m,h,order, opt) + default_arg('opt', '') + + if order == 2 + [H, HI, D1, D2, D3, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher_variable2(m,h); + obj.borrowing.N.S2 = 1.2500; + obj.borrowing.N.S3 = 0.4000; + elseif order == 4 + switch opt + case 'min_boundary_points' + [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher_variable4_min_boundary_points(m,h); + % obj.borrowing.N.S2 = 0.5055; + % obj.borrowing.N.S3 = 0.9290; + otherwise + [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher_variable4(m,h); + obj.borrowing.N.S2 = 0.5055; + obj.borrowing.N.S3 = 0.9290; + end + elseif order == 6 + switch opt + case '2' + [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher_variable6_2(m,h); + % obj.borrowing.N.S2 = 0.3259; + % obj.borrowing.N.S3 = 0.1580; + case '3' + [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher_variable6_3(m,h); + % obj.borrowing.N.S2 = 0.3259; + % obj.borrowing.N.S3 = 0.1580; + case 'min_boundary_points' + [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher_variable6_min_boundary_points(m,h); + % obj.borrowing.N.S2 = 0.3259; + % obj.borrowing.N.S3 = 0.1580; + otherwise + [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher_variable6(m,h); + obj.borrowing.N.S2 = 0.3259; + obj.borrowing.N.S3 = 0.1580; + end + elseif order == 8 + switch opt + case 'min_boundary_points' + [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher_variable8_min_boundary_points(m,h); + % obj.borrowing.N.S2 = 0.3259; + % obj.borrowing.N.S3 = 0.1580; + otherwise + [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher_variable8_higher_boundary_order(m,h); + % obj.borrowing.N.S2 = 0.3259; + % obj.borrowing.N.S3 = 0.1580; + end + + else + error('Invalid operator order.'); + end + + obj.h = h; + obj.m = m; + + obj.norms.H = H; + obj.norms.HI = HI; + obj.norms.N = M4; + + obj.boundary.e_1 = e_1; + obj.boundary.S_1 = S_1; + obj.boundary.S2_1 = S2_1; + obj.boundary.S3_1 = S3_1; + + obj.boundary.e_m = e_m; + obj.boundary.S_m = S_m; + obj.boundary.S2_m = S2_m; + obj.boundary.S3_m = S3_m; + + obj.derivatives.D2 = D2; + obj.derivatives.D4 = D4; + + end + end + + methods (Static) + function lambda = smallestGrid(obj) + error('Not implmented') + end + end + + + +end \ No newline at end of file
--- a/+sbp/higher2_compatible_halfvariable.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,161 +0,0 @@ -% Returns D2 as a function handle -function [H, HI, D1, D2, D3, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = higher2_compatible_halfvariable(m,h) - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - %%% 4:de ordn. SBP Finita differens %%% - %%% operatorer framtagna av Ken Mattsson %%% - %%% %%% - %%% 6 randpunkter, diagonal norm %%% - %%% %%% - %%% Datum: 2013-11-11 %%% - %%% %%% - %%% %%% - %%% H (Normen) %%% - %%% D1 (approx f?rsta derivatan) %%% - %%% D2 (approx andra derivatan) %%% - %%% D3 (approx tredje derivatan) %%% - %%% D2 (approx fj?rde derivatan) %%% - %%% %%% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - % M?ste ange antal punkter (m) och stegl?ngd (h) - % Notera att dessa opetratorer ?r framtagna f?r att anv?ndas n?r - % vi har 3de och 4de derivator i v?r PDE - % I annat fall anv?nd de "traditionella" som har noggrannare - % randsplutningar f?r D1 och D2 - - % Vi b?rjar med normen. Notera att alla SBP operatorer delar samma norm, - % vilket ?r n?dv?ndigt f?r stabilitet - - H=diag(ones(m,1),0);H(1,1)=1/2;H(m,m)=1/2; - - - H=H*h; - HI=inv(H); - - - % First derivative SBP operator, 1st order accurate at first 6 boundary points - - q1=1/2; - Q=q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1)); - - %Q=(-1/12*diag(ones(m-2,1),2)+8/12*diag(ones(m-1,1),1)-8/12*diag(ones(m-1,1),-1)+1/12*diag(ones(m-2,1),-2)); - - - e_1=zeros(m,1);e_1(1)=1; - e_m=zeros(m,1);e_m(m)=1; - - - D1=HI*(Q-1/2*e_1*e_1'+1/2*e_m*e_m') ; - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - - - % Second derivative, 1st order accurate at first boundary points - - %% below for constant coefficients - % m1=-1;m0=2; - % M=m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0);M(1,1)=1;M(m,m)=1; - % M=M/h; - %D2=HI*(-M-e_1*S_1+e_m*S_m); - - %% Below for variable coefficients - %% Require a vector c with the koeffients - - S_U=[-3/2 2 -1/2]/h; - S_1=zeros(1,m); - S_1(1:3)=S_U; - S_m=zeros(1,m); - S_m(m-2:m)=fliplr(-S_U); - - S_1 = S_1'; - S_m = S_m'; - - M=sparse(m,m); - e_1 = sparse(e_1); - e_m = sparse(e_m); - S_1 = sparse(S_1); - S_m = sparse(S_m); - - scheme_width = 3; - scheme_radius = (scheme_width-1)/2; - r = (1+scheme_radius):(m-scheme_radius); - - function D2 = D2_fun(c) - - Mm1 = -c(r-1)/2 - c(r)/2; - M0 = c(r-1)/2 + c(r) + c(r+1)/2; - Mp1 = -c(r)/2 - c(r+1)/2; - - M(r,:) = spdiags([Mm1 M0 Mp1],0:2*scheme_radius,length(r),m); - - - M(1:2,1:2)=[c(1)/2 + c(2)/2 -c(1)/2 - c(2)/2; -c(1)/2 - c(2)/2 c(1)/2 + c(2) + c(3)/2;]; - M(m-1:m,m-1:m)=[c(m-2)/2 + c(m-1) + c(m)/2 -c(m-1)/2 - c(m)/2; -c(m-1)/2 - c(m)/2 c(m-1)/2 + c(m)/2;]; - M=M/h; - - D2=HI*(-M-c(1)*e_1*S_1'+c(m)*e_m*S_m'); - end - D2 = @D2_fun; - - - - - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - - - % Third derivative, 1st order accurate at first 6 boundary points - - q2=1/2;q1=-1; - Q3=q2*(diag(ones(m-2,1),2)-diag(ones(m-2,1),-2))+q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1)); - - %QQ3=(-1/8*diag(ones(m-3,1),3) + 1*diag(ones(m-2,1),2) - 13/8*diag(ones(m-1,1),1) +13/8*diag(ones(m-1,1),-1) -1*diag(ones(m-2,1),-2) + 1/8*diag(ones(m-3,1),-3)); - - - Q3_U = [0 -0.13e2 / 0.16e2 0.7e1 / 0.8e1 -0.1e1 / 0.16e2; 0.13e2 / 0.16e2 0 -0.23e2 / 0.16e2 0.5e1 / 0.8e1; -0.7e1 / 0.8e1 0.23e2 / 0.16e2 0 -0.17e2 / 0.16e2; 0.1e1 / 0.16e2 -0.5e1 / 0.8e1 0.17e2 / 0.16e2 0;]; - Q3(1:4,1:4)=Q3_U; - Q3(m-3:m,m-3:m)=flipud( fliplr( -Q3_U ) ); - Q3=Q3/h^2; - - - - S2_U=[1 -2 1;]/h^2; - S2_1=zeros(1,m); - S2_1(1:3)=S2_U; - S2_m=zeros(1,m); - S2_m(m-2:m)=fliplr(S2_U); - S2_1 = S2_1'; - S2_m = S2_m'; - - - - D3=HI*(Q3 - e_1*S2_1' + e_m*S2_m' +1/2*S_1*S_1' -1/2*S_m*S_m' ) ; - - % Fourth derivative, 0th order accurate at first 6 boundary points (still - % yield 4th order convergence if stable: for example u_tt=-u_xxxx - - m2=1;m1=-4;m0=6; - M4=m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0); - - %M4=(-1/6*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3) ) + 2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2)) -13/2*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)) + 28/3*diag(ones(m,1),0)); - - M4_U=[0.13e2 / 0.10e2 -0.12e2 / 0.5e1 0.9e1 / 0.10e2 0.1e1 / 0.5e1; -0.12e2 / 0.5e1 0.26e2 / 0.5e1 -0.16e2 / 0.5e1 0.2e1 / 0.5e1; 0.9e1 / 0.10e2 -0.16e2 / 0.5e1 0.47e2 / 0.10e2 -0.17e2 / 0.5e1; 0.1e1 / 0.5e1 0.2e1 / 0.5e1 -0.17e2 / 0.5e1 0.29e2 / 0.5e1;]; - - - M4(1:4,1:4)=M4_U; - - M4(m-3:m,m-3:m)=flipud( fliplr( M4_U ) ); - M4=M4/h^3; - - S3_U=[-1 3 -3 1;]/h^3; - S3_1=zeros(1,m); - S3_1(1:4)=S3_U; - S3_m=zeros(1,m); - S3_m(m-3:m)=fliplr(-S3_U); - S3_1 = S3_1'; - S3_m = S3_m'; - - D4=HI*(M4-e_1*S3_1'+e_m*S3_m' + S_1*S2_1'-S_m*S2_m'); -end
--- a/+sbp/higher4_compatible_halfvariable.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,154 +0,0 @@ -function [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = higher4_compatible_halfvariable(m,h) - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - %%% 4:de ordn. SBP Finita differens %%% - %%% %%% - %%% H (Normen) %%% - %%% D1=H^(-1)Q (approx f?rsta derivatan) %%% - %%% D2 (approx andra derivatan) %%% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - - %m=20; %problemstorlek - %h=1/(m-1); - %h=1; - - c=ones(m,1); - - - H=diag(ones(m,1),0); - H(1:4,1:4)=diag([17/48 59/48 43/48 49/48]); - H(m-3:m,m-3:m)=fliplr(flipud(diag([17/48 59/48 43/48 49/48]))); - H=H*h; - HI=inv(H); - HI = sparse(HI); - - - - Q=(-1/12*diag(ones(m-2,1),2)+8/12*diag(ones(m-1,1),1)-8/12*diag(ones(m-1,1),-1)+1/12*diag(ones(m-2,1),-2)); - Q_U = [0 0.59e2 / 0.96e2 -0.1e1 / 0.12e2 -0.1e1 / 0.32e2; -0.59e2 / 0.96e2 0 0.59e2 / 0.96e2 0; 0.1e1 / 0.12e2 -0.59e2 / 0.96e2 0 0.59e2 / 0.96e2; 0.1e1 / 0.32e2 0 -0.59e2 / 0.96e2 0;]; - Q(1:4,1:4)=Q_U; - Q(m-3:m,m-3:m)=flipud( fliplr(-Q_U(1:4,1:4) ) ); - - e_1=zeros(m,1);e_1(1)=1; - e_m=zeros(m,1);e_m(m)=1; - - D1=HI*(Q-1/2*e_1*e_1'+1/2*e_m*e_m') ; - - M_U=[0.9e1 / 0.8e1 -0.59e2 / 0.48e2 0.1e1 / 0.12e2 0.1e1 / 0.48e2; -0.59e2 / 0.48e2 0.59e2 / 0.24e2 -0.59e2 / 0.48e2 0; 0.1e1 / 0.12e2 -0.59e2 / 0.48e2 0.55e2 / 0.24e2 -0.59e2 / 0.48e2; 0.1e1 / 0.48e2 0 -0.59e2 / 0.48e2 0.59e2 / 0.24e2;]; - M=-(-1/12*diag(ones(m-2,1),2)+16/12*diag(ones(m-1,1),1)+16/12*diag(ones(m-1,1),-1)-1/12*diag(ones(m-2,1),-2)-30/12*diag(ones(m,1),0)); - - M(1:4,1:4)=M_U; - - M(m-3:m,m-3:m)=flipud( fliplr( M_U ) ); - M=M/h; - - S_U=[-0.11e2 / 0.6e1 3 -0.3e1 / 0.2e1 0.1e1 / 0.3e1;]/h; - S_1=zeros(1,m); - S_1(1:4)=S_U; - S_m=zeros(1,m); - S_m(m-3:m)=fliplr(-S_U); - S_1 = S_1'; - S_m = S_m'; - - - M=sparse(m,m); - e_1 = sparse(e_1); - e_m = sparse(e_m); - S_1 = sparse(S_1); - S_m = sparse(S_m); - - - scheme_width = 5; - scheme_radius = (scheme_width-1)/2; - r = (1+scheme_radius):(m-scheme_radius); - - function D2 = D2_fun(c) - - %% ALTERNATIVES %%%%%%%%%%%%% - % for i=4:m-3 - % M(i,i-2:i+2)=[-c(i-1) / 0.6e1 + c(i-2) / 0.8e1 + c(i) / 0.8e1 -c(i-2) / 0.6e1 - c(i+1) / 0.6e1 - c(i-1) / 0.2e1 - c(i) / 0.2e1 c(i-2) / 0.24e2 + 0.5e1 / 0.6e1 * c(i-1) + 0.5e1 / 0.6e1 * c(i+1) + c(i+2) / 0.24e2 + 0.3e1 / 0.4e1 * c(i) -c(i-1) / 0.6e1 - c(i+2) / 0.6e1 - c(i) / 0.2e1 - c(i+1) / 0.2e1 -c(i+1) / 0.6e1 + c(i) / 0.8e1 + c(i+2) / 0.8e1;]; - % end - %%%%%%%%%%%%%%%%%%%%%%%%%%%%% - % for i=4:m-3 - % M(i,i-2:i+2)= [ - % c(i-2) / 0.8e1 - c(i-1) / 0.6e1 + c(i) / 0.8e1 , - % -c(i-2) / 0.6e1 - c(i-1) / 0.2e1 - c(i) / 0.2e1 - c(i+1) / 0.6e1 , - % c(i-2) / 2.4e1 + c(i-1) / 1.2e0 + c(i) * 0.3/0.4 + c(i+1) / 1.2e0 + c(i+2) / 2.4e1 , - % -c(i-1) / 0.6e1 - c(i) / 0.2e1 - c(i+1) / 0.2e1 - c(i+2) / 0.6e1 , - % c(i) / 0.8e1 - c(i+1) / 0.6e1 + c(i+2) / 0.8e1 , - % ]; - % end - %%%%%%%%%%%%%%%%%%%%%%%%%%%%% - Mm2 = c(r-2) / 0.8e1 - c(r-1) / 0.6e1 + c(r) / 0.8e1 ; - Mm1 = -c(r-2) / 0.6e1 - c(r-1) / 0.2e1 - c(r) / 0.2e1 - c(r+1) / 0.6e1 ; - M0 = c(r-2) / 2.4e1 + c(r-1) / 1.2e0 + c(r) * 0.3/0.4 + c(r+1) / 1.2e0 + c(r+2) / 2.4e1; - Mp1 = -c(r-1) / 0.6e1 - c(r) / 0.2e1 - c(r+1) / 0.2e1 - c(r+2) / 0.6e1; - Mp2 = c(r) / 0.8e1 - c(r+1) / 0.6e1 + c(r+2) / 0.8e1; - % printSize(Mm2); - % scheme_radius - % m - M(r,:) = spdiags([Mm2 Mm1 M0 Mp1 Mp2],0:2*scheme_radius,length(r),m); - % M(r,:) = spdiags([Mm2 Mm1 M0 Mp1 Mp2],(-2:2)+scheme_radius,M(r,:)); % This is slower - %%%%%%%%%%%%%%%%%%%%%%%%%%%%% - % %% Somthing is wrong here!! - % Mm2 = c(r-2) / 0.8e1 - c(r-1) / 0.6e1 + c(r) / 0.8e1 ; - % Mm1 = -c(r-2) / 0.6e1 - c(r-1) / 0.2e1 - c(r) / 0.2e1 - c(r+1) / 0.6e1 ; - % M0 = c(r-2) / 2.4e1 + c(r-1) / 1.2e0 + c(r) * 0.3/0.4 + c(r+1) / 1.2e0 + c(r+2) / 2.4e1; - % Mp1 = -c(r-1) / 0.6e1 - c(r) / 0.2e1 - c(r+1) / 0.2e1 - c(r+2) / 0.6e1; - % Mp2 = c(r) / 0.8e1 - c(r+1) / 0.6e1 + c(r+2) / 0.8e1; - % % printSize(M_diag_ind); - % % Mdiags = [Mm2 Mm1 M0 Mp1 Mp2]; - % % printSize(Mdiags); - % M(M_diag_ind) = [Mm2 Mm1 M0 Mp1 Mp2]; % This is slightly faster - %%%%%%%%%%%%%%%%%%%%%%%%%%%%% - % Kan man skriva det som en multiplikation av en 3-dim matris? - %%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - - - - M(1:6,1:6)=[0.12e2 / 0.17e2 * c(1) + 0.59e2 / 0.192e3 * c(2) + 0.27010400129e11 / 0.345067064608e12 * c(3) + 0.69462376031e11 / 0.2070402387648e13 * c(4) -0.59e2 / 0.68e2 * c(1) - 0.6025413881e10 / 0.21126554976e11 * c(3) - 0.537416663e9 / 0.7042184992e10 * c(4) 0.2e1 / 0.17e2 * c(1) - 0.59e2 / 0.192e3 * c(2) + 0.213318005e9 / 0.16049630912e11 * c(4) + 0.2083938599e10 / 0.8024815456e10 * c(3) 0.3e1 / 0.68e2 * c(1) - 0.1244724001e10 / 0.21126554976e11 * c(3) + 0.752806667e9 / 0.21126554976e11 * c(4) 0.49579087e8 / 0.10149031312e11 * c(3) - 0.49579087e8 / 0.10149031312e11 * c(4) -c(4) / 0.784e3 + c(3) / 0.784e3; -0.59e2 / 0.68e2 * c(1) - 0.6025413881e10 / 0.21126554976e11 * c(3) - 0.537416663e9 / 0.7042184992e10 * c(4) 0.3481e4 / 0.3264e4 * c(1) + 0.9258282831623875e16 / 0.7669235228057664e16 * c(3) + 0.236024329996203e15 / 0.1278205871342944e16 * c(4) -0.59e2 / 0.408e3 * c(1) - 0.29294615794607e14 / 0.29725717938208e14 * c(3) - 0.2944673881023e13 / 0.29725717938208e14 * c(4) -0.59e2 / 0.1088e4 * c(1) + 0.260297319232891e15 / 0.2556411742685888e16 * c(3) - 0.60834186813841e14 / 0.1278205871342944e16 * c(4) -0.1328188692663e13 / 0.37594290333616e14 * c(3) + 0.1328188692663e13 / 0.37594290333616e14 * c(4) -0.8673e4 / 0.2904112e7 * c(3) + 0.8673e4 / 0.2904112e7 * c(4); 0.2e1 / 0.17e2 * c(1) - 0.59e2 / 0.192e3 * c(2) + 0.213318005e9 / 0.16049630912e11 * c(4) + 0.2083938599e10 / 0.8024815456e10 * c(3) -0.59e2 / 0.408e3 * c(1) - 0.29294615794607e14 / 0.29725717938208e14 * c(3) - 0.2944673881023e13 / 0.29725717938208e14 * c(4) c(1) / 0.51e2 + 0.59e2 / 0.192e3 * c(2) + 0.13777050223300597e17 / 0.26218083221499456e17 * c(4) + 0.564461e6 / 0.13384296e8 * c(5) + 0.378288882302546512209e21 / 0.270764341349677687456e21 * c(3) c(1) / 0.136e3 - 0.125059e6 / 0.743572e6 * c(5) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(3) - 0.17220493277981e14 / 0.89177153814624e14 * c(4) -0.10532412077335e14 / 0.42840005263888e14 * c(4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(3) + 0.564461e6 / 0.4461432e7 * c(5) -0.960119e6 / 0.1280713392e10 * c(4) - 0.3391e4 / 0.6692148e7 * c(5) + 0.33235054191e11 / 0.26452850508784e14 * c(3); 0.3e1 / 0.68e2 * c(1) - 0.1244724001e10 / 0.21126554976e11 * c(3) + 0.752806667e9 / 0.21126554976e11 * c(4) -0.59e2 / 0.1088e4 * c(1) + 0.260297319232891e15 / 0.2556411742685888e16 * c(3) - 0.60834186813841e14 / 0.1278205871342944e16 * c(4) c(1) / 0.136e3 - 0.125059e6 / 0.743572e6 * c(5) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(3) - 0.17220493277981e14 / 0.89177153814624e14 * c(4) 0.3e1 / 0.1088e4 * c(1) + 0.507284006600757858213e21 / 0.475219048083107777984e21 * c(3) + 0.1869103e7 / 0.2230716e7 * c(5) + c(6) / 0.24e2 + 0.1950062198436997e16 / 0.3834617614028832e16 * c(4) -0.4959271814984644613e19 / 0.20965546238960637264e20 * c(3) - c(6) / 0.6e1 - 0.15998714909649e14 / 0.37594290333616e14 * c(4) - 0.375177e6 / 0.743572e6 * c(5) -0.368395e6 / 0.2230716e7 * c(5) + 0.752806667e9 / 0.539854092016e12 * c(3) + 0.1063649e7 / 0.8712336e7 * c(4) + c(6) / 0.8e1; 0.49579087e8 / 0.10149031312e11 * c(3) - 0.49579087e8 / 0.10149031312e11 * c(4) -0.1328188692663e13 / 0.37594290333616e14 * c(3) + 0.1328188692663e13 / 0.37594290333616e14 * c(4) -0.10532412077335e14 / 0.42840005263888e14 * c(4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(3) + 0.564461e6 / 0.4461432e7 * c(5) -0.4959271814984644613e19 / 0.20965546238960637264e20 * c(3) - c(6) / 0.6e1 - 0.15998714909649e14 / 0.37594290333616e14 * c(4) - 0.375177e6 / 0.743572e6 * c(5) 0.8386761355510099813e19 / 0.128413970713633903242e21 * c(3) + 0.2224717261773437e16 / 0.2763180339520776e16 * c(4) + 0.5e1 / 0.6e1 * c(6) + c(7) / 0.24e2 + 0.280535e6 / 0.371786e6 * c(5) -0.35039615e8 / 0.213452232e9 * c(4) - c(7) / 0.6e1 - 0.13091810925e11 / 0.13226425254392e14 * c(3) - 0.1118749e7 / 0.2230716e7 * c(5) - c(6) / 0.2e1; -c(4) / 0.784e3 + c(3) / 0.784e3 -0.8673e4 / 0.2904112e7 * c(3) + 0.8673e4 / 0.2904112e7 * c(4) -0.960119e6 / 0.1280713392e10 * c(4) - 0.3391e4 / 0.6692148e7 * c(5) + 0.33235054191e11 / 0.26452850508784e14 * c(3) -0.368395e6 / 0.2230716e7 * c(5) + 0.752806667e9 / 0.539854092016e12 * c(3) + 0.1063649e7 / 0.8712336e7 * c(4) + c(6) / 0.8e1 -0.35039615e8 / 0.213452232e9 * c(4) - c(7) / 0.6e1 - 0.13091810925e11 / 0.13226425254392e14 * c(3) - 0.1118749e7 / 0.2230716e7 * c(5) - c(6) / 0.2e1 0.3290636e7 / 0.80044587e8 * c(4) + 0.5580181e7 / 0.6692148e7 * c(5) + 0.5e1 / 0.6e1 * c(7) + c(8) / 0.24e2 + 0.660204843e9 / 0.13226425254392e14 * c(3) + 0.3e1 / 0.4e1 * c(6);]; - - M(m-5:m,m-5:m)=[c(m-7) / 0.24e2 + 0.5e1 / 0.6e1 * c(m-6) + 0.5580181e7 / 0.6692148e7 * c(m-4) + 0.4887707739997e13 / 0.119037827289528e15 * c(m-3) + 0.3e1 / 0.4e1 * c(m-5) + 0.660204843e9 / 0.13226425254392e14 * c(m-2) + 0.660204843e9 / 0.13226425254392e14 * c(m-1) -c(m-6) / 0.6e1 - 0.1618585929605e13 / 0.9919818940794e13 * c(m-3) - c(m-5) / 0.2e1 - 0.1118749e7 / 0.2230716e7 * c(m-4) - 0.13091810925e11 / 0.13226425254392e14 * c(m-2) - 0.13091810925e11 / 0.13226425254392e14 * c(m-1) -0.368395e6 / 0.2230716e7 * c(m-4) + c(m-5) / 0.8e1 + 0.48866620889e11 / 0.404890569012e12 * c(m-3) + 0.752806667e9 / 0.539854092016e12 * c(m-2) + 0.752806667e9 / 0.539854092016e12 * c(m-1) -0.3391e4 / 0.6692148e7 * c(m-4) - 0.238797444493e12 / 0.119037827289528e15 * c(m-3) + 0.33235054191e11 / 0.26452850508784e14 * c(m-2) + 0.33235054191e11 / 0.26452850508784e14 * c(m-1) -0.8673e4 / 0.2904112e7 * c(m-2) - 0.8673e4 / 0.2904112e7 * c(m-1) + 0.8673e4 / 0.1452056e7 * c(m-3) -c(m-3) / 0.392e3 + c(m-2) / 0.784e3 + c(m-1) / 0.784e3; -c(m-6) / 0.6e1 - 0.1618585929605e13 / 0.9919818940794e13 * c(m-3) - c(m-5) / 0.2e1 - 0.1118749e7 / 0.2230716e7 * c(m-4) - 0.13091810925e11 / 0.13226425254392e14 * c(m-2) - 0.13091810925e11 / 0.13226425254392e14 * c(m-1) c(m-6) / 0.24e2 + 0.5e1 / 0.6e1 * c(m-5) + 0.3896014498639e13 / 0.4959909470397e13 * c(m-3) + 0.8386761355510099813e19 / 0.128413970713633903242e21 * c(m-2) + 0.280535e6 / 0.371786e6 * c(m-4) + 0.3360696339136261875e19 / 0.171218627618178537656e21 * c(m-1) -c(m-5) / 0.6e1 - 0.4959271814984644613e19 / 0.20965546238960637264e20 * c(m-2) - 0.375177e6 / 0.743572e6 * c(m-4) - 0.13425842714e11 / 0.33740880751e11 * c(m-3) - 0.193247108773400725e18 / 0.6988515412986879088e19 * c(m-1) -0.365281640980e12 / 0.1653303156799e13 * c(m-3) + 0.564461e6 / 0.4461432e7 * c(m-4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(m-2) - 0.198407225513315475e18 / 0.7963657098519931984e19 * c(m-1) -0.1328188692663e13 / 0.37594290333616e14 * c(m-2) + 0.2226377963775e13 / 0.37594290333616e14 * c(m-1) - 0.8673e4 / 0.363014e6 * c(m-3) c(m-3) / 0.49e2 + 0.49579087e8 / 0.10149031312e11 * c(m-2) - 0.256702175e9 / 0.10149031312e11 * c(m-1); -0.368395e6 / 0.2230716e7 * c(m-4) + c(m-5) / 0.8e1 + 0.48866620889e11 / 0.404890569012e12 * c(m-3) + 0.752806667e9 / 0.539854092016e12 * c(m-2) + 0.752806667e9 / 0.539854092016e12 * c(m-1) -c(m-5) / 0.6e1 - 0.4959271814984644613e19 / 0.20965546238960637264e20 * c(m-2) - 0.375177e6 / 0.743572e6 * c(m-4) - 0.13425842714e11 / 0.33740880751e11 * c(m-3) - 0.193247108773400725e18 / 0.6988515412986879088e19 * c(m-1) c(m-5) / 0.24e2 + 0.1869103e7 / 0.2230716e7 * c(m-4) + 0.507284006600757858213e21 / 0.475219048083107777984e21 * c(m-2) + 0.3e1 / 0.1088e4 * c(m) + 0.31688435395e11 / 0.67481761502e11 * c(m-3) + 0.27769176016102795561e20 / 0.712828572124661666976e21 * c(m-1) -0.125059e6 / 0.743572e6 * c(m-4) + c(m) / 0.136e3 - 0.23099342648e11 / 0.101222642253e12 * c(m-3) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(m-2) + 0.193950157930938693e18 / 0.5525802884687299744e19 * c(m-1) 0.260297319232891e15 / 0.2556411742685888e16 * c(m-2) - 0.59e2 / 0.1088e4 * c(m) - 0.106641839640553e15 / 0.1278205871342944e16 * c(m-1) + 0.26019e5 / 0.726028e6 * c(m-3) -0.1244724001e10 / 0.21126554976e11 * c(m-2) + 0.3e1 / 0.68e2 * c(m) + 0.752806667e9 / 0.21126554976e11 * c(m-1); -0.3391e4 / 0.6692148e7 * c(m-4) - 0.238797444493e12 / 0.119037827289528e15 * c(m-3) + 0.33235054191e11 / 0.26452850508784e14 * c(m-2) + 0.33235054191e11 / 0.26452850508784e14 * c(m-1) -0.365281640980e12 / 0.1653303156799e13 * c(m-3) + 0.564461e6 / 0.4461432e7 * c(m-4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(m-2) - 0.198407225513315475e18 / 0.7963657098519931984e19 * c(m-1) -0.125059e6 / 0.743572e6 * c(m-4) + c(m) / 0.136e3 - 0.23099342648e11 / 0.101222642253e12 * c(m-3) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(m-2) + 0.193950157930938693e18 / 0.5525802884687299744e19 * c(m-1) 0.564461e6 / 0.13384296e8 * c(m-4) + 0.470299699916357e15 / 0.952302618316224e15 * c(m-3) + 0.550597048646198778781e21 / 0.1624586048098066124736e22 * c(m-1) + c(m) / 0.51e2 + 0.378288882302546512209e21 / 0.270764341349677687456e21 * c(m-2) -0.59e2 / 0.408e3 * c(m) - 0.29294615794607e14 / 0.29725717938208e14 * c(m-2) - 0.2234477713167e13 / 0.29725717938208e14 * c(m-1) - 0.8673e4 / 0.363014e6 * c(m-3) -0.59e2 / 0.3136e4 * c(m-3) - 0.13249937023e11 / 0.48148892736e11 * c(m-1) + 0.2e1 / 0.17e2 * c(m) + 0.2083938599e10 / 0.8024815456e10 * c(m-2); -0.8673e4 / 0.2904112e7 * c(m-2) - 0.8673e4 / 0.2904112e7 * c(m-1) + 0.8673e4 / 0.1452056e7 * c(m-3) -0.1328188692663e13 / 0.37594290333616e14 * c(m-2) + 0.2226377963775e13 / 0.37594290333616e14 * c(m-1) - 0.8673e4 / 0.363014e6 * c(m-3) 0.260297319232891e15 / 0.2556411742685888e16 * c(m-2) - 0.59e2 / 0.1088e4 * c(m) - 0.106641839640553e15 / 0.1278205871342944e16 * c(m-1) + 0.26019e5 / 0.726028e6 * c(m-3) -0.59e2 / 0.408e3 * c(m) - 0.29294615794607e14 / 0.29725717938208e14 * c(m-2) - 0.2234477713167e13 / 0.29725717938208e14 * c(m-1) - 0.8673e4 / 0.363014e6 * c(m-3) 0.9258282831623875e16 / 0.7669235228057664e16 * c(m-2) + 0.3481e4 / 0.3264e4 * c(m) + 0.228389721191751e15 / 0.1278205871342944e16 * c(m-1) + 0.8673e4 / 0.1452056e7 * c(m-3) -0.6025413881e10 / 0.21126554976e11 * c(m-2) - 0.59e2 / 0.68e2 * c(m) - 0.537416663e9 / 0.7042184992e10 * c(m-1); -c(m-3) / 0.392e3 + c(m-2) / 0.784e3 + c(m-1) / 0.784e3 c(m-3) / 0.49e2 + 0.49579087e8 / 0.10149031312e11 * c(m-2) - 0.256702175e9 / 0.10149031312e11 * c(m-1) -0.1244724001e10 / 0.21126554976e11 * c(m-2) + 0.3e1 / 0.68e2 * c(m) + 0.752806667e9 / 0.21126554976e11 * c(m-1) -0.59e2 / 0.3136e4 * c(m-3) - 0.13249937023e11 / 0.48148892736e11 * c(m-1) + 0.2e1 / 0.17e2 * c(m) + 0.2083938599e10 / 0.8024815456e10 * c(m-2) -0.6025413881e10 / 0.21126554976e11 * c(m-2) - 0.59e2 / 0.68e2 * c(m) - 0.537416663e9 / 0.7042184992e10 * c(m-1) 0.3e1 / 0.3136e4 * c(m-3) + 0.27010400129e11 / 0.345067064608e12 * c(m-2) + 0.234566387291e12 / 0.690134129216e12 * c(m-1) + 0.12e2 / 0.17e2 * c(m);]; - - M=M/h; - D2=HI*(-M-c(1)*e_1*S_1'+c(m)*e_m*S_m'); - end - D2 = @D2_fun; - - - S2_U=[2 -5 4 -1;]/h^2; - S2_1=zeros(1,m); - S2_1(1:4)=S2_U; - S2_m=zeros(1,m); - S2_m(m-3:m)=fliplr(S2_U); - S2_1 = S2_1'; - S2_m = S2_m'; - - m3=-1/6;m2=2;m1=-13/2;m0=28/3; - M4=m3*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3))+m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0); - - %M4=(-1/6*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3) ) + 2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2)) -13/2*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)) + 28/3*diag(ones(m,1),0)); - - M4_U=[0.5762947e7 / 0.2316384e7 -0.6374287e7 / 0.1158192e7 0.573947e6 / 0.165456e6 -0.124637e6 / 0.289548e6 0.67979e5 / 0.2316384e7 -0.60257e5 / 0.1158192e7; -0.6374287e7 / 0.1158192e7 0.30392389e8 / 0.2316384e7 -0.2735053e7 / 0.289548e6 0.273109e6 / 0.165456e6 0.83767e5 / 0.1158192e7 0.245549e6 / 0.2316384e7; 0.573947e6 / 0.165456e6 -0.2735053e7 / 0.289548e6 0.5266855e7 / 0.579096e6 -0.1099715e7 / 0.289548e6 0.869293e6 / 0.1158192e7 -0.10195e5 / 0.144774e6; -0.124637e6 / 0.289548e6 0.273109e6 / 0.165456e6 -0.1099715e7 / 0.289548e6 0.3259225e7 / 0.579096e6 -0.324229e6 / 0.72387e5 0.1847891e7 / 0.1158192e7; 0.67979e5 / 0.2316384e7 0.83767e5 / 0.1158192e7 0.869293e6 / 0.1158192e7 -0.324229e6 / 0.72387e5 0.2626501e7 / 0.330912e6 -0.7115491e7 / 0.1158192e7; -0.60257e5 / 0.1158192e7 0.245549e6 / 0.2316384e7 -0.10195e5 / 0.144774e6 0.1847891e7 / 0.1158192e7 -0.7115491e7 / 0.1158192e7 0.21383077e8 / 0.2316384e7;]; - - M4(1:6,1:6)=M4_U; - - M4(m-5:m,m-5:m)=flipud( fliplr( M4_U ) ); - M4=M4/h^3; - - S3_U=[-1 3 -3 1;]/h^3; - S3_1=zeros(1,m); - S3_1(1:4)=S3_U; - S3_m=zeros(1,m); - S3_m(m-3:m)=fliplr(-S3_U); - S3_1 = S3_1'; - S3_m = S3_m'; - - D4=HI*(M4-e_1*S3_1'+e_m*S3_m' + S_1*S2_1'-S_m*S2_m'); - - - - - -end
--- a/+sbp/higher6_compatible_halfvariable.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,160 +0,0 @@ -function [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = higher6_compatible_halfvariable(m,h) - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - %%% 6:te ordn. SBP Finita differens %%% - %%% operatorer med diagonal norm %%% - %%% Extension to variable koeff %%% - %%% %%% - %%% H (Normen) %%% - %%% D1=H^(-1)Q (approx f?rsta derivatan) %%% - %%% D2 (approx andra derivatan) %%% - %%% D2=HI*(R+C*D*S %%% - %%% %%% - %%% R=-D1'*H*C*D1-RR %%% - %%% %%% - %%% RR ?r dissipation) %%% - %%% Dissipationen uppbyggd av D4: %%% - %%% DI=D4*B*H*D4 %%% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - - %m=10; %problemstorlek - %h=1/(m-1); - - % Variable koefficicients are stored in vector: c, size m, - % with the unknown stored as c(1), c(2), ..., c_m - % x=1:h:m*h;x=x'; - % c=x.^0; - - - H=diag(ones(m,1),0); - H(1:6,1:6)=diag([13649/43200,12013/8640,2711/4320,5359/4320,7877/8640, ... - 43801/43200]); - H(m-5:m,m-5:m)=fliplr(flipud(diag([13649/43200,12013/8640, ... - 2711/4320,5359/4320,7877/8640,43801/43200]))); - - - x1=0.70127127127127; - - - D1=(1/60*diag(ones(m-3,1),3)-9/60*diag(ones(m-2,1),2)+45/60*diag(ones(m-1,1),1)-45/60*diag(ones(m-1,1),-1)+9/60*diag(ones(m-2,1),-2)-1/60*diag(ones(m-3,1),-3)); - - - - D1(1:6,1:9)=[-21600/13649, 43200/13649*x1-7624/40947, -172800/13649*x1+ ... - 715489/81894, 259200/13649*x1-187917/13649, -172800/13649* ... - x1+735635/81894, 43200/13649*x1-89387/40947, 0, 0, 0; ... - -8640/12013*x1+7624/180195, 0, 86400/12013*x1-57139/12013, ... - -172800/12013*x1+745733/72078, 129600/12013*x1-91715/12013, ... - -34560/12013*x1+240569/120130, 0, 0, 0; ... - 17280/2711*x1-715489/162660, -43200/2711*x1+57139/5422, 0, ... - 86400/2711*x1-176839/8133, -86400/2711*x1+242111/10844, ... - 25920/2711*x1-182261/27110, 0, 0, 0; ... - -25920/5359*x1+187917/53590, 86400/5359*x1-745733/64308, ... - -86400/5359*x1+176839/16077, 0, 43200/5359*x1-165041/32154, ... - -17280/5359*x1+710473/321540, 72/5359, 0, 0; ... - 34560/7877*x1-147127/47262, -129600/7877*x1+91715/7877, ... - 172800/7877*x1-242111/15754, -86400/7877*x1+165041/23631, ... - 0, 8640/7877*x1, -1296/7877, 144/7877, 0; ... - -43200/43801*x1+89387/131403, 172800/43801*x1-240569/87602,... - -259200/43801*x1+182261/43801, 172800/43801*x1-710473/262806, ... - -43200/43801*x1, 0, 32400/43801, -6480/43801, 720/43801]; - D1(m-5:m,m-8:m)=flipud( fliplr(-D1(1:6,1:9))); - D1=D1/h; - - e_1=zeros(m,1);e_1(1)=1; - e_m=zeros(m,1);e_m(m)=1; - - S_U=[-25/12, 4, -3, 4/3, -1/4]/h; - S_1=zeros(1,m); - S_1(1:5)=S_U; - S_m=zeros(1,m); - S_m(m-4:m)=fliplr(-S_U); - S_1 = S_1'; - S_m = S_m'; - - - - %DS=zeros(m,m); - %DS(1,1:5)=-[-25/12, 4, -3, 4/3, -1/4]; - %DS(m,m-4:m)=fliplr(-[-25/12, 4, -3, 4/3, -1/4]); - %DS=diag(c)*DS/h; - - - H=h*H; - HI=inv(H); - - - M=sparse(m,m); - e_1 = sparse(e_1); - e_m = sparse(e_m); - S_1 = sparse(S_1); - S_m = sparse(S_m); - - scheme_width = 7; - scheme_radius = (scheme_width-1)/2; - r = (1+scheme_radius):(m-scheme_radius); - - function D2 = D2_fun(c) - - Mm3 = c(r-2) / 0.40e2 + c(r-1) / 0.40e2 - 0.11e2 / 0.360e3 * c(r-3) - 0.11e2 / 0.360e3 * c(r); - Mm2 = c(r-3) / 0.20e2 - 0.3e1 / 0.10e2 * c(r-1) + c(r+1) / 0.20e2 + 0.7e1 / 0.40e2 * c(r) + 0.7e1 / 0.40e2 * c(r-2); - Mm1 = -c(r-3) / 0.40e2 - 0.3e1 / 0.10e2 * c(r-2) - 0.3e1 / 0.10e2 * c(r+1) - c(r+2) / 0.40e2 - 0.17e2 / 0.40e2 * c(r) - 0.17e2 / 0.40e2 * c(r-1); - M0 = c(r-3) / 0.180e3 + c(r-2) / 0.8e1 + 0.19e2 / 0.20e2 * c(r-1) + 0.19e2 / 0.20e2 * c(r+1) + c(r+2) / 0.8e1 + c(r+3) / 0.180e3 + 0.101e3 / 0.180e3 * c(r); - Mp1 = -c(r-2) / 0.40e2 - 0.3e1 / 0.10e2 * c(r-1) - 0.3e1 / 0.10e2 * c(r+2) - c(r+3) / 0.40e2 - 0.17e2 / 0.40e2 * c(r) - 0.17e2 / 0.40e2 * c(r+1); - Mp2 = c(r-1) / 0.20e2 - 0.3e1 / 0.10e2 * c(r+1) + c(r+3) / 0.20e2 + 0.7e1 / 0.40e2 * c(r) + 0.7e1 / 0.40e2 * c(r+2); - Mp3 = c(r+1) / 0.40e2 + c(r+2) / 0.40e2 - 0.11e2 / 0.360e3 * c(r) - 0.11e2 / 0.360e3 * c(r+3); - - M(r,:) = spdiags([Mm3 Mm2 Mm1 M0 Mp1 Mp2 Mp3],0:2*scheme_radius,length(r),m); - - - M(1:9,1:9)=[0.7912667594695582093926295e0 * c(1) + 0.2968472090638000742888467e0 * c(2) + 0.3185519088796429015220016e-2 * c(3) + 0.1632404042590951953384672e-1 * c(4) + 0.3160302244094415087693968e-1 * c(5) + 0.3167964748016105299646518e-1 * c(6) + 0.3148577733947253920469418e-1 * c(7) -0.1016689339350338144430605e1 * c(1) - 0.2845627370491611369031341e-1 * c(3) - 0.4128029838349298819825156e-1 * c(4) - 0.1392281451620140507549866e0 * c(5) - 0.1195777325611201766551392e0 * c(6) - 0.1194267756529333410855186e0 * c(7) 0.7075642937243715046279337e-1 * c(1) - 0.1845476106024151050283847e0 * c(2) - 0.4364163147111892346990101e-1 * c(4) + 0.2432367907207732460874765e0 * c(5) + 0.1582127073537215443965653e0 * c(6) + 0.1602348578364786307613271e0 * c(7) 0.2251991532891353212689574e0 * c(1) - 0.1662748711097054895317080e0 * c(2) + 0.2710530961648671297733465e-1 * c(3) - 0.1916646185968439909125616e0 * c(5) - 0.7684117160199014594442072e-1 * c(6) - 0.8219586949831697575883635e-1 * c(7) -0.5224403464202056316702078e-1 * c(1) + 0.4440063948509876221050939e-1 * c(2) - 0.1023976547309387874453988e-2 * c(3) + 0.7403484645316174090533193e-1 * c(4) + 0.1241625568998496895352046e-1 * c(6) + 0.7188652847892601282652228e-1 * c(5) + 0.1379362997104735503447960e-1 * c(7) -0.1828896813877197352675410e-1 * c(1) + 0.9574633163221758060736592e-2 * c(2) - 0.8105784530576404277872603e-3 * c(3) - 0.7348845587775519698437916e-2 * c(4) + 0.1063601949723906997026904e-1 * c(5) - 0.1315967038382618382356495e-1 * c(6) - 0.2117936478838753524581943e-1 * c(7) 0.1911888563316170927411831e-2 * c(4) - 0.4068130355529149936100229e-1 * c(5) + 0.1319674981073749167009902e-1 * c(6) + 0.2557266518123783676349144e-1 * c(7) 0.1559652871136785763960685e-1 * c(5) - 0.6486184157331537899459796e-2 * c(6) - 0.9110344554036319740147054e-2 * c(7) 0.5593983696629863059347067e-3 * c(6) - 0.1384822535100796372263822e-2 * c(5) + 0.8254241654378100663291154e-3 * c(7); -0.1016689339350338144430605e1 * c(1) - 0.2845627370491611369031341e-1 * c(3) - 0.4128029838349298819825156e-1 * c(4) - 0.1392281451620140507549866e0 * c(5) - 0.1195777325611201766551392e0 * c(6) - 0.1194267756529333410855186e0 * c(7) 0.1306332157111667628555907e1 * c(1) + 0.2542001760457345743492403e0 * c(3) + 0.1043897828092562609502636e0 * c(4) + 0.6672328021032112950919876e0 * c(5) + 0.4681819359722749441073885e0 * c(6) + 0.4676415410195836920069412e0 * c(7) -0.9091410269992464604926176e-1 * c(1) + 0.1103611313171476425250639e0 * c(4) - 0.1290397544997518887000350e1 * c(5) - 0.6639605248735044787146222e0 * c(6) - 0.6615974464005206184151509e0 * c(7) -0.2893557395653431666593814e0 * c(1) - 0.2421320004064592721552708e0 * c(3) + 0.1187670255028031027693374e1 * c(5) + 0.3956598149904136332753521e0 * c(6) + 0.3860048921755800000681479e0 * c(7) 0.6712774475803763988977355e-1 * c(1) + 0.9147192682075630179962131e-2 * c(3) - 0.1872196143003808021730728e0 * c(4) - 0.1319358558853174530078498e0 * c(6) - 0.4871575736811911887376923e0 * c(5) - 0.1047516312275448138054418e0 * c(7) 0.2349927974590068869356781e-1 * c(1) + 0.7240905383565181316381731e-2 * c(3) + 0.1858378996391679448655070e-1 * c(4) - 0.9289616133938676174345208e-1 * c(5) + 0.1223513270418807666970488e0 * c(6) + 0.1113520320436295033894092e0 * c(7) -0.4834791406446907590553793e-2 * c(4) + 0.2310683832687820403062715e0 * c(5) - 0.1080774142196007991746827e0 * c(6) - 0.1181561776427343335410349e0 * c(7) -0.8368141434403455353724691e-1 * c(5) + 0.4093499466767054661591066e-1 * c(6) + 0.4274641967636400692133626e-1 * c(7) -0.3576545132696983143406173e-2 * c(6) + 0.7389399124121078682094445e-2 * c(5) - 0.3812853991424095538688273e-2 * c(7); 0.7075642937243715046279337e-1 * c(1) - 0.1845476106024151050283847e0 * c(2) - 0.4364163147111892346990101e-1 * c(4) + 0.2432367907207732460874765e0 * c(5) + 0.1582127073537215443965653e0 * c(6) + 0.1602348578364786307613271e0 * c(7) -0.9091410269992464604926176e-1 * c(1) + 0.1103611313171476425250639e0 * c(4) - 0.1290397544997518887000350e1 * c(5) - 0.6639605248735044787146222e0 * c(6) - 0.6615974464005206184151509e0 * c(7) 0.6327161147136873807796515e-2 * c(1) + 0.1147318200715868527529827e0 * c(2) + 0.1166740554279680007487795e0 * c(4) + 0.2766610808285444037240703e1 * c(5) + 0.1070920689960817104203947e1 * c(6) + 0.1013161391032973057171717e1 * c(7) 0.2013769413884797246646959e-1 * c(1) + 0.1033717994630886401730470e0 * c(2) - 0.2913221621151742724258117e1 * c(5) - 0.8755807343482262259774782e0 * c(6) - 0.6909957183488812426508351e0 * c(7) -0.4671751091575462868310238e-2 * c(1) - 0.2760353365637712827793337e-1 * c(2) - 0.1979290298620869974478871e0 * c(4) + 0.5402985338373433052255418e0 * c(6) + 0.1239177593031911077924537e1 * c(5) + 0.2628038050247358227280031e0 * c(7) -0.1635430866921887819487473e-2 * c(1) - 0.5952475275883259619711594e-2 * c(2) + 0.1964682777744275219350831e-1 * c(4) + 0.3236640012639046600590714e0 * c(5) - 0.4659516693228870973898560e0 * c(6) - 0.2217272720941736859420432e0 * c(7) -0.5111353189352474549563559e-2 * c(4) - 0.5355878163774754346032096e0 * c(5) + 0.3328335104489738933610597e0 * c(6) + 0.2078656591178540157917135e0 * c(7) 0.1824328174134289562208038e0 * c(5) - 0.1059816030196818445908057e0 * c(6) - 0.7645121439374711162999809e-1 * c(7) 0.9209089963443799485648361e-2 * c(6) - 0.1591502818872493167091475e-1 * c(5) + 0.6705938225281132185266388e-2 * c(7); 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0.1559652871136785763960685e-1 * c(5) - 0.6486184157331537899459796e-2 * c(6) - 0.9110344554036319740147054e-2 * c(7) -0.8368141434403455353724691e-1 * c(5) + 0.4093499466767054661591066e-1 * c(6) + 0.4274641967636400692133626e-1 * c(7) 0.1824328174134289562208038e0 * c(5) - 0.1059816030196818445908057e0 * c(6) - 0.7645121439374711162999809e-1 * c(7) -0.2022909060111751565150958e0 * c(5) + 0.1453421858063658498587377e0 * c(6) + 0.5694872020480930665635812e-1 * c(7) -0.8631683980217122275970376e-1 * c(6) + 0.2698842360470999243492629e-1 * c(7) + 0.8098194147715651085292754e-1 * c(5) - 0.3276463639080639163926118e-1 * c(8) 0.3354661791693352108660900e-1 * c(5) - 0.3343620022386971405018586e0 * c(7) + 0.5000000000000000000000000e-1 * c(9) + 0.2169790609807602750804271e0 * c(6) + 0.1838363233410033443348225e0 * c(8) -0.3784113973033012949863031e-1 * c(5) - 0.2997556885134827361576001e0 * c(6) - 0.3000000000000000000000000e0 * c(9) - 0.2500000000000000000000000e-1 * c(10) - 0.3991486867446821178415359e0 * c(7) - 0.4382544850115050165022338e0 * c(8) 0.1230328942716804455358698e-1 * c(5) + 0.1183647529645898332481833e0 * c(6) + 0.9410511898227943334189628e0 * c(7) + 0.9500000000000000000000000e0 * c(9) + 0.1250000000000000000000000e0 * c(10) + 0.5555555555555555555555556e-2 * c(11) + 0.5699474344521144554459336e0 * c(8) -0.2308067892671916339568942e-1 * c(6) - 0.2986625053775149497180439e0 * c(7) - 0.3000000000000000000000000e0 * c(10) - 0.2500000000000000000000000e-1 * c(11) - 0.1047734860515050802561078e-2 * c(5) - 0.4272090808352508360837056e0 * c(8) - 0.4250000000000000000000000e0 * c(9); 0.5593983696629863059347067e-3 * c(6) - 0.1384822535100796372263822e-2 * c(5) + 0.8254241654378100663291154e-3 * c(7) -0.3576545132696983143406173e-2 * c(6) + 0.7389399124121078682094445e-2 * c(5) - 0.3812853991424095538688273e-2 * c(7) 0.9209089963443799485648361e-2 * c(6) - 0.1591502818872493167091475e-1 * c(5) + 0.6705938225281132185266388e-2 * c(7) -0.1200429618441003833696998e-1 * c(6) - 0.4776915669385923841535432e-2 * c(7) + 0.1678121185379596217850541e-1 * c(5) 0.7462059484530855073291365e-2 * c(6) - 0.8121640361668678949573496e-3 * c(7) + 0.5522702088127090209264064e-3 * c(8) - 0.7202165657176696199260422e-2 * c(5) 0.2912518476823004642951502e-1 * c(7) + 0.2279091916474916391629437e-1 * c(8) - 0.3068985997518740530511593e-1 * c(6) - 0.1781799513347360596249022e-2 * c(5) - 0.3055555555555555555555556e-1 * c(9) 0.4698146218022683933926520e-1 * c(6) - 0.2966863787471237458744416e0 * c(8) + 0.5000000000000000000000000e-1 * c(10) + 0.1716355704146006481727960e0 * c(7) + 0.3069346152296258362380356e-2 * c(5) + 0.1750000000000000000000000e0 * c(9) -0.2308067892671916339568942e-1 * c(6) - 0.2986625053775149497180439e0 * c(7) - 0.3000000000000000000000000e0 * c(10) - 0.2500000000000000000000000e-1 * c(11) - 0.1047734860515050802561078e-2 * c(5) - 0.4272090808352508360837056e0 * c(8) - 0.4250000000000000000000000e0 * c(9) 0.5139370221149109977041877e-2 * c(6) + 0.1247723215009422001393184e0 * c(7) + 0.9505522702088127090209264e0 * c(8) + 0.9500000000000000000000000e0 * c(10) + 0.1250000000000000000000000e0 * c(11) + 0.5555555555555555555555556e-2 * c(12) + 0.9159362465153641826887659e-4 * c(5) + 0.5611111111111111111111111e0 * c(9);]; - - M(m-8:m,m-8:m)=[0.5555555555555555555555556e-2 * c(m-11) + 0.1250000000000000000000000e0 * c(m-10) + 0.9500000000000000000000000e0 * c(m-9) + 0.9505522702088127090209264e0 * c(m-7) + 0.1247205076844361998744053e0 * c(m-6) + 0.5139370221149109977041877e-2 * c(m-5) + 0.5611111111111111111111111e0 * c(m-8) + 0.1434074411575366831819799e-3 * c(m-4) -0.2500000000000000000000000e-1 * c(m-10) - 0.3000000000000000000000000e0 * c(m-9) - 0.2980649679116425253322056e0 * c(m-6) - 0.2308067892671916339568942e-1 * c(m-5) - 0.4250000000000000000000000e0 * c(m-8) - 0.4272090808352508360837056e0 * c(m-7) - 0.1645272326387475188399322e-2 * c(m-4) 0.5000000000000000000000000e-1 * c(m-9) - 0.2966863787471237458744416e0 * c(m-7) + 0.4698146218022683933926520e-1 * c(m-5) + 0.1750000000000000000000000e0 * c(m-8) + 0.1700291833903489463825077e0 * c(m-6) + 0.4675733176547960152668626e-2 * c(m-4) 0.2279091916474916391629437e-1 * c(m-7) + 0.3097763128598982561225538e-1 * c(m-6) - 0.3055555555555555555555556e-1 * c(m-8) - 0.3068985997518740530511593e-1 * c(m-5) - 0.3634246031107139778989373e-2 * c(m-4) 0.5522702088127090209264064e-3 * c(m-7) - 0.3265435411305071914756373e-2 * c(m-6) + 0.7462059484530855073291365e-2 * c(m-5) - 0.4748894282038492179461399e-2 * c(m-4) 0.6272075574042975468177820e-3 * c(m-6) - 0.1200429618441003833696998e-1 * c(m-5) + 0.1137708862700574079015220e-1 * c(m-4) 0.9209089963443799485648361e-2 * c(m-5) - 0.3129629392354775191148163e-3 * c(m-6) - 0.8896127024208321966533544e-2 * c(m-4) -0.3576545132696983143406173e-2 * c(m-5) + 0.4335019854436220306755673e-3 * c(m-6) + 0.3143043147253361112730605e-2 * c(m-4) 0.5593983696629863059347067e-3 * c(m-5) - 0.1446656414398166805849327e-3 * c(m-6) - 0.4147327282231696253497740e-3 * c(m-4); -0.2500000000000000000000000e-1 * c(m-10) - 0.3000000000000000000000000e0 * c(m-9) - 0.2980649679116425253322056e0 * c(m-6) - 0.2308067892671916339568942e-1 * c(m-5) - 0.4250000000000000000000000e0 * c(m-8) - 0.4272090808352508360837056e0 * c(m-7) - 0.1645272326387475188399322e-2 * c(m-4) 0.5555555555555555555555556e-2 * c(m-10) + 0.1250000000000000000000000e0 * c(m-9) + 0.9500000000000000000000000e0 * c(m-8) + 0.9341601509609901526962449e0 * c(m-6) + 0.1183647529645898332481833e0 * c(m-5) + 0.1919432828897222527630486e-1 * c(m-4) + 0.5699474344521144554459336e0 * c(m-7) -0.2500000000000000000000000e-1 * c(m-9) - 0.3000000000000000000000000e0 * c(m-8) - 0.2997556885134827361576001e0 * c(m-5) - 0.5636663150858098975790317e-1 * c(m-4) - 0.4382544850115050165022338e0 * c(m-7) - 0.3806231949664312575822630e0 * c(m-6) 0.5000000000000000000000000e-1 * c(m-8) - 0.3557251496099816106154206e0 * c(m-6) + 0.5490976528821799120017102e-1 * c(m-4) + 0.1838363233410033443348225e0 * c(m-7) + 0.2169790609807602750804271e0 * c(m-5) 0.5528052133944605740009217e-1 * c(m-6) - 0.8631683980217122275970376e-1 * c(m-5) - 0.3276463639080639163926118e-1 * c(m-7) + 0.5268984374242044588776166e-1 * c(m-4) -0.5373770512016897565958305e-2 * c(m-6) + 0.1453421858063658498587377e0 * c(m-5) - 0.1399684152943489522927794e0 * c(m-4) -0.1059816030196818445908057e0 * c(m-5) + 0.1014880675788250237247178e0 * c(m-4) + 0.4493535440856820866087846e-2 * c(m-6) 0.4093499466767054661591066e-1 * c(m-5) - 0.3471075437892810033585296e-1 * c(m-4) - 0.6224240288742446280057699e-2 * c(m-6) -0.6486184157331537899459796e-2 * c(m-5) + 0.4409068609809831485979484e-2 * c(m-4) + 0.2077115547521706413480312e-2 * c(m-6); 0.5000000000000000000000000e-1 * c(m-9) - 0.2966863787471237458744416e0 * c(m-7) + 0.4698146218022683933926520e-1 * c(m-5) + 0.1750000000000000000000000e0 * c(m-8) + 0.1700291833903489463825077e0 * c(m-6) + 0.4675733176547960152668626e-2 * c(m-4) -0.2500000000000000000000000e-1 * c(m-9) - 0.3000000000000000000000000e0 * c(m-8) - 0.2997556885134827361576001e0 * c(m-5) - 0.5636663150858098975790317e-1 * c(m-4) - 0.4382544850115050165022338e0 * c(m-7) - 0.3806231949664312575822630e0 * c(m-6) 0.5555555555555555555555556e-2 * c(m-9) + 0.1250000000000000000000000e0 * c(m-8) + 0.9698817275172575247533506e0 * c(m-7) + 0.1011699483929608164601067e1 * c(m-5) + 0.1773466968705924819112984e0 * c(m-4) + 0.2239223735771599178951297e-3 * c(m-3) + 0.4325148359756313354830552e0 * c(m-6) -0.2500000000000000000000000e-1 * c(m-8) - 0.3132544850115050165022338e0 * c(m-7) - 0.2322389872063761557916742e0 * c(m-4) - 0.8607044252686413302647675e-3 * c(m-3) - 0.2594851141920572702679681e0 * c(m-6) - 0.6691607091647929161078591e0 * c(m-5) 0.5331362125287625412555844e-1 * c(m-7) + 0.1736073411355428563685818e0 * c(m-5) + 0.8671038084174692625075159e-2 * c(m-3) + 0.8084259844422177692569663e-1 * c(m-6) - 0.1664345989168155800449120e0 * c(m-4) -0.5013247356072127938999311e0 * c(m-5) + 0.5021853752328231128475915e0 * c(m-4) - 0.1197175073672143005877150e-1 * c(m-6) 0.3328335104489738933610597e0 * c(m-5) - 0.3179803804558436283847901e0 * c(m-4) - 0.5111353189352474549563559e-2 * c(m-3) - 0.9741776803777790426705996e-2 * c(m-6) -0.1080774142196007991746827e0 * c(m-5) + 0.9941834083648937298100811e-1 * c(m-4) - 0.4834791406446907590553793e-2 * c(m-3) + 0.1349386478955833378422842e-1 * c(m-6) 0.1319674981073749167009902e-1 * c(m-5) - 0.1060554802883657391328704e-1 * c(m-4) + 0.1911888563316170927411831e-2 * c(m-3) - 0.4503090345217088684223814e-2 * c(m-6); 0.2279091916474916391629437e-1 * c(m-7) + 0.3097763128598982561225538e-1 * c(m-6) - 0.3055555555555555555555556e-1 * c(m-8) - 0.3068985997518740530511593e-1 * c(m-5) - 0.3634246031107139778989373e-2 * c(m-4) 0.5000000000000000000000000e-1 * c(m-8) - 0.3557251496099816106154206e0 * c(m-6) + 0.5490976528821799120017102e-1 * c(m-4) + 0.1838363233410033443348225e0 * c(m-7) + 0.2169790609807602750804271e0 * c(m-5) -0.2500000000000000000000000e-1 * c(m-8) - 0.3132544850115050165022338e0 * c(m-7) - 0.2322389872063761557916742e0 * c(m-4) - 0.8607044252686413302647675e-3 * c(m-3) - 0.2594851141920572702679681e0 * c(m-6) - 0.6691607091647929161078591e0 * c(m-5) 0.5555555555555555555555556e-2 * c(m-8) + 0.1338363233410033443348225e0 * c(m-7) + 0.7391887916719206077121040e0 * c(m-6) + 0.6490333320052011212240632e0 * c(m-4) + 0.3308343404200968256656458e-2 * c(m-3) + 0.2062575706647430620228133e-3 * c(m-2) + 0.3088241944378964404772302e-3 * c(m-1) + 0.4227226173449345042468960e-3 * c(m) + 0.1190362071861893051132274e1 * c(m-5) -0.2720908083525083608370563e-1 * c(m-7) - 0.1931148612480615118957263e0 * c(m-6) - 0.3332941113251635390801278e-1 * c(m-3) + 0.2605582646183255957264249e-3 * c(m-2) + 0.1432116665752147607469646e-2 * c(m-1) + 0.1207544072304193806052558e-2 * c(m) - 0.1348436986667115543203552e1 * c(m-5) + 0.1687723507780044227927853e-1 * c(m-4) 0.3590669644811151307464697e-1 * c(m-6) - 0.5925443480724830632401754e0 * c(m-4) - 0.6897142765790609546343709e-2 * c(m-2) - 0.5363098747528542488971874e-2 * c(m-1) - 0.5205147429855955657625694e-2 * c(m) + 0.9977064356292750529201981e0 * c(m-5) 0.7272438906214475928744770e-1 * c(m-4) + 0.1964682777744275219350831e-1 * c(m-3) - 0.5952475275883259619711594e-2 * c(m-1) - 0.1635430866921887819487473e-2 * c(m) + 0.2921234010758621482958052e-1 * c(m-6) - 0.4659516693228870973898560e0 * c(m-5) 0.5891947149681041048896399e-1 * c(m-4) + 0.1858378996391679448655070e-1 * c(m-3) + 0.7240905383565181316381731e-2 * c(m-2) + 0.2349927974590068869356781e-1 * c(m) - 0.4046360079256766884300687e-1 * c(m-6) + 0.1223513270418807666970488e0 * c(m-5) -0.2404661162020836566908542e-1 * c(m-4) - 0.7348845587775519698437916e-2 * c(m-3) - 0.8105784530576404277872603e-3 * c(m-2) + 0.9574633163221758060736592e-2 * c(m-1) - 0.1828896813877197352675410e-1 * c(m) + 0.1350326632905990039353503e-1 * c(m-6) - 0.1315967038382618382356495e-1 * c(m-5); 0.5522702088127090209264064e-3 * c(m-7) - 0.3265435411305071914756373e-2 * c(m-6) + 0.7462059484530855073291365e-2 * c(m-5) - 0.4748894282038492179461399e-2 * c(m-4) 0.5528052133944605740009217e-1 * c(m-6) - 0.8631683980217122275970376e-1 * c(m-5) - 0.3276463639080639163926118e-1 * c(m-7) + 0.5268984374242044588776166e-1 * c(m-4) 0.5331362125287625412555844e-1 * c(m-7) + 0.1736073411355428563685818e0 * c(m-5) + 0.8671038084174692625075159e-2 * c(m-3) + 0.8084259844422177692569663e-1 * c(m-6) - 0.1664345989168155800449120e0 * c(m-4) -0.2720908083525083608370563e-1 * c(m-7) - 0.1931148612480615118957263e0 * c(m-6) - 0.3332941113251635390801278e-1 * c(m-3) + 0.2605582646183255957264249e-3 * c(m-2) + 0.1432116665752147607469646e-2 * c(m-1) + 0.1207544072304193806052558e-2 * c(m) - 0.1348436986667115543203552e1 * c(m-5) + 0.1687723507780044227927853e-1 * c(m-4) 0.6107825764368264576481962e-2 * c(m-7) + 0.1155752633643216628010304e0 * c(m-6) + 0.2096413329579026439044119e1 * c(m-5) + 0.3357721707576477199985656e0 * c(m-3) + 0.3291545083271862858501887e-3 * c(m-2) + 0.6641183499427826101618457e-2 * c(m-1) + 0.3449455095910233625229891e-2 * c(m) + 0.8270696421223286922584620e0 * c(m-4) -0.4995827370863505253765970e-1 * c(m-6) - 0.1263507837371824205693950e1 * c(m-5) - 0.8712928907711754187084757e-2 * c(m-2) - 0.2487040599390160764166412e-1 * c(m-1) - 0.1486895819265604128572498e-1 * c(m) - 0.1726565392121567634950213e1 * c(m-4) 0.5402985338373433052255418e0 * c(m-5) - 0.1979290298620869974478871e0 * c(m-3) - 0.2760353365637712827793337e-1 * c(m-1) - 0.4671751091575462868310238e-2 * c(m) - 0.6952587985456154591014641e-1 * c(m-6) + 0.1571507277911208446562686e1 * c(m-4) -0.1319358558853174530078498e0 * c(m-5) - 0.1872196143003808021730728e0 * c(m-3) + 0.9147192682075630179962131e-2 * c(m-2) + 0.6712774475803763988977355e-1 * c(m) + 0.9630407686703666967100804e-1 * c(m-6) - 0.6882132817757726722141421e0 * c(m-4) 0.1241625568998496895352046e-1 * c(m-5) + 0.7403484645316174090533193e-1 * c(m-3) - 0.1023976547309387874453988e-2 * c(m-2) + 0.4440063948509876221050939e-1 * c(m-1) - 0.5224403464202056316702078e-1 * c(m) - 0.3213800979246298453953842e-1 * c(m-6) + 0.1178181682424363524005403e0 * c(m-4); 0.6272075574042975468177820e-3 * c(m-6) - 0.1200429618441003833696998e-1 * c(m-5) + 0.1137708862700574079015220e-1 * c(m-4) -0.5373770512016897565958305e-2 * c(m-6) + 0.1453421858063658498587377e0 * c(m-5) - 0.1399684152943489522927794e0 * c(m-4) -0.5013247356072127938999311e0 * c(m-5) + 0.5021853752328231128475915e0 * c(m-4) - 0.1197175073672143005877150e-1 * c(m-6) 0.3590669644811151307464697e-1 * c(m-6) - 0.5925443480724830632401754e0 * c(m-4) - 0.6897142765790609546343709e-2 * c(m-2) - 0.5363098747528542488971874e-2 * c(m-1) - 0.5205147429855955657625694e-2 * c(m) + 0.9977064356292750529201981e0 * c(m-5) -0.4995827370863505253765970e-1 * c(m-6) - 0.1263507837371824205693950e1 * c(m-5) - 0.8712928907711754187084757e-2 * c(m-2) - 0.2487040599390160764166412e-1 * c(m-1) - 0.1486895819265604128572498e-1 * c(m) - 0.1726565392121567634950213e1 * c(m-4) 0.2760393423824887721078848e-1 * c(m-6) + 0.1190550338687608873798462e1 * c(m-5) + 0.4253084328734353394994388e1 * c(m-4) + 0.2306367624634749229113646e0 * c(m-2) + 0.9313657638804699948929701e-1 * c(m-1) + 0.6409299775987186986730499e-1 * c(m) -0.8755807343482262259774782e0 * c(m-5) - 0.3645285178085761821545207e1 * c(m-4) + 0.1033717994630886401730470e0 * c(m-1) + 0.2013769413884797246646959e-1 * c(m) + 0.4106783858513785463625543e-1 * c(m-6) 0.3956598149904136332753521e0 * c(m-5) + 0.1630560443616104907615866e1 * c(m-4) - 0.2421320004064592721552708e0 * c(m-2) - 0.2893557395653431666593814e0 * c(m) - 0.5688529641249387985434413e-1 * c(m-6) -0.7684117160199014594442072e-1 * c(m-5) - 0.2928439026361256842196229e0 * c(m-4) + 0.2710530961648671297733465e-1 * c(m-2) - 0.1662748711097054895317080e0 * c(m-1) + 0.2251991532891353212689574e0 * c(m) + 0.1898341454096471754822498e-1 * c(m-6); 0.9209089963443799485648361e-2 * c(m-5) - 0.3129629392354775191148163e-3 * c(m-6) - 0.8896127024208321966533544e-2 * c(m-4) -0.1059816030196818445908057e0 * c(m-5) + 0.1014880675788250237247178e0 * c(m-4) + 0.4493535440856820866087846e-2 * c(m-6) 0.3328335104489738933610597e0 * c(m-5) - 0.3179803804558436283847901e0 * c(m-4) - 0.5111353189352474549563559e-2 * c(m-3) - 0.9741776803777790426705996e-2 * c(m-6) 0.7272438906214475928744770e-1 * c(m-4) + 0.1964682777744275219350831e-1 * c(m-3) - 0.5952475275883259619711594e-2 * c(m-1) - 0.1635430866921887819487473e-2 * c(m) + 0.2921234010758621482958052e-1 * c(m-6) - 0.4659516693228870973898560e0 * c(m-5) 0.5402985338373433052255418e0 * c(m-5) - 0.1979290298620869974478871e0 * c(m-3) - 0.2760353365637712827793337e-1 * c(m-1) - 0.4671751091575462868310238e-2 * c(m) - 0.6952587985456154591014641e-1 * c(m-6) + 0.1571507277911208446562686e1 * c(m-4) -0.8755807343482262259774782e0 * c(m-5) - 0.3645285178085761821545207e1 * c(m-4) + 0.1033717994630886401730470e0 * c(m-1) + 0.2013769413884797246646959e-1 * c(m) + 0.4106783858513785463625543e-1 * c(m-6) 0.1070920689960817104203947e1 * c(m-5) + 0.3717418466925056542408153e1 * c(m-4) + 0.1166740554279680007487795e0 * c(m-3) + 0.1147318200715868527529827e0 * c(m-1) + 0.6327161147136873807796515e-2 * c(m) + 0.6235373239336055200426697e-1 * c(m-6) -0.6639605248735044787146222e0 * c(m-5) - 0.1865625445986772763641423e1 * c(m-4) + 0.1103611313171476425250639e0 * c(m-3) - 0.9091410269992464604926176e-1 * c(m) - 0.8636954541126674177407762e-1 * c(m-6) 0.1582127073537215443965653e0 * c(m-5) + 0.3746489300753517635549495e0 * c(m-4) - 0.4364163147111892346990101e-1 * c(m-3) - 0.1845476106024151050283847e0 * c(m-1) + 0.7075642937243715046279337e-1 * c(m) + 0.2882271848190011329385407e-1 * c(m-6); -0.3576545132696983143406173e-2 * c(m-5) + 0.4335019854436220306755673e-3 * c(m-6) + 0.3143043147253361112730605e-2 * c(m-4) 0.4093499466767054661591066e-1 * c(m-5) - 0.3471075437892810033585296e-1 * c(m-4) - 0.6224240288742446280057699e-2 * c(m-6) -0.1080774142196007991746827e0 * c(m-5) + 0.9941834083648937298100811e-1 * c(m-4) - 0.4834791406446907590553793e-2 * c(m-3) + 0.1349386478955833378422842e-1 * c(m-6) 0.5891947149681041048896399e-1 * c(m-4) + 0.1858378996391679448655070e-1 * c(m-3) + 0.7240905383565181316381731e-2 * c(m-2) + 0.2349927974590068869356781e-1 * c(m) - 0.4046360079256766884300687e-1 * c(m-6) + 0.1223513270418807666970488e0 * c(m-5) -0.1319358558853174530078498e0 * c(m-5) - 0.1872196143003808021730728e0 * c(m-3) + 0.9147192682075630179962131e-2 * c(m-2) + 0.6712774475803763988977355e-1 * c(m) + 0.9630407686703666967100804e-1 * c(m-6) - 0.6882132817757726722141421e0 * c(m-4) 0.3956598149904136332753521e0 * c(m-5) + 0.1630560443616104907615866e1 * c(m-4) - 0.2421320004064592721552708e0 * c(m-2) - 0.2893557395653431666593814e0 * c(m) - 0.5688529641249387985434413e-1 * c(m-6) -0.6639605248735044787146222e0 * c(m-5) - 0.1865625445986772763641423e1 * c(m-4) + 0.1103611313171476425250639e0 * c(m-3) - 0.9091410269992464604926176e-1 * c(m) - 0.8636954541126674177407762e-1 * c(m-6) 0.4681819359722749441073885e0 * c(m-5) + 0.1015239189167790053447110e1 * c(m-4) + 0.1043897828092562609502636e0 * c(m-3) + 0.2542001760457345743492403e0 * c(m-2) + 0.1306332157111667628555907e1 * c(m) + 0.1196351539550049336518187e0 * c(m-6) -0.1195777325611201766551392e0 * c(m-5) - 0.2187310061229745694542609e0 * c(m-4) - 0.4128029838349298819825156e-1 * c(m-3) - 0.2845627370491611369031341e-1 * c(m-2) - 0.1016689339350338144430605e1 * c(m) - 0.3992391469197282238624438e-1 * c(m-6); 0.5593983696629863059347067e-3 * c(m-5) - 0.1446656414398166805849327e-3 * c(m-6) - 0.4147327282231696253497740e-3 * c(m-4) -0.6486184157331537899459796e-2 * c(m-5) + 0.4409068609809831485979484e-2 * c(m-4) + 0.2077115547521706413480312e-2 * c(m-6) 0.1319674981073749167009902e-1 * c(m-5) - 0.1060554802883657391328704e-1 * c(m-4) + 0.1911888563316170927411831e-2 * c(m-3) - 0.4503090345217088684223814e-2 * c(m-6) -0.2404661162020836566908542e-1 * c(m-4) - 0.7348845587775519698437916e-2 * c(m-3) - 0.8105784530576404277872603e-3 * c(m-2) + 0.9574633163221758060736592e-2 * c(m-1) - 0.1828896813877197352675410e-1 * c(m) + 0.1350326632905990039353503e-1 * c(m-6) - 0.1315967038382618382356495e-1 * c(m-5) 0.1241625568998496895352046e-1 * c(m-5) + 0.7403484645316174090533193e-1 * c(m-3) - 0.1023976547309387874453988e-2 * c(m-2) + 0.4440063948509876221050939e-1 * c(m-1) - 0.5224403464202056316702078e-1 * c(m) - 0.3213800979246298453953842e-1 * c(m-6) + 0.1178181682424363524005403e0 * c(m-4) -0.7684117160199014594442072e-1 * c(m-5) - 0.2928439026361256842196229e0 * c(m-4) + 0.2710530961648671297733465e-1 * c(m-2) - 0.1662748711097054895317080e0 * c(m-1) + 0.2251991532891353212689574e0 * c(m) + 0.1898341454096471754822498e-1 * c(m-6) 0.1582127073537215443965653e0 * c(m-5) + 0.3746489300753517635549495e0 * c(m-4) - 0.4364163147111892346990101e-1 * c(m-3) - 0.1845476106024151050283847e0 * c(m-1) + 0.7075642937243715046279337e-1 * c(m) + 0.2882271848190011329385407e-1 * c(m-6) -0.1195777325611201766551392e0 * c(m-5) - 0.2187310061229745694542609e0 * c(m-4) - 0.4128029838349298819825156e-1 * c(m-3) - 0.2845627370491611369031341e-1 * c(m-2) - 0.1016689339350338144430605e1 * c(m) - 0.3992391469197282238624438e-1 * c(m-6) 0.3167964748016105299646518e-1 * c(m-5) + 0.4976563420877041544013670e-1 * c(m-4) + 0.1632404042590951953384672e-1 * c(m-3) + 0.3185519088796429015220016e-2 * c(m-2) + 0.2968472090638000742888467e0 * c(m-1) + 0.7912667594695582093926295e0 * c(m) + 0.1332316557164627464149716e-1 * c(m-6);]; - - M(5,10)=M(10,5); - M(m-4,m-9)=M(m-9,m-4); - - M=M/h; - - D2=HI*(-M-diag(c)*e_1*S_1'+diag(c)*e_m*S_m'); - end - D2 = @D2_fun; - - S2_U=[0.35e2 / 0.12e2 -0.26e2 / 0.3e1 0.19e2 / 0.2e1 -0.14e2 / 0.3e1 0.11e2 / 0.12e2;]/h^2; - S2_1=zeros(1,m); - S2_1(1:5)=S2_U; - S2_m=zeros(1,m); - S2_m(m-4:m)=fliplr(S2_U); - S2_1 = S2_1'; - S2_m = S2_m'; - - - - - - % Fourth derivative, 1th order accurate at first 8 boundary points (still - % yield 5th order convergence if stable: for example u_tt=-u_xxxx - - m4=7/240;m3=-2/5;m2=169/60;m1=-122/15;m0=91/8; - M4=m4*(diag(ones(m-4,1),4)+diag(ones(m-4,1),-4))+m3*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3))+m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0); - - %M4=(-1/6*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3) ) + 2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2)) -13/2*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)) + 28/3*diag(ones(m,1),0)); - - M4_U=[0.1394226315049e13 / 0.367201486080e12 -0.1137054563243e13 / 0.114750464400e12 0.16614189027367e14 / 0.1836007430400e13 -0.1104821700277e13 / 0.306001238400e12 0.1355771086763e13 / 0.1836007430400e13 -0.27818686453e11 / 0.459001857600e12 -0.40671054239e11 / 0.1836007430400e13 0.5442887371e10 / 0.306001238400e12; -0.1137054563243e13 / 0.114750464400e12 0.70616795535409e14 / 0.2570410402560e13 -0.173266854731041e15 / 0.6426026006400e13 0.28938615291031e14 / 0.2570410402560e13 -0.146167361863e12 / 0.71400288960e11 0.2793470836571e13 / 0.12852052012800e14 0.6219558097e10 / 0.428401733760e12 -0.7313844559e10 / 0.166909766400e12; 0.16614189027367e14 / 0.1836007430400e13 -0.173266854731041e15 / 0.6426026006400e13 0.378613061504779e15 / 0.12852052012800e14 -0.9117069604217e13 / 0.642602600640e12 0.632177582849e12 / 0.233673672960e12 -0.1057776382577e13 / 0.6426026006400e13 0.443019868399e12 / 0.4284017337600e13 -0.3707981e7 / 0.2318191200e10; -0.1104821700277e13 / 0.306001238400e12 0.28938615291031e14 / 0.2570410402560e13 -0.9117069604217e13 / 0.642602600640e12 0.5029150721885e13 / 0.514082080512e12 -0.5209119714341e13 / 0.1285205201280e13 0.12235427457469e14 / 0.12852052012800e14 -0.13731270505e11 / 0.64260260064e11 0.2933596129e10 / 0.40800165120e11; 0.1355771086763e13 / 0.1836007430400e13 -0.146167361863e12 / 0.71400288960e11 0.632177582849e12 / 0.233673672960e12 -0.5209119714341e13 / 0.1285205201280e13 0.14871726798559e14 / 0.2570410402560e13 -0.7504337615347e13 / 0.1606506501600e13 0.310830296467e12 / 0.171360693504e12 -0.55284274391e11 / 0.183600743040e12; -0.27818686453e11 / 0.459001857600e12 0.2793470836571e13 / 0.12852052012800e14 -0.1057776382577e13 / 0.6426026006400e13 0.12235427457469e14 / 0.12852052012800e14 -0.7504337615347e13 / 0.1606506501600e13 0.106318657014853e15 / 0.12852052012800e14 -0.14432772918527e14 / 0.2142008668800e13 0.58102695589e11 / 0.22666758400e11; -0.40671054239e11 / 0.1836007430400e13 0.6219558097e10 / 0.428401733760e12 0.443019868399e12 / 0.4284017337600e13 -0.13731270505e11 / 0.64260260064e11 0.310830296467e12 / 0.171360693504e12 -0.14432772918527e14 / 0.2142008668800e13 0.27102479467823e14 / 0.2570410402560e13 -0.1216032192203e13 / 0.153000619200e12; 0.5442887371e10 / 0.306001238400e12 -0.7313844559e10 / 0.166909766400e12 -0.3707981e7 / 0.2318191200e10 0.2933596129e10 / 0.40800165120e11 -0.55284274391e11 / 0.183600743040e12 0.58102695589e11 / 0.22666758400e11 -0.1216032192203e13 / 0.153000619200e12 0.20799922829107e14 / 0.1836007430400e13;]; - - M4(1:8,1:8)=M4_U; - - M4(m-7:m,m-7:m)=flipud( fliplr( M4_U ) ); - M4=M4/h^3; - - S3_U=[-0.5e1 / 0.2e1 9 -12 7 -0.3e1 / 0.2e1;]/h^3; - S3_1=zeros(1,m); - S3_1(1:5)=S3_U; - S3_m=zeros(1,m); - S3_m(m-4:m)=fliplr(-S3_U); - S3_1 = S3_1'; - S3_m = S3_m'; - - D4=HI*(M4-e_1*S3_1'+e_m*S3_m' + S_1*S2_1'-S_m*S2_m'); - -end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/higher_variable2.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,161 @@ +% Returns D2 as a function handle +function [H, HI, D1, D2, D3, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = higher_variable2(m,h) + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + %%% 4:de ordn. SBP Finita differens %%% + %%% operatorer framtagna av Ken Mattsson %%% + %%% %%% + %%% 6 randpunkter, diagonal norm %%% + %%% %%% + %%% Datum: 2013-11-11 %%% + %%% %%% + %%% %%% + %%% H (Normen) %%% + %%% D1 (approx f?rsta derivatan) %%% + %%% D2 (approx andra derivatan) %%% + %%% D3 (approx tredje derivatan) %%% + %%% D2 (approx fj?rde derivatan) %%% + %%% %%% + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % M?ste ange antal punkter (m) och stegl?ngd (h) + % Notera att dessa opetratorer ?r framtagna f?r att anv?ndas n?r + % vi har 3de och 4de derivator i v?r PDE + % I annat fall anv?nd de "traditionella" som har noggrannare + % randsplutningar f?r D1 och D2 + + % Vi b?rjar med normen. Notera att alla SBP operatorer delar samma norm, + % vilket ?r n?dv?ndigt f?r stabilitet + + H=diag(ones(m,1),0);H(1,1)=1/2;H(m,m)=1/2; + + + H=H*h; + HI=inv(H); + + + % First derivative SBP operator, 1st order accurate at first 6 boundary points + + q1=1/2; + Q=q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1)); + + %Q=(-1/12*diag(ones(m-2,1),2)+8/12*diag(ones(m-1,1),1)-8/12*diag(ones(m-1,1),-1)+1/12*diag(ones(m-2,1),-2)); + + + e_1=zeros(m,1);e_1(1)=1; + e_m=zeros(m,1);e_m(m)=1; + + + D1=HI*(Q-1/2*e_1*e_1'+1/2*e_m*e_m') ; + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + + % Second derivative, 1st order accurate at first boundary points + + %% below for constant coefficients + % m1=-1;m0=2; + % M=m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0);M(1,1)=1;M(m,m)=1; + % M=M/h; + %D2=HI*(-M-e_1*S_1+e_m*S_m); + + %% Below for variable coefficients + %% Require a vector c with the koeffients + + S_U=[-3/2 2 -1/2]/h; + S_1=zeros(1,m); + S_1(1:3)=S_U; + S_m=zeros(1,m); + S_m(m-2:m)=fliplr(-S_U); + + S_1 = S_1'; + S_m = S_m'; + + M=sparse(m,m); + e_1 = sparse(e_1); + e_m = sparse(e_m); + S_1 = sparse(S_1); + S_m = sparse(S_m); + + scheme_width = 3; + scheme_radius = (scheme_width-1)/2; + r = (1+scheme_radius):(m-scheme_radius); + + function D2 = D2_fun(c) + + Mm1 = -c(r-1)/2 - c(r)/2; + M0 = c(r-1)/2 + c(r) + c(r+1)/2; + Mp1 = -c(r)/2 - c(r+1)/2; + + M(r,:) = spdiags([Mm1 M0 Mp1],0:2*scheme_radius,length(r),m); + + + M(1:2,1:2)=[c(1)/2 + c(2)/2 -c(1)/2 - c(2)/2; -c(1)/2 - c(2)/2 c(1)/2 + c(2) + c(3)/2;]; + M(m-1:m,m-1:m)=[c(m-2)/2 + c(m-1) + c(m)/2 -c(m-1)/2 - c(m)/2; -c(m-1)/2 - c(m)/2 c(m-1)/2 + c(m)/2;]; + M=M/h; + + D2=HI*(-M-c(1)*e_1*S_1'+c(m)*e_m*S_m'); + end + D2 = @D2_fun; + + + + + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + + % Third derivative, 1st order accurate at first 6 boundary points + + q2=1/2;q1=-1; + Q3=q2*(diag(ones(m-2,1),2)-diag(ones(m-2,1),-2))+q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1)); + + %QQ3=(-1/8*diag(ones(m-3,1),3) + 1*diag(ones(m-2,1),2) - 13/8*diag(ones(m-1,1),1) +13/8*diag(ones(m-1,1),-1) -1*diag(ones(m-2,1),-2) + 1/8*diag(ones(m-3,1),-3)); + + + Q3_U = [0 -0.13e2 / 0.16e2 0.7e1 / 0.8e1 -0.1e1 / 0.16e2; 0.13e2 / 0.16e2 0 -0.23e2 / 0.16e2 0.5e1 / 0.8e1; -0.7e1 / 0.8e1 0.23e2 / 0.16e2 0 -0.17e2 / 0.16e2; 0.1e1 / 0.16e2 -0.5e1 / 0.8e1 0.17e2 / 0.16e2 0;]; + Q3(1:4,1:4)=Q3_U; + Q3(m-3:m,m-3:m)=flipud( fliplr( -Q3_U ) ); + Q3=Q3/h^2; + + + + S2_U=[1 -2 1;]/h^2; + S2_1=zeros(1,m); + S2_1(1:3)=S2_U; + S2_m=zeros(1,m); + S2_m(m-2:m)=fliplr(S2_U); + S2_1 = S2_1'; + S2_m = S2_m'; + + + + D3=HI*(Q3 - e_1*S2_1' + e_m*S2_m' +1/2*S_1*S_1' -1/2*S_m*S_m' ) ; + + % Fourth derivative, 0th order accurate at first 6 boundary points (still + % yield 4th order convergence if stable: for example u_tt=-u_xxxx + + m2=1;m1=-4;m0=6; + M4=m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0); + + %M4=(-1/6*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3) ) + 2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2)) -13/2*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)) + 28/3*diag(ones(m,1),0)); + + M4_U=[0.13e2 / 0.10e2 -0.12e2 / 0.5e1 0.9e1 / 0.10e2 0.1e1 / 0.5e1; -0.12e2 / 0.5e1 0.26e2 / 0.5e1 -0.16e2 / 0.5e1 0.2e1 / 0.5e1; 0.9e1 / 0.10e2 -0.16e2 / 0.5e1 0.47e2 / 0.10e2 -0.17e2 / 0.5e1; 0.1e1 / 0.5e1 0.2e1 / 0.5e1 -0.17e2 / 0.5e1 0.29e2 / 0.5e1;]; + + + M4(1:4,1:4)=M4_U; + + M4(m-3:m,m-3:m)=flipud( fliplr( M4_U ) ); + M4=M4/h^3; + + S3_U=[-1 3 -3 1;]/h^3; + S3_1=zeros(1,m); + S3_1(1:4)=S3_U; + S3_m=zeros(1,m); + S3_m(m-3:m)=fliplr(-S3_U); + S3_1 = S3_1'; + S3_m = S3_m'; + + D4=HI*(M4-e_1*S3_1'+e_m*S3_m' + S_1*S2_1'-S_m*S2_m'); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/higher_variable4.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,143 @@ +function [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = higher_variable4(m,h) + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + %%% 4:de ordn. SBP Finita differens %%% + %%% %%% + %%% H (Normen) %%% + %%% D1=H^(-1)Q (approx f?rsta derivatan) %%% + %%% D2 (approx andra derivatan) %%% + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + Hv = ones(m,1); + Hv(1:4) = [17/48 59/48 43/48 49/48]; + Hv(m-3:m) = rot90(Hv(1:4),2); + Hv = h * Hv; + H = spdiag(Hv,0); + HI = spdiag(1./Hv,0); + + e_1 = sparse(m, 1); + e_1(1) = 1; + e_m = rot90(e_1, 2); + + S_1 = sparse(m, 1); + S_1(1:4) = [-0.11e2/0.6e1 3 -0.3e1/0.2e1 0.1e1/0.3e1]/h; + S_m = -rot90(S_1, 2); + + S2_1 = sparse(m, 1); + S2_1(1:4) = [2 -5 4 -1;]/h^2; + S2_m = rot90(S2_1, 2); + + S3_1 = sparse(m, 1); + S3_1(1:4) = [-1 3 -3 1;]/h^3; + S3_m = rot90(S3_1, 2); + + + q_diags = [-2, -1, 0, 1, 2]; + q_stencil = [-1/12, 8/12, 0, -8/12, 1/12]; + Q = stripeMatrix(q_stencil, q_diags,m); + + Q_U = [ + 0 0.59e2/0.96e2 -0.1e1/0.12e2 -0.1e1/0.32e2; + -0.59e2/0.96e2 0 0.59e2/0.96e2 0; + 0.1e1/0.12e2 -0.59e2/0.96e2 0 0.59e2/0.96e2; + 0.1e1/0.32e2 0 -0.59e2/0.96e2 0; + ]; + + Q(1:4,1:4)=Q_U; + Q(m-3:m,m-3:m)=-rot90(Q_U, 2); + + + D1=HI*(Q-1/2*e_1*e_1'+1/2*e_m*e_m') ; + + m_diags = [-2, -1, 0, 1, 2]; + m_stencil = [-1/12, 16/12, -30/12, -16/12, -1/12]; + + M_U=[ + 0.9e1 / 0.8e1 -0.59e2 / 0.48e2 0.1e1 / 0.12e2 0.1e1 / 0.48e2; + -0.59e2 / 0.48e2 0.59e2 / 0.24e2 -0.59e2 / 0.48e2 0; + 0.1e1 / 0.12e2 -0.59e2 / 0.48e2 0.55e2 / 0.24e2 -0.59e2 / 0.48e2; + 0.1e1 / 0.48e2 0 -0.59e2 / 0.48e2 0.59e2 / 0.24e2; + ]; + + M(1:4,1:4)=M_U; + M(m-3:m,m-3:m)=rot90(M_U, 2); + M=M/h; + + + scheme_width = 5; + scheme_radius = (scheme_width-1)/2; + r = (1+scheme_radius):(m-scheme_radius); + + function D2 = D2_fun(c) + + %% ALTERNATIVES %%%%%%%%%%%%% + % for i=4:m-3 + % M(i,i-2:i+2)=[-c(i-1) / 0.6e1 + c(i-2) / 0.8e1 + c(i) / 0.8e1 -c(i-2) / 0.6e1 - c(i+1) / 0.6e1 - c(i-1) / 0.2e1 - c(i) / 0.2e1 c(i-2) / 0.24e2 + 0.5e1 / 0.6e1 * c(i-1) + 0.5e1 / 0.6e1 * c(i+1) + c(i+2) / 0.24e2 + 0.3e1 / 0.4e1 * c(i) -c(i-1) / 0.6e1 - c(i+2) / 0.6e1 - c(i) / 0.2e1 - c(i+1) / 0.2e1 -c(i+1) / 0.6e1 + c(i) / 0.8e1 + c(i+2) / 0.8e1;]; + % end + %%%%%%%%%%%%%%%%%%%%%%%%%%%%% + % for i=4:m-3 + % M(i,i-2:i+2)= [ + % c(i-2) / 0.8e1 - c(i-1) / 0.6e1 + c(i) / 0.8e1 , + % -c(i-2) / 0.6e1 - c(i-1) / 0.2e1 - c(i) / 0.2e1 - c(i+1) / 0.6e1 , + % c(i-2) / 2.4e1 + c(i-1) / 1.2e0 + c(i) * 0.3/0.4 + c(i+1) / 1.2e0 + c(i+2) / 2.4e1 , + % -c(i-1) / 0.6e1 - c(i) / 0.2e1 - c(i+1) / 0.2e1 - c(i+2) / 0.6e1 , + % c(i) / 0.8e1 - c(i+1) / 0.6e1 + c(i+2) / 0.8e1 , + % ]; + % end + %%%%%%%%%%%%%%%%%%%%%%%%%%%%% + Mm2 = c(r-2) / 0.8e1 - c(r-1) / 0.6e1 + c(r) / 0.8e1 ; + Mm1 = -c(r-2) / 0.6e1 - c(r-1) / 0.2e1 - c(r) / 0.2e1 - c(r+1) / 0.6e1 ; + M0 = c(r-2) / 2.4e1 + c(r-1) / 1.2e0 + c(r) * 0.3/0.4 + c(r+1) / 1.2e0 + c(r+2) / 2.4e1; + Mp1 = -c(r-1) / 0.6e1 - c(r) / 0.2e1 - c(r+1) / 0.2e1 - c(r+2) / 0.6e1; + Mp2 = c(r) / 0.8e1 - c(r+1) / 0.6e1 + c(r+2) / 0.8e1; + % printSize(Mm2); + % scheme_radius + % m + M(r,:) = spdiags([Mm2 Mm1 M0 Mp1 Mp2],0:2*scheme_radius,length(r),m); + % M(r,:) = spdiags([Mm2 Mm1 M0 Mp1 Mp2],(-2:2)+scheme_radius,M(r,:)); % This is slower + %%%%%%%%%%%%%%%%%%%%%%%%%%%%% + % %% Somthing is wrong here!! + % Mm2 = c(r-2) / 0.8e1 - c(r-1) / 0.6e1 + c(r) / 0.8e1 ; + % Mm1 = -c(r-2) / 0.6e1 - c(r-1) / 0.2e1 - c(r) / 0.2e1 - c(r+1) / 0.6e1 ; + % M0 = c(r-2) / 2.4e1 + c(r-1) / 1.2e0 + c(r) * 0.3/0.4 + c(r+1) / 1.2e0 + c(r+2) / 2.4e1; + % Mp1 = -c(r-1) / 0.6e1 - c(r) / 0.2e1 - c(r+1) / 0.2e1 - c(r+2) / 0.6e1; + % Mp2 = c(r) / 0.8e1 - c(r+1) / 0.6e1 + c(r+2) / 0.8e1; + % % printSize(M_diag_ind); + % % Mdiags = [Mm2 Mm1 M0 Mp1 Mp2]; + % % printSize(Mdiags); + % M(M_diag_ind) = [Mm2 Mm1 M0 Mp1 Mp2]; % This is slightly faster + %%%%%%%%%%%%%%%%%%%%%%%%%%%%% + % Kan man skriva det som en multiplikation av en 3-dim matris? + %%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + + + M(1:6,1:6)=[0.12e2 / 0.17e2 * c(1) + 0.59e2 / 0.192e3 * c(2) + 0.27010400129e11 / 0.345067064608e12 * c(3) + 0.69462376031e11 / 0.2070402387648e13 * c(4) -0.59e2 / 0.68e2 * c(1) - 0.6025413881e10 / 0.21126554976e11 * c(3) - 0.537416663e9 / 0.7042184992e10 * c(4) 0.2e1 / 0.17e2 * c(1) - 0.59e2 / 0.192e3 * c(2) + 0.213318005e9 / 0.16049630912e11 * c(4) + 0.2083938599e10 / 0.8024815456e10 * c(3) 0.3e1 / 0.68e2 * c(1) - 0.1244724001e10 / 0.21126554976e11 * c(3) + 0.752806667e9 / 0.21126554976e11 * c(4) 0.49579087e8 / 0.10149031312e11 * c(3) - 0.49579087e8 / 0.10149031312e11 * c(4) -c(4) / 0.784e3 + c(3) / 0.784e3; -0.59e2 / 0.68e2 * c(1) - 0.6025413881e10 / 0.21126554976e11 * c(3) - 0.537416663e9 / 0.7042184992e10 * c(4) 0.3481e4 / 0.3264e4 * c(1) + 0.9258282831623875e16 / 0.7669235228057664e16 * c(3) + 0.236024329996203e15 / 0.1278205871342944e16 * c(4) -0.59e2 / 0.408e3 * c(1) - 0.29294615794607e14 / 0.29725717938208e14 * c(3) - 0.2944673881023e13 / 0.29725717938208e14 * c(4) -0.59e2 / 0.1088e4 * c(1) + 0.260297319232891e15 / 0.2556411742685888e16 * c(3) - 0.60834186813841e14 / 0.1278205871342944e16 * c(4) -0.1328188692663e13 / 0.37594290333616e14 * c(3) + 0.1328188692663e13 / 0.37594290333616e14 * c(4) -0.8673e4 / 0.2904112e7 * c(3) + 0.8673e4 / 0.2904112e7 * c(4); 0.2e1 / 0.17e2 * c(1) - 0.59e2 / 0.192e3 * c(2) + 0.213318005e9 / 0.16049630912e11 * c(4) + 0.2083938599e10 / 0.8024815456e10 * c(3) -0.59e2 / 0.408e3 * c(1) - 0.29294615794607e14 / 0.29725717938208e14 * c(3) - 0.2944673881023e13 / 0.29725717938208e14 * c(4) c(1) / 0.51e2 + 0.59e2 / 0.192e3 * c(2) + 0.13777050223300597e17 / 0.26218083221499456e17 * c(4) + 0.564461e6 / 0.13384296e8 * c(5) + 0.378288882302546512209e21 / 0.270764341349677687456e21 * c(3) c(1) / 0.136e3 - 0.125059e6 / 0.743572e6 * c(5) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(3) - 0.17220493277981e14 / 0.89177153814624e14 * c(4) -0.10532412077335e14 / 0.42840005263888e14 * c(4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(3) + 0.564461e6 / 0.4461432e7 * c(5) -0.960119e6 / 0.1280713392e10 * c(4) - 0.3391e4 / 0.6692148e7 * c(5) + 0.33235054191e11 / 0.26452850508784e14 * c(3); 0.3e1 / 0.68e2 * c(1) - 0.1244724001e10 / 0.21126554976e11 * c(3) + 0.752806667e9 / 0.21126554976e11 * c(4) -0.59e2 / 0.1088e4 * c(1) + 0.260297319232891e15 / 0.2556411742685888e16 * c(3) - 0.60834186813841e14 / 0.1278205871342944e16 * c(4) c(1) / 0.136e3 - 0.125059e6 / 0.743572e6 * c(5) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(3) - 0.17220493277981e14 / 0.89177153814624e14 * c(4) 0.3e1 / 0.1088e4 * c(1) + 0.507284006600757858213e21 / 0.475219048083107777984e21 * c(3) + 0.1869103e7 / 0.2230716e7 * c(5) + c(6) / 0.24e2 + 0.1950062198436997e16 / 0.3834617614028832e16 * c(4) -0.4959271814984644613e19 / 0.20965546238960637264e20 * c(3) - c(6) / 0.6e1 - 0.15998714909649e14 / 0.37594290333616e14 * c(4) - 0.375177e6 / 0.743572e6 * c(5) -0.368395e6 / 0.2230716e7 * c(5) + 0.752806667e9 / 0.539854092016e12 * c(3) + 0.1063649e7 / 0.8712336e7 * c(4) + c(6) / 0.8e1; 0.49579087e8 / 0.10149031312e11 * c(3) - 0.49579087e8 / 0.10149031312e11 * c(4) -0.1328188692663e13 / 0.37594290333616e14 * c(3) + 0.1328188692663e13 / 0.37594290333616e14 * c(4) -0.10532412077335e14 / 0.42840005263888e14 * c(4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(3) + 0.564461e6 / 0.4461432e7 * c(5) -0.4959271814984644613e19 / 0.20965546238960637264e20 * c(3) - c(6) / 0.6e1 - 0.15998714909649e14 / 0.37594290333616e14 * c(4) - 0.375177e6 / 0.743572e6 * c(5) 0.8386761355510099813e19 / 0.128413970713633903242e21 * c(3) + 0.2224717261773437e16 / 0.2763180339520776e16 * c(4) + 0.5e1 / 0.6e1 * c(6) + c(7) / 0.24e2 + 0.280535e6 / 0.371786e6 * c(5) -0.35039615e8 / 0.213452232e9 * c(4) - c(7) / 0.6e1 - 0.13091810925e11 / 0.13226425254392e14 * c(3) - 0.1118749e7 / 0.2230716e7 * c(5) - c(6) / 0.2e1; -c(4) / 0.784e3 + c(3) / 0.784e3 -0.8673e4 / 0.2904112e7 * c(3) + 0.8673e4 / 0.2904112e7 * c(4) -0.960119e6 / 0.1280713392e10 * c(4) - 0.3391e4 / 0.6692148e7 * c(5) + 0.33235054191e11 / 0.26452850508784e14 * c(3) -0.368395e6 / 0.2230716e7 * c(5) + 0.752806667e9 / 0.539854092016e12 * c(3) + 0.1063649e7 / 0.8712336e7 * c(4) + c(6) / 0.8e1 -0.35039615e8 / 0.213452232e9 * c(4) - c(7) / 0.6e1 - 0.13091810925e11 / 0.13226425254392e14 * c(3) - 0.1118749e7 / 0.2230716e7 * c(5) - c(6) / 0.2e1 0.3290636e7 / 0.80044587e8 * c(4) + 0.5580181e7 / 0.6692148e7 * c(5) + 0.5e1 / 0.6e1 * c(7) + c(8) / 0.24e2 + 0.660204843e9 / 0.13226425254392e14 * c(3) + 0.3e1 / 0.4e1 * c(6);]; + + M(m-5:m,m-5:m)=[c(m-7) / 0.24e2 + 0.5e1 / 0.6e1 * c(m-6) + 0.5580181e7 / 0.6692148e7 * c(m-4) + 0.4887707739997e13 / 0.119037827289528e15 * c(m-3) + 0.3e1 / 0.4e1 * c(m-5) + 0.660204843e9 / 0.13226425254392e14 * c(m-2) + 0.660204843e9 / 0.13226425254392e14 * c(m-1) -c(m-6) / 0.6e1 - 0.1618585929605e13 / 0.9919818940794e13 * c(m-3) - c(m-5) / 0.2e1 - 0.1118749e7 / 0.2230716e7 * c(m-4) - 0.13091810925e11 / 0.13226425254392e14 * c(m-2) - 0.13091810925e11 / 0.13226425254392e14 * c(m-1) -0.368395e6 / 0.2230716e7 * c(m-4) + c(m-5) / 0.8e1 + 0.48866620889e11 / 0.404890569012e12 * c(m-3) + 0.752806667e9 / 0.539854092016e12 * c(m-2) + 0.752806667e9 / 0.539854092016e12 * c(m-1) -0.3391e4 / 0.6692148e7 * c(m-4) - 0.238797444493e12 / 0.119037827289528e15 * c(m-3) + 0.33235054191e11 / 0.26452850508784e14 * c(m-2) + 0.33235054191e11 / 0.26452850508784e14 * c(m-1) -0.8673e4 / 0.2904112e7 * c(m-2) - 0.8673e4 / 0.2904112e7 * c(m-1) + 0.8673e4 / 0.1452056e7 * c(m-3) -c(m-3) / 0.392e3 + c(m-2) / 0.784e3 + c(m-1) / 0.784e3; -c(m-6) / 0.6e1 - 0.1618585929605e13 / 0.9919818940794e13 * c(m-3) - c(m-5) / 0.2e1 - 0.1118749e7 / 0.2230716e7 * c(m-4) - 0.13091810925e11 / 0.13226425254392e14 * c(m-2) - 0.13091810925e11 / 0.13226425254392e14 * c(m-1) c(m-6) / 0.24e2 + 0.5e1 / 0.6e1 * c(m-5) + 0.3896014498639e13 / 0.4959909470397e13 * c(m-3) + 0.8386761355510099813e19 / 0.128413970713633903242e21 * c(m-2) + 0.280535e6 / 0.371786e6 * c(m-4) + 0.3360696339136261875e19 / 0.171218627618178537656e21 * c(m-1) -c(m-5) / 0.6e1 - 0.4959271814984644613e19 / 0.20965546238960637264e20 * c(m-2) - 0.375177e6 / 0.743572e6 * c(m-4) - 0.13425842714e11 / 0.33740880751e11 * c(m-3) - 0.193247108773400725e18 / 0.6988515412986879088e19 * c(m-1) -0.365281640980e12 / 0.1653303156799e13 * c(m-3) + 0.564461e6 / 0.4461432e7 * c(m-4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(m-2) - 0.198407225513315475e18 / 0.7963657098519931984e19 * c(m-1) -0.1328188692663e13 / 0.37594290333616e14 * c(m-2) + 0.2226377963775e13 / 0.37594290333616e14 * c(m-1) - 0.8673e4 / 0.363014e6 * c(m-3) c(m-3) / 0.49e2 + 0.49579087e8 / 0.10149031312e11 * c(m-2) - 0.256702175e9 / 0.10149031312e11 * c(m-1); -0.368395e6 / 0.2230716e7 * c(m-4) + c(m-5) / 0.8e1 + 0.48866620889e11 / 0.404890569012e12 * c(m-3) + 0.752806667e9 / 0.539854092016e12 * c(m-2) + 0.752806667e9 / 0.539854092016e12 * c(m-1) -c(m-5) / 0.6e1 - 0.4959271814984644613e19 / 0.20965546238960637264e20 * c(m-2) - 0.375177e6 / 0.743572e6 * c(m-4) - 0.13425842714e11 / 0.33740880751e11 * c(m-3) - 0.193247108773400725e18 / 0.6988515412986879088e19 * c(m-1) c(m-5) / 0.24e2 + 0.1869103e7 / 0.2230716e7 * c(m-4) + 0.507284006600757858213e21 / 0.475219048083107777984e21 * c(m-2) + 0.3e1 / 0.1088e4 * c(m) + 0.31688435395e11 / 0.67481761502e11 * c(m-3) + 0.27769176016102795561e20 / 0.712828572124661666976e21 * c(m-1) -0.125059e6 / 0.743572e6 * c(m-4) + c(m) / 0.136e3 - 0.23099342648e11 / 0.101222642253e12 * c(m-3) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(m-2) + 0.193950157930938693e18 / 0.5525802884687299744e19 * c(m-1) 0.260297319232891e15 / 0.2556411742685888e16 * c(m-2) - 0.59e2 / 0.1088e4 * c(m) - 0.106641839640553e15 / 0.1278205871342944e16 * c(m-1) + 0.26019e5 / 0.726028e6 * c(m-3) -0.1244724001e10 / 0.21126554976e11 * c(m-2) + 0.3e1 / 0.68e2 * c(m) + 0.752806667e9 / 0.21126554976e11 * c(m-1); -0.3391e4 / 0.6692148e7 * c(m-4) - 0.238797444493e12 / 0.119037827289528e15 * c(m-3) + 0.33235054191e11 / 0.26452850508784e14 * c(m-2) + 0.33235054191e11 / 0.26452850508784e14 * c(m-1) -0.365281640980e12 / 0.1653303156799e13 * c(m-3) + 0.564461e6 / 0.4461432e7 * c(m-4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(m-2) - 0.198407225513315475e18 / 0.7963657098519931984e19 * c(m-1) -0.125059e6 / 0.743572e6 * c(m-4) + c(m) / 0.136e3 - 0.23099342648e11 / 0.101222642253e12 * c(m-3) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(m-2) + 0.193950157930938693e18 / 0.5525802884687299744e19 * c(m-1) 0.564461e6 / 0.13384296e8 * c(m-4) + 0.470299699916357e15 / 0.952302618316224e15 * c(m-3) + 0.550597048646198778781e21 / 0.1624586048098066124736e22 * c(m-1) + c(m) / 0.51e2 + 0.378288882302546512209e21 / 0.270764341349677687456e21 * c(m-2) -0.59e2 / 0.408e3 * c(m) - 0.29294615794607e14 / 0.29725717938208e14 * c(m-2) - 0.2234477713167e13 / 0.29725717938208e14 * c(m-1) - 0.8673e4 / 0.363014e6 * c(m-3) -0.59e2 / 0.3136e4 * c(m-3) - 0.13249937023e11 / 0.48148892736e11 * c(m-1) + 0.2e1 / 0.17e2 * c(m) + 0.2083938599e10 / 0.8024815456e10 * c(m-2); -0.8673e4 / 0.2904112e7 * c(m-2) - 0.8673e4 / 0.2904112e7 * c(m-1) + 0.8673e4 / 0.1452056e7 * c(m-3) -0.1328188692663e13 / 0.37594290333616e14 * c(m-2) + 0.2226377963775e13 / 0.37594290333616e14 * c(m-1) - 0.8673e4 / 0.363014e6 * c(m-3) 0.260297319232891e15 / 0.2556411742685888e16 * c(m-2) - 0.59e2 / 0.1088e4 * c(m) - 0.106641839640553e15 / 0.1278205871342944e16 * c(m-1) + 0.26019e5 / 0.726028e6 * c(m-3) -0.59e2 / 0.408e3 * c(m) - 0.29294615794607e14 / 0.29725717938208e14 * c(m-2) - 0.2234477713167e13 / 0.29725717938208e14 * c(m-1) - 0.8673e4 / 0.363014e6 * c(m-3) 0.9258282831623875e16 / 0.7669235228057664e16 * c(m-2) + 0.3481e4 / 0.3264e4 * c(m) + 0.228389721191751e15 / 0.1278205871342944e16 * c(m-1) + 0.8673e4 / 0.1452056e7 * c(m-3) -0.6025413881e10 / 0.21126554976e11 * c(m-2) - 0.59e2 / 0.68e2 * c(m) - 0.537416663e9 / 0.7042184992e10 * c(m-1); -c(m-3) / 0.392e3 + c(m-2) / 0.784e3 + c(m-1) / 0.784e3 c(m-3) / 0.49e2 + 0.49579087e8 / 0.10149031312e11 * c(m-2) - 0.256702175e9 / 0.10149031312e11 * c(m-1) -0.1244724001e10 / 0.21126554976e11 * c(m-2) + 0.3e1 / 0.68e2 * c(m) + 0.752806667e9 / 0.21126554976e11 * c(m-1) -0.59e2 / 0.3136e4 * c(m-3) - 0.13249937023e11 / 0.48148892736e11 * c(m-1) + 0.2e1 / 0.17e2 * c(m) + 0.2083938599e10 / 0.8024815456e10 * c(m-2) -0.6025413881e10 / 0.21126554976e11 * c(m-2) - 0.59e2 / 0.68e2 * c(m) - 0.537416663e9 / 0.7042184992e10 * c(m-1) 0.3e1 / 0.3136e4 * c(m-3) + 0.27010400129e11 / 0.345067064608e12 * c(m-2) + 0.234566387291e12 / 0.690134129216e12 * c(m-1) + 0.12e2 / 0.17e2 * c(m);]; + + M=M/h; + D2=HI*(-M-c(1)*e_1*S_1'+c(m)*e_m*S_m'); + end + D2 = @D2_fun; + + m3=-1/6;m2=2;m1=-13/2;m0=28/3; + M4=m3*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3))+m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0); + + %M4=(-1/6*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3) ) + 2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2)) -13/2*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)) + 28/3*diag(ones(m,1),0)); + + M4_U=[0.5762947e7 / 0.2316384e7 -0.6374287e7 / 0.1158192e7 0.573947e6 / 0.165456e6 -0.124637e6 / 0.289548e6 0.67979e5 / 0.2316384e7 -0.60257e5 / 0.1158192e7; -0.6374287e7 / 0.1158192e7 0.30392389e8 / 0.2316384e7 -0.2735053e7 / 0.289548e6 0.273109e6 / 0.165456e6 0.83767e5 / 0.1158192e7 0.245549e6 / 0.2316384e7; 0.573947e6 / 0.165456e6 -0.2735053e7 / 0.289548e6 0.5266855e7 / 0.579096e6 -0.1099715e7 / 0.289548e6 0.869293e6 / 0.1158192e7 -0.10195e5 / 0.144774e6; -0.124637e6 / 0.289548e6 0.273109e6 / 0.165456e6 -0.1099715e7 / 0.289548e6 0.3259225e7 / 0.579096e6 -0.324229e6 / 0.72387e5 0.1847891e7 / 0.1158192e7; 0.67979e5 / 0.2316384e7 0.83767e5 / 0.1158192e7 0.869293e6 / 0.1158192e7 -0.324229e6 / 0.72387e5 0.2626501e7 / 0.330912e6 -0.7115491e7 / 0.1158192e7; -0.60257e5 / 0.1158192e7 0.245549e6 / 0.2316384e7 -0.10195e5 / 0.144774e6 0.1847891e7 / 0.1158192e7 -0.7115491e7 / 0.1158192e7 0.21383077e8 / 0.2316384e7;]; + + M4(1:6,1:6)=M4_U; + + M4(m-5:m,m-5:m)=flipud( fliplr( M4_U ) ); + M4=M4/h^3; + + + D4=HI*(M4-e_1*S3_1'+e_m*S3_m' + S_1*S2_1'-S_m*S2_m'); + + + + + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/higher_variable4_min_boundary_points.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,104 @@ +function [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = higher_variable4_min_boundary_points(m,h) +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% 4:de ordn. SBP Finita differens %%% +%%% operatorer framtagna av Mark Carpenter %%% +%%% %%% +%%% H (Normen) %%% +%%% D1=H^(-1)Q (approx f?rsta derivatan) %%% +%%% D2 (approx andra derivatan) %%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%H?r med endast 4 randpunkter + +%m=20; %problemstorlek +%h=1/(m-1); +%h=1; + +Hv = ones(m,1) +Hv(1:4) = [17/48 59/48 43/48 49/48]; +Hv(m-3:m) = rot90(Hv(1:4),2); +Hv = h * Hv; +H = spdiag(Hv,0); +HI = spdiag(1./Hv,0); + + +% Boundary operators +e_1 = sparse(m,1); +e_1(1) = 1; +e_m = rot90(e_1, 2); + +S_U=[-0.11e2 / 0.6e1 3 -0.3e1 / 0.2e1 0.1e1 / 0.3e1;]/h; +S_1=sparse(1,m); +S_1(1:4)=S_U; +S_m = -rot90(S_1, 2); + +S2_U=[2 -5 4 -1;]/h^2; +S2_1=sparse(1,m); +S2_1(1:4)=S2_U; +S2_m = rot90(S2_1, 2); + +S3_U=[-1 3 -3 1;]/h^3; +S3_1=sparse(1,m); +S3_1(1:4)=S3_U; +S3_m = rot90(S3_1, 2); + + + + + + +q_diags = [-1, 0, 1, 2]; + q_stencil = [-1/3 -1/2 1 -1/6]; + Qp = stripeMatrix(q_stencil, q_diags,m); + + Q_U = [ + -1/24 17/24 -1/6; + -13/24 -1/4 23/24; + 1/12 -11/24 -11/24; + ]; + Qp(1:3,1:3)=Q_U; + Qp(m-2:m,m-2:m)=rot90(Q_U,2)'; %%% This is different from standard SBP + + Qm=-Qp'; + + +Q=(-1/12*diag(ones(m-2,1),2)+8/12*diag(ones(m-1,1),1)-8/12*diag(ones(m-1,1),-1)+1/12*diag(ones(m-2,1),-2)); +Q_U = [0 0.59e2 / 0.96e2 -0.1e1 / 0.12e2 -0.1e1 / 0.32e2; -0.59e2 / 0.96e2 0 0.59e2 / 0.96e2 0; 0.1e1 / 0.12e2 -0.59e2 / 0.96e2 0 0.59e2 / 0.96e2; 0.1e1 / 0.32e2 0 -0.59e2 / 0.96e2 0;]; +Q(1:4,1:4)=Q_U; +Q(m-3:m,m-3:m)=flipud( fliplr(-Q_U(1:4,1:4) ) ); + + + + +D1=HI*(Q-1/2*e_1*e_1'+1/2*e_m*e_m') ; + +M_U=[0.9e1 / 0.8e1 -0.59e2 / 0.48e2 0.1e1 / 0.12e2 0.1e1 / 0.48e2; -0.59e2 / 0.48e2 0.59e2 / 0.24e2 -0.59e2 / 0.48e2 0; 0.1e1 / 0.12e2 -0.59e2 / 0.48e2 0.55e2 / 0.24e2 -0.59e2 / 0.48e2; 0.1e1 / 0.48e2 0 -0.59e2 / 0.48e2 0.59e2 / 0.24e2;]; +M=-(-1/12*diag(ones(m-2,1),2)+16/12*diag(ones(m-1,1),1)+16/12*diag(ones(m-1,1),-1)-1/12*diag(ones(m-2,1),-2)-30/12*diag(ones(m,1),0)); + +M(1:4,1:4)=M_U; + +M(m-3:m,m-3:m)=flipud( fliplr( M_U ) ); +M=M/h; +M=zeros(m,m); + +for i=4:m-3 +M(i,i-2:i+2)=[-c(i-1) / 0.6e1 + c(i-2) / 0.8e1 + c(i) / 0.8e1 -c(i-2) / 0.6e1 - c(i+1) / 0.6e1 - c(i-1) / 0.2e1 - c(i) / 0.2e1 c(i-2) / 0.24e2 + 0.5e1 / 0.6e1 * c(i-1) + 0.5e1 / 0.6e1 * c(i+1) + c(i+2) / 0.24e2 + 0.3e1 / 0.4e1 * c(i) -c(i-1) / 0.6e1 - c(i+2) / 0.6e1 - c(i) / 0.2e1 - c(i+1) / 0.2e1 -c(i+1) / 0.6e1 + c(i) / 0.8e1 + c(i+2) / 0.8e1;]; +end + +M(1:6,1:6)=[0.12e2 / 0.17e2 * c(1) + 0.59e2 / 0.192e3 * c(2) + 0.27010400129e11 / 0.345067064608e12 * c(3) + 0.69462376031e11 / 0.2070402387648e13 * c(4) -0.59e2 / 0.68e2 * c(1) - 0.6025413881e10 / 0.21126554976e11 * c(3) - 0.537416663e9 / 0.7042184992e10 * c(4) 0.2e1 / 0.17e2 * c(1) - 0.59e2 / 0.192e3 * c(2) + 0.213318005e9 / 0.16049630912e11 * c(4) + 0.2083938599e10 / 0.8024815456e10 * c(3) 0.3e1 / 0.68e2 * c(1) - 0.1244724001e10 / 0.21126554976e11 * c(3) + 0.752806667e9 / 0.21126554976e11 * c(4) 0.49579087e8 / 0.10149031312e11 * c(3) - 0.49579087e8 / 0.10149031312e11 * c(4) -c(4) / 0.784e3 + c(3) / 0.784e3; -0.59e2 / 0.68e2 * c(1) - 0.6025413881e10 / 0.21126554976e11 * c(3) - 0.537416663e9 / 0.7042184992e10 * c(4) 0.3481e4 / 0.3264e4 * c(1) + 0.9258282831623875e16 / 0.7669235228057664e16 * c(3) + 0.236024329996203e15 / 0.1278205871342944e16 * c(4) -0.59e2 / 0.408e3 * c(1) - 0.29294615794607e14 / 0.29725717938208e14 * c(3) - 0.2944673881023e13 / 0.29725717938208e14 * c(4) -0.59e2 / 0.1088e4 * c(1) + 0.260297319232891e15 / 0.2556411742685888e16 * c(3) - 0.60834186813841e14 / 0.1278205871342944e16 * c(4) -0.1328188692663e13 / 0.37594290333616e14 * c(3) + 0.1328188692663e13 / 0.37594290333616e14 * c(4) -0.8673e4 / 0.2904112e7 * c(3) + 0.8673e4 / 0.2904112e7 * c(4); 0.2e1 / 0.17e2 * c(1) - 0.59e2 / 0.192e3 * c(2) + 0.213318005e9 / 0.16049630912e11 * c(4) + 0.2083938599e10 / 0.8024815456e10 * c(3) -0.59e2 / 0.408e3 * c(1) - 0.29294615794607e14 / 0.29725717938208e14 * c(3) - 0.2944673881023e13 / 0.29725717938208e14 * c(4) c(1) / 0.51e2 + 0.59e2 / 0.192e3 * c(2) + 0.13777050223300597e17 / 0.26218083221499456e17 * c(4) + 0.564461e6 / 0.13384296e8 * c(5) + 0.378288882302546512209e21 / 0.270764341349677687456e21 * c(3) c(1) / 0.136e3 - 0.125059e6 / 0.743572e6 * c(5) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(3) - 0.17220493277981e14 / 0.89177153814624e14 * c(4) -0.10532412077335e14 / 0.42840005263888e14 * c(4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(3) + 0.564461e6 / 0.4461432e7 * c(5) -0.960119e6 / 0.1280713392e10 * c(4) - 0.3391e4 / 0.6692148e7 * c(5) + 0.33235054191e11 / 0.26452850508784e14 * c(3); 0.3e1 / 0.68e2 * c(1) - 0.1244724001e10 / 0.21126554976e11 * c(3) + 0.752806667e9 / 0.21126554976e11 * c(4) -0.59e2 / 0.1088e4 * c(1) + 0.260297319232891e15 / 0.2556411742685888e16 * c(3) - 0.60834186813841e14 / 0.1278205871342944e16 * c(4) c(1) / 0.136e3 - 0.125059e6 / 0.743572e6 * c(5) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(3) - 0.17220493277981e14 / 0.89177153814624e14 * c(4) 0.3e1 / 0.1088e4 * c(1) + 0.507284006600757858213e21 / 0.475219048083107777984e21 * c(3) + 0.1869103e7 / 0.2230716e7 * c(5) + c(6) / 0.24e2 + 0.1950062198436997e16 / 0.3834617614028832e16 * c(4) -0.4959271814984644613e19 / 0.20965546238960637264e20 * c(3) - c(6) / 0.6e1 - 0.15998714909649e14 / 0.37594290333616e14 * c(4) - 0.375177e6 / 0.743572e6 * c(5) -0.368395e6 / 0.2230716e7 * c(5) + 0.752806667e9 / 0.539854092016e12 * c(3) + 0.1063649e7 / 0.8712336e7 * c(4) + c(6) / 0.8e1; 0.49579087e8 / 0.10149031312e11 * c(3) - 0.49579087e8 / 0.10149031312e11 * c(4) -0.1328188692663e13 / 0.37594290333616e14 * c(3) + 0.1328188692663e13 / 0.37594290333616e14 * c(4) -0.10532412077335e14 / 0.42840005263888e14 * c(4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(3) + 0.564461e6 / 0.4461432e7 * c(5) -0.4959271814984644613e19 / 0.20965546238960637264e20 * c(3) - c(6) / 0.6e1 - 0.15998714909649e14 / 0.37594290333616e14 * c(4) - 0.375177e6 / 0.743572e6 * c(5) 0.8386761355510099813e19 / 0.128413970713633903242e21 * c(3) + 0.2224717261773437e16 / 0.2763180339520776e16 * c(4) + 0.5e1 / 0.6e1 * c(6) + c(7) / 0.24e2 + 0.280535e6 / 0.371786e6 * c(5) -0.35039615e8 / 0.213452232e9 * c(4) - c(7) / 0.6e1 - 0.13091810925e11 / 0.13226425254392e14 * c(3) - 0.1118749e7 / 0.2230716e7 * c(5) - c(6) / 0.2e1; -c(4) / 0.784e3 + c(3) / 0.784e3 -0.8673e4 / 0.2904112e7 * c(3) + 0.8673e4 / 0.2904112e7 * c(4) -0.960119e6 / 0.1280713392e10 * c(4) - 0.3391e4 / 0.6692148e7 * c(5) + 0.33235054191e11 / 0.26452850508784e14 * c(3) -0.368395e6 / 0.2230716e7 * c(5) + 0.752806667e9 / 0.539854092016e12 * c(3) + 0.1063649e7 / 0.8712336e7 * c(4) + c(6) / 0.8e1 -0.35039615e8 / 0.213452232e9 * c(4) - c(7) / 0.6e1 - 0.13091810925e11 / 0.13226425254392e14 * c(3) - 0.1118749e7 / 0.2230716e7 * c(5) - c(6) / 0.2e1 0.3290636e7 / 0.80044587e8 * c(4) + 0.5580181e7 / 0.6692148e7 * c(5) + 0.5e1 / 0.6e1 * c(7) + c(8) / 0.24e2 + 0.660204843e9 / 0.13226425254392e14 * c(3) + 0.3e1 / 0.4e1 * c(6);]; + +M(m-5:m,m-5:m)=[c(m-7) / 0.24e2 + 0.5e1 / 0.6e1 * c(m-6) + 0.5580181e7 / 0.6692148e7 * c(m-4) + 0.4887707739997e13 / 0.119037827289528e15 * c(m-3) + 0.3e1 / 0.4e1 * c(m-5) + 0.660204843e9 / 0.13226425254392e14 * c(m-2) + 0.660204843e9 / 0.13226425254392e14 * c(m-1) -c(m-6) / 0.6e1 - 0.1618585929605e13 / 0.9919818940794e13 * c(m-3) - c(m-5) / 0.2e1 - 0.1118749e7 / 0.2230716e7 * c(m-4) - 0.13091810925e11 / 0.13226425254392e14 * c(m-2) - 0.13091810925e11 / 0.13226425254392e14 * c(m-1) -0.368395e6 / 0.2230716e7 * c(m-4) + c(m-5) / 0.8e1 + 0.48866620889e11 / 0.404890569012e12 * c(m-3) + 0.752806667e9 / 0.539854092016e12 * c(m-2) + 0.752806667e9 / 0.539854092016e12 * c(m-1) -0.3391e4 / 0.6692148e7 * c(m-4) - 0.238797444493e12 / 0.119037827289528e15 * c(m-3) + 0.33235054191e11 / 0.26452850508784e14 * c(m-2) + 0.33235054191e11 / 0.26452850508784e14 * c(m-1) -0.8673e4 / 0.2904112e7 * c(m-2) - 0.8673e4 / 0.2904112e7 * c(m-1) + 0.8673e4 / 0.1452056e7 * c(m-3) -c(m-3) / 0.392e3 + c(m-2) / 0.784e3 + c(m-1) / 0.784e3; -c(m-6) / 0.6e1 - 0.1618585929605e13 / 0.9919818940794e13 * c(m-3) - c(m-5) / 0.2e1 - 0.1118749e7 / 0.2230716e7 * c(m-4) - 0.13091810925e11 / 0.13226425254392e14 * c(m-2) - 0.13091810925e11 / 0.13226425254392e14 * c(m-1) c(m-6) / 0.24e2 + 0.5e1 / 0.6e1 * c(m-5) + 0.3896014498639e13 / 0.4959909470397e13 * c(m-3) + 0.8386761355510099813e19 / 0.128413970713633903242e21 * c(m-2) + 0.280535e6 / 0.371786e6 * c(m-4) + 0.3360696339136261875e19 / 0.171218627618178537656e21 * c(m-1) -c(m-5) / 0.6e1 - 0.4959271814984644613e19 / 0.20965546238960637264e20 * c(m-2) - 0.375177e6 / 0.743572e6 * c(m-4) - 0.13425842714e11 / 0.33740880751e11 * c(m-3) - 0.193247108773400725e18 / 0.6988515412986879088e19 * c(m-1) -0.365281640980e12 / 0.1653303156799e13 * c(m-3) + 0.564461e6 / 0.4461432e7 * c(m-4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(m-2) - 0.198407225513315475e18 / 0.7963657098519931984e19 * c(m-1) -0.1328188692663e13 / 0.37594290333616e14 * c(m-2) + 0.2226377963775e13 / 0.37594290333616e14 * c(m-1) - 0.8673e4 / 0.363014e6 * c(m-3) c(m-3) / 0.49e2 + 0.49579087e8 / 0.10149031312e11 * c(m-2) - 0.256702175e9 / 0.10149031312e11 * c(m-1); -0.368395e6 / 0.2230716e7 * c(m-4) + c(m-5) / 0.8e1 + 0.48866620889e11 / 0.404890569012e12 * c(m-3) + 0.752806667e9 / 0.539854092016e12 * c(m-2) + 0.752806667e9 / 0.539854092016e12 * c(m-1) -c(m-5) / 0.6e1 - 0.4959271814984644613e19 / 0.20965546238960637264e20 * c(m-2) - 0.375177e6 / 0.743572e6 * c(m-4) - 0.13425842714e11 / 0.33740880751e11 * c(m-3) - 0.193247108773400725e18 / 0.6988515412986879088e19 * c(m-1) c(m-5) / 0.24e2 + 0.1869103e7 / 0.2230716e7 * c(m-4) + 0.507284006600757858213e21 / 0.475219048083107777984e21 * c(m-2) + 0.3e1 / 0.1088e4 * c(m) + 0.31688435395e11 / 0.67481761502e11 * c(m-3) + 0.27769176016102795561e20 / 0.712828572124661666976e21 * c(m-1) -0.125059e6 / 0.743572e6 * c(m-4) + c(m) / 0.136e3 - 0.23099342648e11 / 0.101222642253e12 * c(m-3) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(m-2) + 0.193950157930938693e18 / 0.5525802884687299744e19 * c(m-1) 0.260297319232891e15 / 0.2556411742685888e16 * c(m-2) - 0.59e2 / 0.1088e4 * c(m) - 0.106641839640553e15 / 0.1278205871342944e16 * c(m-1) + 0.26019e5 / 0.726028e6 * c(m-3) -0.1244724001e10 / 0.21126554976e11 * c(m-2) + 0.3e1 / 0.68e2 * c(m) + 0.752806667e9 / 0.21126554976e11 * c(m-1); -0.3391e4 / 0.6692148e7 * c(m-4) - 0.238797444493e12 / 0.119037827289528e15 * c(m-3) + 0.33235054191e11 / 0.26452850508784e14 * c(m-2) + 0.33235054191e11 / 0.26452850508784e14 * c(m-1) -0.365281640980e12 / 0.1653303156799e13 * c(m-3) + 0.564461e6 / 0.4461432e7 * c(m-4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(m-2) - 0.198407225513315475e18 / 0.7963657098519931984e19 * c(m-1) -0.125059e6 / 0.743572e6 * c(m-4) + c(m) / 0.136e3 - 0.23099342648e11 / 0.101222642253e12 * c(m-3) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(m-2) + 0.193950157930938693e18 / 0.5525802884687299744e19 * c(m-1) 0.564461e6 / 0.13384296e8 * c(m-4) + 0.470299699916357e15 / 0.952302618316224e15 * c(m-3) + 0.550597048646198778781e21 / 0.1624586048098066124736e22 * c(m-1) + c(m) / 0.51e2 + 0.378288882302546512209e21 / 0.270764341349677687456e21 * c(m-2) -0.59e2 / 0.408e3 * c(m) - 0.29294615794607e14 / 0.29725717938208e14 * c(m-2) - 0.2234477713167e13 / 0.29725717938208e14 * c(m-1) - 0.8673e4 / 0.363014e6 * c(m-3) -0.59e2 / 0.3136e4 * c(m-3) - 0.13249937023e11 / 0.48148892736e11 * c(m-1) + 0.2e1 / 0.17e2 * c(m) + 0.2083938599e10 / 0.8024815456e10 * c(m-2); -0.8673e4 / 0.2904112e7 * c(m-2) - 0.8673e4 / 0.2904112e7 * c(m-1) + 0.8673e4 / 0.1452056e7 * c(m-3) -0.1328188692663e13 / 0.37594290333616e14 * c(m-2) + 0.2226377963775e13 / 0.37594290333616e14 * c(m-1) - 0.8673e4 / 0.363014e6 * c(m-3) 0.260297319232891e15 / 0.2556411742685888e16 * c(m-2) - 0.59e2 / 0.1088e4 * c(m) - 0.106641839640553e15 / 0.1278205871342944e16 * c(m-1) + 0.26019e5 / 0.726028e6 * c(m-3) -0.59e2 / 0.408e3 * c(m) - 0.29294615794607e14 / 0.29725717938208e14 * c(m-2) - 0.2234477713167e13 / 0.29725717938208e14 * c(m-1) - 0.8673e4 / 0.363014e6 * c(m-3) 0.9258282831623875e16 / 0.7669235228057664e16 * c(m-2) + 0.3481e4 / 0.3264e4 * c(m) + 0.228389721191751e15 / 0.1278205871342944e16 * c(m-1) + 0.8673e4 / 0.1452056e7 * c(m-3) -0.6025413881e10 / 0.21126554976e11 * c(m-2) - 0.59e2 / 0.68e2 * c(m) - 0.537416663e9 / 0.7042184992e10 * c(m-1); -c(m-3) / 0.392e3 + c(m-2) / 0.784e3 + c(m-1) / 0.784e3 c(m-3) / 0.49e2 + 0.49579087e8 / 0.10149031312e11 * c(m-2) - 0.256702175e9 / 0.10149031312e11 * c(m-1) -0.1244724001e10 / 0.21126554976e11 * c(m-2) + 0.3e1 / 0.68e2 * c(m) + 0.752806667e9 / 0.21126554976e11 * c(m-1) -0.59e2 / 0.3136e4 * c(m-3) - 0.13249937023e11 / 0.48148892736e11 * c(m-1) + 0.2e1 / 0.17e2 * c(m) + 0.2083938599e10 / 0.8024815456e10 * c(m-2) -0.6025413881e10 / 0.21126554976e11 * c(m-2) - 0.59e2 / 0.68e2 * c(m) - 0.537416663e9 / 0.7042184992e10 * c(m-1) 0.3e1 / 0.3136e4 * c(m-3) + 0.27010400129e11 / 0.345067064608e12 * c(m-2) + 0.234566387291e12 / 0.690134129216e12 * c(m-1) + 0.12e2 / 0.17e2 * c(m);]; + +M=M/h; + +D2=HI*(-M-c(1)*e_1*S_1+c(m)*e_m*S_m); +m3=-1/6;m2=2;m1=-13/2;m0=28/3; +M4=m3*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3))+m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0); + + +M4_U=[0.8e1 / 0.3e1 -0.37e2 / 0.6e1 0.13e2 / 0.3e1 -0.5e1 / 0.6e1; -0.37e2 / 0.6e1 0.47e2 / 0.3e1 -13 0.11e2 / 0.3e1; 0.13e2 / 0.3e1 -13 0.44e2 / 0.3e1 -0.47e2 / 0.6e1; -0.5e1 / 0.6e1 0.11e2 / 0.3e1 -0.47e2 / 0.6e1 0.29e2 / 0.3e1;]; +M4(1:4,1:4)=M4_U; + +M4(m-3:m,m-3:m)=flipud( fliplr( M4_U ) ); +M4=M4/h^3; + +D4=HI*(M4-e_1*S3_1+e_m*S3_m + S_1'*S2_1-S_m'*S2_m);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/higher_variable6.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,160 @@ +function [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = higher_variable6(m,h) + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + %%% 6:te ordn. SBP Finita differens %%% + %%% operatorer med diagonal norm %%% + %%% Extension to variable koeff %%% + %%% %%% + %%% H (Normen) %%% + %%% D1=H^(-1)Q (approx f?rsta derivatan) %%% + %%% D2 (approx andra derivatan) %%% + %%% D2=HI*(R+C*D*S %%% + %%% %%% + %%% R=-D1'*H*C*D1-RR %%% + %%% %%% + %%% RR ?r dissipation) %%% + %%% Dissipationen uppbyggd av D4: %%% + %%% DI=D4*B*H*D4 %%% + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + %m=10; %problemstorlek + %h=1/(m-1); + + % Variable koefficicients are stored in vector: c, size m, + % with the unknown stored as c(1), c(2), ..., c_m + % x=1:h:m*h;x=x'; + % c=x.^0; + + + H=diag(ones(m,1),0); + H(1:6,1:6)=diag([13649/43200,12013/8640,2711/4320,5359/4320,7877/8640, ... + 43801/43200]); + H(m-5:m,m-5:m)=fliplr(flipud(diag([13649/43200,12013/8640, ... + 2711/4320,5359/4320,7877/8640,43801/43200]))); + + + x1=0.70127127127127; + + + D1=(1/60*diag(ones(m-3,1),3)-9/60*diag(ones(m-2,1),2)+45/60*diag(ones(m-1,1),1)-45/60*diag(ones(m-1,1),-1)+9/60*diag(ones(m-2,1),-2)-1/60*diag(ones(m-3,1),-3)); + + + + D1(1:6,1:9)=[-21600/13649, 43200/13649*x1-7624/40947, -172800/13649*x1+ ... + 715489/81894, 259200/13649*x1-187917/13649, -172800/13649* ... + x1+735635/81894, 43200/13649*x1-89387/40947, 0, 0, 0; ... + -8640/12013*x1+7624/180195, 0, 86400/12013*x1-57139/12013, ... + -172800/12013*x1+745733/72078, 129600/12013*x1-91715/12013, ... + -34560/12013*x1+240569/120130, 0, 0, 0; ... + 17280/2711*x1-715489/162660, -43200/2711*x1+57139/5422, 0, ... + 86400/2711*x1-176839/8133, -86400/2711*x1+242111/10844, ... + 25920/2711*x1-182261/27110, 0, 0, 0; ... + -25920/5359*x1+187917/53590, 86400/5359*x1-745733/64308, ... + -86400/5359*x1+176839/16077, 0, 43200/5359*x1-165041/32154, ... + -17280/5359*x1+710473/321540, 72/5359, 0, 0; ... + 34560/7877*x1-147127/47262, -129600/7877*x1+91715/7877, ... + 172800/7877*x1-242111/15754, -86400/7877*x1+165041/23631, ... + 0, 8640/7877*x1, -1296/7877, 144/7877, 0; ... + -43200/43801*x1+89387/131403, 172800/43801*x1-240569/87602,... + -259200/43801*x1+182261/43801, 172800/43801*x1-710473/262806, ... + -43200/43801*x1, 0, 32400/43801, -6480/43801, 720/43801]; + D1(m-5:m,m-8:m)=flipud( fliplr(-D1(1:6,1:9))); + D1=D1/h; + + e_1=zeros(m,1);e_1(1)=1; + e_m=zeros(m,1);e_m(m)=1; + + S_U=[-25/12, 4, -3, 4/3, -1/4]/h; + S_1=zeros(1,m); + S_1(1:5)=S_U; + S_m=zeros(1,m); + S_m(m-4:m)=fliplr(-S_U); + S_1 = S_1'; + S_m = S_m'; + + + + %DS=zeros(m,m); + %DS(1,1:5)=-[-25/12, 4, -3, 4/3, -1/4]; + %DS(m,m-4:m)=fliplr(-[-25/12, 4, -3, 4/3, -1/4]); + %DS=diag(c)*DS/h; + + + H=h*H; + HI=inv(H); + + + M=sparse(m,m); + e_1 = sparse(e_1); + e_m = sparse(e_m); + S_1 = sparse(S_1); + S_m = sparse(S_m); + + scheme_width = 7; + scheme_radius = (scheme_width-1)/2; + r = (1+scheme_radius):(m-scheme_radius); + + function D2 = D2_fun(c) + + Mm3 = c(r-2) / 0.40e2 + c(r-1) / 0.40e2 - 0.11e2 / 0.360e3 * c(r-3) - 0.11e2 / 0.360e3 * c(r); + Mm2 = c(r-3) / 0.20e2 - 0.3e1 / 0.10e2 * c(r-1) + c(r+1) / 0.20e2 + 0.7e1 / 0.40e2 * c(r) + 0.7e1 / 0.40e2 * c(r-2); + Mm1 = -c(r-3) / 0.40e2 - 0.3e1 / 0.10e2 * c(r-2) - 0.3e1 / 0.10e2 * c(r+1) - c(r+2) / 0.40e2 - 0.17e2 / 0.40e2 * c(r) - 0.17e2 / 0.40e2 * c(r-1); + M0 = c(r-3) / 0.180e3 + c(r-2) / 0.8e1 + 0.19e2 / 0.20e2 * c(r-1) + 0.19e2 / 0.20e2 * c(r+1) + c(r+2) / 0.8e1 + c(r+3) / 0.180e3 + 0.101e3 / 0.180e3 * c(r); + Mp1 = -c(r-2) / 0.40e2 - 0.3e1 / 0.10e2 * c(r-1) - 0.3e1 / 0.10e2 * c(r+2) - c(r+3) / 0.40e2 - 0.17e2 / 0.40e2 * c(r) - 0.17e2 / 0.40e2 * c(r+1); + Mp2 = c(r-1) / 0.20e2 - 0.3e1 / 0.10e2 * c(r+1) + c(r+3) / 0.20e2 + 0.7e1 / 0.40e2 * c(r) + 0.7e1 / 0.40e2 * c(r+2); + Mp3 = c(r+1) / 0.40e2 + c(r+2) / 0.40e2 - 0.11e2 / 0.360e3 * c(r) - 0.11e2 / 0.360e3 * c(r+3); + + M(r,:) = spdiags([Mm3 Mm2 Mm1 M0 Mp1 Mp2 Mp3],0:2*scheme_radius,length(r),m); + + + M(1:9,1:9)=[0.7912667594695582093926295e0 * c(1) + 0.2968472090638000742888467e0 * c(2) + 0.3185519088796429015220016e-2 * c(3) + 0.1632404042590951953384672e-1 * c(4) + 0.3160302244094415087693968e-1 * c(5) + 0.3167964748016105299646518e-1 * c(6) + 0.3148577733947253920469418e-1 * c(7) -0.1016689339350338144430605e1 * c(1) - 0.2845627370491611369031341e-1 * c(3) - 0.4128029838349298819825156e-1 * c(4) - 0.1392281451620140507549866e0 * c(5) - 0.1195777325611201766551392e0 * c(6) - 0.1194267756529333410855186e0 * c(7) 0.7075642937243715046279337e-1 * c(1) - 0.1845476106024151050283847e0 * c(2) - 0.4364163147111892346990101e-1 * c(4) + 0.2432367907207732460874765e0 * c(5) + 0.1582127073537215443965653e0 * c(6) + 0.1602348578364786307613271e0 * c(7) 0.2251991532891353212689574e0 * c(1) - 0.1662748711097054895317080e0 * c(2) + 0.2710530961648671297733465e-1 * c(3) - 0.1916646185968439909125616e0 * c(5) - 0.7684117160199014594442072e-1 * c(6) - 0.8219586949831697575883635e-1 * c(7) -0.5224403464202056316702078e-1 * c(1) + 0.4440063948509876221050939e-1 * c(2) - 0.1023976547309387874453988e-2 * c(3) + 0.7403484645316174090533193e-1 * c(4) + 0.1241625568998496895352046e-1 * c(6) + 0.7188652847892601282652228e-1 * c(5) + 0.1379362997104735503447960e-1 * c(7) -0.1828896813877197352675410e-1 * c(1) + 0.9574633163221758060736592e-2 * c(2) - 0.8105784530576404277872603e-3 * c(3) - 0.7348845587775519698437916e-2 * c(4) + 0.1063601949723906997026904e-1 * c(5) - 0.1315967038382618382356495e-1 * c(6) - 0.2117936478838753524581943e-1 * c(7) 0.1911888563316170927411831e-2 * c(4) - 0.4068130355529149936100229e-1 * c(5) + 0.1319674981073749167009902e-1 * c(6) + 0.2557266518123783676349144e-1 * c(7) 0.1559652871136785763960685e-1 * c(5) - 0.6486184157331537899459796e-2 * c(6) - 0.9110344554036319740147054e-2 * c(7) 0.5593983696629863059347067e-3 * c(6) - 0.1384822535100796372263822e-2 * c(5) + 0.8254241654378100663291154e-3 * c(7); -0.1016689339350338144430605e1 * c(1) - 0.2845627370491611369031341e-1 * c(3) - 0.4128029838349298819825156e-1 * c(4) - 0.1392281451620140507549866e0 * c(5) - 0.1195777325611201766551392e0 * c(6) - 0.1194267756529333410855186e0 * c(7) 0.1306332157111667628555907e1 * c(1) + 0.2542001760457345743492403e0 * c(3) + 0.1043897828092562609502636e0 * c(4) + 0.6672328021032112950919876e0 * c(5) + 0.4681819359722749441073885e0 * c(6) + 0.4676415410195836920069412e0 * c(7) -0.9091410269992464604926176e-1 * c(1) + 0.1103611313171476425250639e0 * c(4) - 0.1290397544997518887000350e1 * c(5) - 0.6639605248735044787146222e0 * c(6) - 0.6615974464005206184151509e0 * c(7) -0.2893557395653431666593814e0 * c(1) - 0.2421320004064592721552708e0 * c(3) + 0.1187670255028031027693374e1 * c(5) + 0.3956598149904136332753521e0 * c(6) + 0.3860048921755800000681479e0 * c(7) 0.6712774475803763988977355e-1 * c(1) + 0.9147192682075630179962131e-2 * c(3) - 0.1872196143003808021730728e0 * c(4) - 0.1319358558853174530078498e0 * c(6) - 0.4871575736811911887376923e0 * c(5) - 0.1047516312275448138054418e0 * c(7) 0.2349927974590068869356781e-1 * c(1) + 0.7240905383565181316381731e-2 * c(3) + 0.1858378996391679448655070e-1 * c(4) - 0.9289616133938676174345208e-1 * c(5) + 0.1223513270418807666970488e0 * c(6) + 0.1113520320436295033894092e0 * c(7) -0.4834791406446907590553793e-2 * c(4) + 0.2310683832687820403062715e0 * c(5) - 0.1080774142196007991746827e0 * c(6) - 0.1181561776427343335410349e0 * c(7) -0.8368141434403455353724691e-1 * c(5) + 0.4093499466767054661591066e-1 * c(6) + 0.4274641967636400692133626e-1 * c(7) -0.3576545132696983143406173e-2 * c(6) + 0.7389399124121078682094445e-2 * c(5) - 0.3812853991424095538688273e-2 * c(7); 0.7075642937243715046279337e-1 * c(1) - 0.1845476106024151050283847e0 * c(2) - 0.4364163147111892346990101e-1 * c(4) + 0.2432367907207732460874765e0 * c(5) + 0.1582127073537215443965653e0 * c(6) + 0.1602348578364786307613271e0 * c(7) -0.9091410269992464604926176e-1 * c(1) + 0.1103611313171476425250639e0 * c(4) - 0.1290397544997518887000350e1 * c(5) - 0.6639605248735044787146222e0 * c(6) - 0.6615974464005206184151509e0 * c(7) 0.6327161147136873807796515e-2 * c(1) + 0.1147318200715868527529827e0 * c(2) + 0.1166740554279680007487795e0 * c(4) + 0.2766610808285444037240703e1 * c(5) + 0.1070920689960817104203947e1 * c(6) + 0.1013161391032973057171717e1 * c(7) 0.2013769413884797246646959e-1 * c(1) + 0.1033717994630886401730470e0 * c(2) - 0.2913221621151742724258117e1 * c(5) - 0.8755807343482262259774782e0 * c(6) - 0.6909957183488812426508351e0 * c(7) -0.4671751091575462868310238e-2 * c(1) - 0.2760353365637712827793337e-1 * c(2) - 0.1979290298620869974478871e0 * c(4) + 0.5402985338373433052255418e0 * c(6) + 0.1239177593031911077924537e1 * c(5) + 0.2628038050247358227280031e0 * c(7) -0.1635430866921887819487473e-2 * c(1) - 0.5952475275883259619711594e-2 * c(2) + 0.1964682777744275219350831e-1 * c(4) + 0.3236640012639046600590714e0 * c(5) - 0.4659516693228870973898560e0 * c(6) - 0.2217272720941736859420432e0 * c(7) -0.5111353189352474549563559e-2 * c(4) - 0.5355878163774754346032096e0 * c(5) + 0.3328335104489738933610597e0 * c(6) + 0.2078656591178540157917135e0 * c(7) 0.1824328174134289562208038e0 * c(5) - 0.1059816030196818445908057e0 * c(6) - 0.7645121439374711162999809e-1 * c(7) 0.9209089963443799485648361e-2 * c(6) - 0.1591502818872493167091475e-1 * c(5) + 0.6705938225281132185266388e-2 * c(7); 0.2251991532891353212689574e0 * c(1) - 0.1662748711097054895317080e0 * c(2) + 0.2710530961648671297733465e-1 * c(3) - 0.1916646185968439909125616e0 * c(5) - 0.7684117160199014594442072e-1 * c(6) - 0.8219586949831697575883635e-1 * c(7) -0.2893557395653431666593814e0 * c(1) - 0.2421320004064592721552708e0 * c(3) + 0.1187670255028031027693374e1 * c(5) + 0.3956598149904136332753521e0 * c(6) + 0.3860048921755800000681479e0 * c(7) 0.2013769413884797246646959e-1 * c(1) + 0.1033717994630886401730470e0 * c(2) - 0.2913221621151742724258117e1 * c(5) - 0.8755807343482262259774782e0 * c(6) - 0.6909957183488812426508351e0 * c(7) 0.6409299775987186986730499e-1 * c(1) + 0.9313657638804699948929701e-1 * c(2) + 0.2306367624634749229113646e0 * c(3) + 0.3689440308283716620260816e1 * c(5) + 0.1190550338687608873798462e1 * c(6) + 0.5912479546888856519443605e0 * c(7) -0.1486895819265604128572498e-1 * c(1) - 0.2487040599390160764166412e-1 * c(2) - 0.8712928907711754187084757e-2 * c(3) - 0.1263507837371824205693950e1 * c(6) - 0.3058317397843997326920898e0 * c(7) - 0.1470691926045802954795783e1 * c(5) -0.5205147429855955657625694e-2 * c(1) - 0.5363098747528542488971874e-2 * c(2) - 0.6897142765790609546343709e-2 * c(3) - 0.7857524521667450101721993e0 * c(5) + 0.2291148005423734600066709e0 * c(7) + 0.9977064356292750529201981e0 * c(6) 0.6697297488067662265210608e0 * c(5) - 0.5013247356072127938999311e0 * c(6) - 0.1795161243106645437322408e0 * c(7) -0.2022909060111751565150958e0 * c(5) + 0.1453421858063658498587377e0 * c(6) + 0.5694872020480930665635812e-1 * c(7) -0.1200429618441003833696998e-1 * c(6) - 0.4776915669385923841535432e-2 * c(7) + 0.1678121185379596217850541e-1 * c(5); -0.5224403464202056316702078e-1 * c(1) + 0.4440063948509876221050939e-1 * c(2) - 0.1023976547309387874453988e-2 * c(3) + 0.7403484645316174090533193e-1 * c(4) + 0.1241625568998496895352046e-1 * c(6) + 0.7188652847892601282652228e-1 * c(5) + 0.1379362997104735503447960e-1 * c(7) 0.6712774475803763988977355e-1 * c(1) + 0.9147192682075630179962131e-2 * c(3) - 0.1872196143003808021730728e0 * c(4) - 0.1319358558853174530078498e0 * c(6) - 0.4871575736811911887376923e0 * c(5) - 0.1047516312275448138054418e0 * c(7) -0.4671751091575462868310238e-2 * c(1) - 0.2760353365637712827793337e-1 * c(2) - 0.1979290298620869974478871e0 * c(4) + 0.5402985338373433052255418e0 * c(6) + 0.1239177593031911077924537e1 * c(5) + 0.2628038050247358227280031e0 * c(7) -0.1486895819265604128572498e-1 * c(1) - 0.2487040599390160764166412e-1 * c(2) - 0.8712928907711754187084757e-2 * c(3) - 0.1263507837371824205693950e1 * c(6) - 0.3058317397843997326920898e0 * c(7) - 0.1470691926045802954795783e1 * c(5) 0.3449455095910233625229891e-2 * c(1) + 0.6641183499427826101618457e-2 * c(2) + 0.3291545083271862858501887e-3 * c(3) + 0.3357721707576477199985656e0 * c(4) + 0.2096413329579026439044119e1 * c(6) + 0.2317323204183126854954203e0 * c(7) + 0.6107825764368264576481962e-2 * c(8) + 0.7109125850683376695640722e0 * c(5) 0.1207544072304193806052558e-2 * c(1) + 0.1432116665752147607469646e-2 * c(2) + 0.2605582646183255957264249e-3 * c(3) - 0.3332941113251635390801278e-1 * c(4) - 0.2808241697385532683612407e0 * c(7) - 0.2720908083525083608370563e-1 * c(8) + 0.1045865435682921987447929e0 * c(5) - 0.1348436986667115543203552e1 * c(6) 0.8671038084174692625075159e-2 * c(4) + 0.1736073411355428563685818e0 * c(6) + 0.5331362125287625412555844e-1 * c(8) - 0.2424935262404526301801157e0 * c(5) + 0.1569015257678588270609004e0 * c(7) -0.8631683980217122275970376e-1 * c(6) + 0.2698842360470999243492629e-1 * c(7) + 0.8098194147715651085292754e-1 * c(5) - 0.3276463639080639163926118e-1 * c(8) 0.7462059484530855073291365e-2 * c(6) - 0.8121640361668678949573496e-3 * c(7) + 0.5522702088127090209264064e-3 * c(8) - 0.7202165657176696199260422e-2 * c(5); -0.1828896813877197352675410e-1 * c(1) + 0.9574633163221758060736592e-2 * c(2) - 0.8105784530576404277872603e-3 * c(3) - 0.7348845587775519698437916e-2 * c(4) + 0.1063601949723906997026904e-1 * c(5) - 0.1315967038382618382356495e-1 * c(6) - 0.2117936478838753524581943e-1 * c(7) 0.2349927974590068869356781e-1 * c(1) + 0.7240905383565181316381731e-2 * c(3) + 0.1858378996391679448655070e-1 * c(4) - 0.9289616133938676174345208e-1 * c(5) + 0.1223513270418807666970488e0 * c(6) + 0.1113520320436295033894092e0 * c(7) -0.1635430866921887819487473e-2 * c(1) - 0.5952475275883259619711594e-2 * c(2) + 0.1964682777744275219350831e-1 * c(4) + 0.3236640012639046600590714e0 * c(5) - 0.4659516693228870973898560e0 * c(6) - 0.2217272720941736859420432e0 * c(7) -0.5205147429855955657625694e-2 * c(1) - 0.5363098747528542488971874e-2 * c(2) - 0.6897142765790609546343709e-2 * c(3) - 0.7857524521667450101721993e0 * c(5) + 0.2291148005423734600066709e0 * c(7) + 0.9977064356292750529201981e0 * c(6) 0.1207544072304193806052558e-2 * c(1) + 0.1432116665752147607469646e-2 * c(2) + 0.2605582646183255957264249e-3 * c(3) - 0.3332941113251635390801278e-1 * c(4) - 0.2808241697385532683612407e0 * c(7) - 0.2720908083525083608370563e-1 * c(8) + 0.1045865435682921987447929e0 * c(5) - 0.1348436986667115543203552e1 * c(6) 0.4227226173449345042468960e-3 * c(1) + 0.3088241944378964404772302e-3 * c(2) + 0.2062575706647430620228133e-3 * c(3) + 0.3308343404200968256656458e-2 * c(4) + 0.5828047016405001815804837e0 * c(5) + 0.8054174220366215473556835e0 * c(7) + 0.1338363233410033443348225e0 * c(8) + 0.5555555555555555555555556e-2 * c(9) + 0.1190362071861893051132274e1 * c(6) -0.8607044252686413302647675e-3 * c(4) - 0.1748074708673904989293256e0 * c(5) - 0.3132544850115050165022338e0 * c(8) - 0.2500000000000000000000000e-1 * c(9) - 0.3169166305310429271303167e0 * c(7) - 0.6691607091647929161078591e0 * c(6) 0.3354661791693352108660900e-1 * c(5) - 0.3343620022386971405018586e0 * c(7) + 0.5000000000000000000000000e-1 * c(9) + 0.2169790609807602750804271e0 * c(6) + 0.1838363233410033443348225e0 * c(8) 0.2912518476823004642951502e-1 * c(7) + 0.2279091916474916391629437e-1 * c(8) - 0.3068985997518740530511593e-1 * c(6) - 0.1781799513347360596249022e-2 * c(5) - 0.3055555555555555555555556e-1 * c(9); 0.1911888563316170927411831e-2 * c(4) - 0.4068130355529149936100229e-1 * c(5) + 0.1319674981073749167009902e-1 * c(6) + 0.2557266518123783676349144e-1 * c(7) -0.4834791406446907590553793e-2 * c(4) + 0.2310683832687820403062715e0 * c(5) - 0.1080774142196007991746827e0 * c(6) - 0.1181561776427343335410349e0 * c(7) -0.5111353189352474549563559e-2 * c(4) - 0.5355878163774754346032096e0 * c(5) + 0.3328335104489738933610597e0 * c(6) + 0.2078656591178540157917135e0 * c(7) 0.6697297488067662265210608e0 * c(5) - 0.5013247356072127938999311e0 * c(6) - 0.1795161243106645437322408e0 * c(7) 0.8671038084174692625075159e-2 * c(4) + 0.1736073411355428563685818e0 * c(6) + 0.5331362125287625412555844e-1 * c(8) - 0.2424935262404526301801157e0 * c(5) + 0.1569015257678588270609004e0 * c(7) -0.8607044252686413302647675e-3 * c(4) - 0.1748074708673904989293256e0 * c(5) - 0.3132544850115050165022338e0 * c(8) - 0.2500000000000000000000000e-1 * c(9) - 0.3169166305310429271303167e0 * c(7) - 0.6691607091647929161078591e0 * c(6) 0.2239223735771599178951297e-3 * c(4) + 0.1275437785430956673825710e0 * c(5) + 0.1011699483929608164601067e1 * c(6) + 0.9698817275172575247533506e0 * c(8) + 0.1250000000000000000000000e0 * c(9) + 0.5555555555555555555555556e-2 * c(10) + 0.4823177543031281500117826e0 * c(7) -0.3784113973033012949863031e-1 * c(5) - 0.2997556885134827361576001e0 * c(6) - 0.3000000000000000000000000e0 * c(9) - 0.2500000000000000000000000e-1 * c(10) - 0.3991486867446821178415359e0 * c(7) - 0.4382544850115050165022338e0 * c(8) 0.4698146218022683933926520e-1 * c(6) - 0.2966863787471237458744416e0 * c(8) + 0.5000000000000000000000000e-1 * c(10) + 0.1716355704146006481727960e0 * c(7) + 0.3069346152296258362380356e-2 * c(5) + 0.1750000000000000000000000e0 * c(9); 0.1559652871136785763960685e-1 * c(5) - 0.6486184157331537899459796e-2 * c(6) - 0.9110344554036319740147054e-2 * c(7) -0.8368141434403455353724691e-1 * c(5) + 0.4093499466767054661591066e-1 * c(6) + 0.4274641967636400692133626e-1 * c(7) 0.1824328174134289562208038e0 * c(5) - 0.1059816030196818445908057e0 * c(6) - 0.7645121439374711162999809e-1 * c(7) -0.2022909060111751565150958e0 * c(5) + 0.1453421858063658498587377e0 * c(6) + 0.5694872020480930665635812e-1 * c(7) -0.8631683980217122275970376e-1 * c(6) + 0.2698842360470999243492629e-1 * c(7) + 0.8098194147715651085292754e-1 * c(5) - 0.3276463639080639163926118e-1 * c(8) 0.3354661791693352108660900e-1 * c(5) - 0.3343620022386971405018586e0 * c(7) + 0.5000000000000000000000000e-1 * c(9) + 0.2169790609807602750804271e0 * c(6) + 0.1838363233410033443348225e0 * c(8) -0.3784113973033012949863031e-1 * c(5) - 0.2997556885134827361576001e0 * c(6) - 0.3000000000000000000000000e0 * c(9) - 0.2500000000000000000000000e-1 * c(10) - 0.3991486867446821178415359e0 * c(7) - 0.4382544850115050165022338e0 * c(8) 0.1230328942716804455358698e-1 * c(5) + 0.1183647529645898332481833e0 * c(6) + 0.9410511898227943334189628e0 * c(7) + 0.9500000000000000000000000e0 * c(9) + 0.1250000000000000000000000e0 * c(10) + 0.5555555555555555555555556e-2 * c(11) + 0.5699474344521144554459336e0 * c(8) -0.2308067892671916339568942e-1 * c(6) - 0.2986625053775149497180439e0 * c(7) - 0.3000000000000000000000000e0 * c(10) - 0.2500000000000000000000000e-1 * c(11) - 0.1047734860515050802561078e-2 * c(5) - 0.4272090808352508360837056e0 * c(8) - 0.4250000000000000000000000e0 * c(9); 0.5593983696629863059347067e-3 * c(6) - 0.1384822535100796372263822e-2 * c(5) + 0.8254241654378100663291154e-3 * c(7) -0.3576545132696983143406173e-2 * c(6) + 0.7389399124121078682094445e-2 * c(5) - 0.3812853991424095538688273e-2 * c(7) 0.9209089963443799485648361e-2 * c(6) - 0.1591502818872493167091475e-1 * c(5) + 0.6705938225281132185266388e-2 * c(7) -0.1200429618441003833696998e-1 * c(6) - 0.4776915669385923841535432e-2 * c(7) + 0.1678121185379596217850541e-1 * c(5) 0.7462059484530855073291365e-2 * c(6) - 0.8121640361668678949573496e-3 * c(7) + 0.5522702088127090209264064e-3 * c(8) - 0.7202165657176696199260422e-2 * c(5) 0.2912518476823004642951502e-1 * c(7) + 0.2279091916474916391629437e-1 * c(8) - 0.3068985997518740530511593e-1 * c(6) - 0.1781799513347360596249022e-2 * c(5) - 0.3055555555555555555555556e-1 * c(9) 0.4698146218022683933926520e-1 * c(6) - 0.2966863787471237458744416e0 * c(8) + 0.5000000000000000000000000e-1 * c(10) + 0.1716355704146006481727960e0 * c(7) + 0.3069346152296258362380356e-2 * c(5) + 0.1750000000000000000000000e0 * c(9) -0.2308067892671916339568942e-1 * c(6) - 0.2986625053775149497180439e0 * c(7) - 0.3000000000000000000000000e0 * c(10) - 0.2500000000000000000000000e-1 * c(11) - 0.1047734860515050802561078e-2 * c(5) - 0.4272090808352508360837056e0 * c(8) - 0.4250000000000000000000000e0 * c(9) 0.5139370221149109977041877e-2 * c(6) + 0.1247723215009422001393184e0 * c(7) + 0.9505522702088127090209264e0 * c(8) + 0.9500000000000000000000000e0 * c(10) + 0.1250000000000000000000000e0 * c(11) + 0.5555555555555555555555556e-2 * c(12) + 0.9159362465153641826887659e-4 * c(5) + 0.5611111111111111111111111e0 * c(9);]; + + M(m-8:m,m-8:m)=[0.5555555555555555555555556e-2 * c(m-11) + 0.1250000000000000000000000e0 * c(m-10) + 0.9500000000000000000000000e0 * c(m-9) + 0.9505522702088127090209264e0 * c(m-7) + 0.1247205076844361998744053e0 * c(m-6) + 0.5139370221149109977041877e-2 * c(m-5) + 0.5611111111111111111111111e0 * c(m-8) + 0.1434074411575366831819799e-3 * c(m-4) -0.2500000000000000000000000e-1 * c(m-10) - 0.3000000000000000000000000e0 * c(m-9) - 0.2980649679116425253322056e0 * c(m-6) - 0.2308067892671916339568942e-1 * c(m-5) - 0.4250000000000000000000000e0 * c(m-8) - 0.4272090808352508360837056e0 * c(m-7) - 0.1645272326387475188399322e-2 * c(m-4) 0.5000000000000000000000000e-1 * c(m-9) - 0.2966863787471237458744416e0 * c(m-7) + 0.4698146218022683933926520e-1 * c(m-5) + 0.1750000000000000000000000e0 * c(m-8) + 0.1700291833903489463825077e0 * c(m-6) + 0.4675733176547960152668626e-2 * c(m-4) 0.2279091916474916391629437e-1 * c(m-7) + 0.3097763128598982561225538e-1 * c(m-6) - 0.3055555555555555555555556e-1 * c(m-8) - 0.3068985997518740530511593e-1 * c(m-5) - 0.3634246031107139778989373e-2 * c(m-4) 0.5522702088127090209264064e-3 * c(m-7) - 0.3265435411305071914756373e-2 * c(m-6) + 0.7462059484530855073291365e-2 * c(m-5) - 0.4748894282038492179461399e-2 * c(m-4) 0.6272075574042975468177820e-3 * c(m-6) - 0.1200429618441003833696998e-1 * c(m-5) + 0.1137708862700574079015220e-1 * c(m-4) 0.9209089963443799485648361e-2 * c(m-5) - 0.3129629392354775191148163e-3 * c(m-6) - 0.8896127024208321966533544e-2 * c(m-4) -0.3576545132696983143406173e-2 * c(m-5) + 0.4335019854436220306755673e-3 * c(m-6) + 0.3143043147253361112730605e-2 * c(m-4) 0.5593983696629863059347067e-3 * c(m-5) - 0.1446656414398166805849327e-3 * c(m-6) - 0.4147327282231696253497740e-3 * c(m-4); -0.2500000000000000000000000e-1 * c(m-10) - 0.3000000000000000000000000e0 * c(m-9) - 0.2980649679116425253322056e0 * c(m-6) - 0.2308067892671916339568942e-1 * c(m-5) - 0.4250000000000000000000000e0 * c(m-8) - 0.4272090808352508360837056e0 * c(m-7) - 0.1645272326387475188399322e-2 * c(m-4) 0.5555555555555555555555556e-2 * c(m-10) + 0.1250000000000000000000000e0 * c(m-9) + 0.9500000000000000000000000e0 * c(m-8) + 0.9341601509609901526962449e0 * c(m-6) + 0.1183647529645898332481833e0 * c(m-5) + 0.1919432828897222527630486e-1 * c(m-4) + 0.5699474344521144554459336e0 * c(m-7) -0.2500000000000000000000000e-1 * c(m-9) - 0.3000000000000000000000000e0 * c(m-8) - 0.2997556885134827361576001e0 * c(m-5) - 0.5636663150858098975790317e-1 * c(m-4) - 0.4382544850115050165022338e0 * c(m-7) - 0.3806231949664312575822630e0 * c(m-6) 0.5000000000000000000000000e-1 * c(m-8) - 0.3557251496099816106154206e0 * c(m-6) + 0.5490976528821799120017102e-1 * c(m-4) + 0.1838363233410033443348225e0 * c(m-7) + 0.2169790609807602750804271e0 * c(m-5) 0.5528052133944605740009217e-1 * c(m-6) - 0.8631683980217122275970376e-1 * c(m-5) - 0.3276463639080639163926118e-1 * c(m-7) + 0.5268984374242044588776166e-1 * c(m-4) -0.5373770512016897565958305e-2 * c(m-6) + 0.1453421858063658498587377e0 * c(m-5) - 0.1399684152943489522927794e0 * c(m-4) -0.1059816030196818445908057e0 * c(m-5) + 0.1014880675788250237247178e0 * c(m-4) + 0.4493535440856820866087846e-2 * c(m-6) 0.4093499466767054661591066e-1 * c(m-5) - 0.3471075437892810033585296e-1 * c(m-4) - 0.6224240288742446280057699e-2 * c(m-6) -0.6486184157331537899459796e-2 * c(m-5) + 0.4409068609809831485979484e-2 * c(m-4) + 0.2077115547521706413480312e-2 * c(m-6); 0.5000000000000000000000000e-1 * c(m-9) - 0.2966863787471237458744416e0 * c(m-7) + 0.4698146218022683933926520e-1 * c(m-5) + 0.1750000000000000000000000e0 * c(m-8) + 0.1700291833903489463825077e0 * c(m-6) + 0.4675733176547960152668626e-2 * c(m-4) -0.2500000000000000000000000e-1 * c(m-9) - 0.3000000000000000000000000e0 * c(m-8) - 0.2997556885134827361576001e0 * c(m-5) - 0.5636663150858098975790317e-1 * c(m-4) - 0.4382544850115050165022338e0 * c(m-7) - 0.3806231949664312575822630e0 * c(m-6) 0.5555555555555555555555556e-2 * c(m-9) + 0.1250000000000000000000000e0 * c(m-8) + 0.9698817275172575247533506e0 * c(m-7) + 0.1011699483929608164601067e1 * c(m-5) + 0.1773466968705924819112984e0 * c(m-4) + 0.2239223735771599178951297e-3 * c(m-3) + 0.4325148359756313354830552e0 * c(m-6) -0.2500000000000000000000000e-1 * c(m-8) - 0.3132544850115050165022338e0 * c(m-7) - 0.2322389872063761557916742e0 * c(m-4) - 0.8607044252686413302647675e-3 * c(m-3) - 0.2594851141920572702679681e0 * c(m-6) - 0.6691607091647929161078591e0 * c(m-5) 0.5331362125287625412555844e-1 * c(m-7) + 0.1736073411355428563685818e0 * c(m-5) + 0.8671038084174692625075159e-2 * c(m-3) + 0.8084259844422177692569663e-1 * c(m-6) - 0.1664345989168155800449120e0 * c(m-4) -0.5013247356072127938999311e0 * c(m-5) + 0.5021853752328231128475915e0 * c(m-4) - 0.1197175073672143005877150e-1 * c(m-6) 0.3328335104489738933610597e0 * c(m-5) - 0.3179803804558436283847901e0 * c(m-4) - 0.5111353189352474549563559e-2 * c(m-3) - 0.9741776803777790426705996e-2 * c(m-6) -0.1080774142196007991746827e0 * c(m-5) + 0.9941834083648937298100811e-1 * c(m-4) - 0.4834791406446907590553793e-2 * c(m-3) + 0.1349386478955833378422842e-1 * c(m-6) 0.1319674981073749167009902e-1 * c(m-5) - 0.1060554802883657391328704e-1 * c(m-4) + 0.1911888563316170927411831e-2 * c(m-3) - 0.4503090345217088684223814e-2 * c(m-6); 0.2279091916474916391629437e-1 * c(m-7) + 0.3097763128598982561225538e-1 * c(m-6) - 0.3055555555555555555555556e-1 * c(m-8) - 0.3068985997518740530511593e-1 * c(m-5) - 0.3634246031107139778989373e-2 * c(m-4) 0.5000000000000000000000000e-1 * c(m-8) - 0.3557251496099816106154206e0 * c(m-6) + 0.5490976528821799120017102e-1 * c(m-4) + 0.1838363233410033443348225e0 * c(m-7) + 0.2169790609807602750804271e0 * c(m-5) -0.2500000000000000000000000e-1 * c(m-8) - 0.3132544850115050165022338e0 * c(m-7) - 0.2322389872063761557916742e0 * c(m-4) - 0.8607044252686413302647675e-3 * c(m-3) - 0.2594851141920572702679681e0 * c(m-6) - 0.6691607091647929161078591e0 * c(m-5) 0.5555555555555555555555556e-2 * c(m-8) + 0.1338363233410033443348225e0 * c(m-7) + 0.7391887916719206077121040e0 * c(m-6) + 0.6490333320052011212240632e0 * c(m-4) + 0.3308343404200968256656458e-2 * c(m-3) + 0.2062575706647430620228133e-3 * c(m-2) + 0.3088241944378964404772302e-3 * c(m-1) + 0.4227226173449345042468960e-3 * c(m) + 0.1190362071861893051132274e1 * c(m-5) -0.2720908083525083608370563e-1 * c(m-7) - 0.1931148612480615118957263e0 * c(m-6) - 0.3332941113251635390801278e-1 * c(m-3) + 0.2605582646183255957264249e-3 * c(m-2) + 0.1432116665752147607469646e-2 * c(m-1) + 0.1207544072304193806052558e-2 * c(m) - 0.1348436986667115543203552e1 * c(m-5) + 0.1687723507780044227927853e-1 * c(m-4) 0.3590669644811151307464697e-1 * c(m-6) - 0.5925443480724830632401754e0 * c(m-4) - 0.6897142765790609546343709e-2 * c(m-2) - 0.5363098747528542488971874e-2 * c(m-1) - 0.5205147429855955657625694e-2 * c(m) + 0.9977064356292750529201981e0 * c(m-5) 0.7272438906214475928744770e-1 * c(m-4) + 0.1964682777744275219350831e-1 * c(m-3) - 0.5952475275883259619711594e-2 * c(m-1) - 0.1635430866921887819487473e-2 * c(m) + 0.2921234010758621482958052e-1 * c(m-6) - 0.4659516693228870973898560e0 * c(m-5) 0.5891947149681041048896399e-1 * c(m-4) + 0.1858378996391679448655070e-1 * c(m-3) + 0.7240905383565181316381731e-2 * c(m-2) + 0.2349927974590068869356781e-1 * c(m) - 0.4046360079256766884300687e-1 * c(m-6) + 0.1223513270418807666970488e0 * c(m-5) -0.2404661162020836566908542e-1 * c(m-4) - 0.7348845587775519698437916e-2 * c(m-3) - 0.8105784530576404277872603e-3 * c(m-2) + 0.9574633163221758060736592e-2 * c(m-1) - 0.1828896813877197352675410e-1 * c(m) + 0.1350326632905990039353503e-1 * c(m-6) - 0.1315967038382618382356495e-1 * c(m-5); 0.5522702088127090209264064e-3 * c(m-7) - 0.3265435411305071914756373e-2 * c(m-6) + 0.7462059484530855073291365e-2 * c(m-5) - 0.4748894282038492179461399e-2 * c(m-4) 0.5528052133944605740009217e-1 * c(m-6) - 0.8631683980217122275970376e-1 * c(m-5) - 0.3276463639080639163926118e-1 * c(m-7) + 0.5268984374242044588776166e-1 * c(m-4) 0.5331362125287625412555844e-1 * c(m-7) + 0.1736073411355428563685818e0 * c(m-5) + 0.8671038084174692625075159e-2 * c(m-3) + 0.8084259844422177692569663e-1 * c(m-6) - 0.1664345989168155800449120e0 * c(m-4) -0.2720908083525083608370563e-1 * c(m-7) - 0.1931148612480615118957263e0 * c(m-6) - 0.3332941113251635390801278e-1 * c(m-3) + 0.2605582646183255957264249e-3 * c(m-2) + 0.1432116665752147607469646e-2 * c(m-1) + 0.1207544072304193806052558e-2 * c(m) - 0.1348436986667115543203552e1 * c(m-5) + 0.1687723507780044227927853e-1 * c(m-4) 0.6107825764368264576481962e-2 * c(m-7) + 0.1155752633643216628010304e0 * c(m-6) + 0.2096413329579026439044119e1 * c(m-5) + 0.3357721707576477199985656e0 * c(m-3) + 0.3291545083271862858501887e-3 * c(m-2) + 0.6641183499427826101618457e-2 * c(m-1) + 0.3449455095910233625229891e-2 * c(m) + 0.8270696421223286922584620e0 * c(m-4) -0.4995827370863505253765970e-1 * c(m-6) - 0.1263507837371824205693950e1 * c(m-5) - 0.8712928907711754187084757e-2 * c(m-2) - 0.2487040599390160764166412e-1 * c(m-1) - 0.1486895819265604128572498e-1 * c(m) - 0.1726565392121567634950213e1 * c(m-4) 0.5402985338373433052255418e0 * c(m-5) - 0.1979290298620869974478871e0 * c(m-3) - 0.2760353365637712827793337e-1 * c(m-1) - 0.4671751091575462868310238e-2 * c(m) - 0.6952587985456154591014641e-1 * c(m-6) + 0.1571507277911208446562686e1 * c(m-4) -0.1319358558853174530078498e0 * c(m-5) - 0.1872196143003808021730728e0 * c(m-3) + 0.9147192682075630179962131e-2 * c(m-2) + 0.6712774475803763988977355e-1 * c(m) + 0.9630407686703666967100804e-1 * c(m-6) - 0.6882132817757726722141421e0 * c(m-4) 0.1241625568998496895352046e-1 * c(m-5) + 0.7403484645316174090533193e-1 * c(m-3) - 0.1023976547309387874453988e-2 * c(m-2) + 0.4440063948509876221050939e-1 * c(m-1) - 0.5224403464202056316702078e-1 * c(m) - 0.3213800979246298453953842e-1 * c(m-6) + 0.1178181682424363524005403e0 * c(m-4); 0.6272075574042975468177820e-3 * c(m-6) - 0.1200429618441003833696998e-1 * c(m-5) + 0.1137708862700574079015220e-1 * c(m-4) -0.5373770512016897565958305e-2 * c(m-6) + 0.1453421858063658498587377e0 * c(m-5) - 0.1399684152943489522927794e0 * c(m-4) -0.5013247356072127938999311e0 * c(m-5) + 0.5021853752328231128475915e0 * c(m-4) - 0.1197175073672143005877150e-1 * c(m-6) 0.3590669644811151307464697e-1 * c(m-6) - 0.5925443480724830632401754e0 * c(m-4) - 0.6897142765790609546343709e-2 * c(m-2) - 0.5363098747528542488971874e-2 * c(m-1) - 0.5205147429855955657625694e-2 * c(m) + 0.9977064356292750529201981e0 * c(m-5) -0.4995827370863505253765970e-1 * c(m-6) - 0.1263507837371824205693950e1 * c(m-5) - 0.8712928907711754187084757e-2 * c(m-2) - 0.2487040599390160764166412e-1 * c(m-1) - 0.1486895819265604128572498e-1 * c(m) - 0.1726565392121567634950213e1 * c(m-4) 0.2760393423824887721078848e-1 * c(m-6) + 0.1190550338687608873798462e1 * c(m-5) + 0.4253084328734353394994388e1 * c(m-4) + 0.2306367624634749229113646e0 * c(m-2) + 0.9313657638804699948929701e-1 * c(m-1) + 0.6409299775987186986730499e-1 * c(m) -0.8755807343482262259774782e0 * c(m-5) - 0.3645285178085761821545207e1 * c(m-4) + 0.1033717994630886401730470e0 * c(m-1) + 0.2013769413884797246646959e-1 * c(m) + 0.4106783858513785463625543e-1 * c(m-6) 0.3956598149904136332753521e0 * c(m-5) + 0.1630560443616104907615866e1 * c(m-4) - 0.2421320004064592721552708e0 * c(m-2) - 0.2893557395653431666593814e0 * c(m) - 0.5688529641249387985434413e-1 * c(m-6) -0.7684117160199014594442072e-1 * c(m-5) - 0.2928439026361256842196229e0 * c(m-4) + 0.2710530961648671297733465e-1 * c(m-2) - 0.1662748711097054895317080e0 * c(m-1) + 0.2251991532891353212689574e0 * c(m) + 0.1898341454096471754822498e-1 * c(m-6); 0.9209089963443799485648361e-2 * c(m-5) - 0.3129629392354775191148163e-3 * c(m-6) - 0.8896127024208321966533544e-2 * c(m-4) -0.1059816030196818445908057e0 * c(m-5) + 0.1014880675788250237247178e0 * c(m-4) + 0.4493535440856820866087846e-2 * c(m-6) 0.3328335104489738933610597e0 * c(m-5) - 0.3179803804558436283847901e0 * c(m-4) - 0.5111353189352474549563559e-2 * c(m-3) - 0.9741776803777790426705996e-2 * c(m-6) 0.7272438906214475928744770e-1 * c(m-4) + 0.1964682777744275219350831e-1 * c(m-3) - 0.5952475275883259619711594e-2 * c(m-1) - 0.1635430866921887819487473e-2 * c(m) + 0.2921234010758621482958052e-1 * c(m-6) - 0.4659516693228870973898560e0 * c(m-5) 0.5402985338373433052255418e0 * c(m-5) - 0.1979290298620869974478871e0 * c(m-3) - 0.2760353365637712827793337e-1 * c(m-1) - 0.4671751091575462868310238e-2 * c(m) - 0.6952587985456154591014641e-1 * c(m-6) + 0.1571507277911208446562686e1 * c(m-4) -0.8755807343482262259774782e0 * c(m-5) - 0.3645285178085761821545207e1 * c(m-4) + 0.1033717994630886401730470e0 * c(m-1) + 0.2013769413884797246646959e-1 * c(m) + 0.4106783858513785463625543e-1 * c(m-6) 0.1070920689960817104203947e1 * c(m-5) + 0.3717418466925056542408153e1 * c(m-4) + 0.1166740554279680007487795e0 * c(m-3) + 0.1147318200715868527529827e0 * c(m-1) + 0.6327161147136873807796515e-2 * c(m) + 0.6235373239336055200426697e-1 * c(m-6) -0.6639605248735044787146222e0 * c(m-5) - 0.1865625445986772763641423e1 * c(m-4) + 0.1103611313171476425250639e0 * c(m-3) - 0.9091410269992464604926176e-1 * c(m) - 0.8636954541126674177407762e-1 * c(m-6) 0.1582127073537215443965653e0 * c(m-5) + 0.3746489300753517635549495e0 * c(m-4) - 0.4364163147111892346990101e-1 * c(m-3) - 0.1845476106024151050283847e0 * c(m-1) + 0.7075642937243715046279337e-1 * c(m) + 0.2882271848190011329385407e-1 * c(m-6); -0.3576545132696983143406173e-2 * c(m-5) + 0.4335019854436220306755673e-3 * c(m-6) + 0.3143043147253361112730605e-2 * c(m-4) 0.4093499466767054661591066e-1 * c(m-5) - 0.3471075437892810033585296e-1 * c(m-4) - 0.6224240288742446280057699e-2 * c(m-6) -0.1080774142196007991746827e0 * c(m-5) + 0.9941834083648937298100811e-1 * c(m-4) - 0.4834791406446907590553793e-2 * c(m-3) + 0.1349386478955833378422842e-1 * c(m-6) 0.5891947149681041048896399e-1 * c(m-4) + 0.1858378996391679448655070e-1 * c(m-3) + 0.7240905383565181316381731e-2 * c(m-2) + 0.2349927974590068869356781e-1 * c(m) - 0.4046360079256766884300687e-1 * c(m-6) + 0.1223513270418807666970488e0 * c(m-5) -0.1319358558853174530078498e0 * c(m-5) - 0.1872196143003808021730728e0 * c(m-3) + 0.9147192682075630179962131e-2 * c(m-2) + 0.6712774475803763988977355e-1 * c(m) + 0.9630407686703666967100804e-1 * c(m-6) - 0.6882132817757726722141421e0 * c(m-4) 0.3956598149904136332753521e0 * c(m-5) + 0.1630560443616104907615866e1 * c(m-4) - 0.2421320004064592721552708e0 * c(m-2) - 0.2893557395653431666593814e0 * c(m) - 0.5688529641249387985434413e-1 * c(m-6) -0.6639605248735044787146222e0 * c(m-5) - 0.1865625445986772763641423e1 * c(m-4) + 0.1103611313171476425250639e0 * c(m-3) - 0.9091410269992464604926176e-1 * c(m) - 0.8636954541126674177407762e-1 * c(m-6) 0.4681819359722749441073885e0 * c(m-5) + 0.1015239189167790053447110e1 * c(m-4) + 0.1043897828092562609502636e0 * c(m-3) + 0.2542001760457345743492403e0 * c(m-2) + 0.1306332157111667628555907e1 * c(m) + 0.1196351539550049336518187e0 * c(m-6) -0.1195777325611201766551392e0 * c(m-5) - 0.2187310061229745694542609e0 * c(m-4) - 0.4128029838349298819825156e-1 * c(m-3) - 0.2845627370491611369031341e-1 * c(m-2) - 0.1016689339350338144430605e1 * c(m) - 0.3992391469197282238624438e-1 * c(m-6); 0.5593983696629863059347067e-3 * c(m-5) - 0.1446656414398166805849327e-3 * c(m-6) - 0.4147327282231696253497740e-3 * c(m-4) -0.6486184157331537899459796e-2 * c(m-5) + 0.4409068609809831485979484e-2 * c(m-4) + 0.2077115547521706413480312e-2 * c(m-6) 0.1319674981073749167009902e-1 * c(m-5) - 0.1060554802883657391328704e-1 * c(m-4) + 0.1911888563316170927411831e-2 * c(m-3) - 0.4503090345217088684223814e-2 * c(m-6) -0.2404661162020836566908542e-1 * c(m-4) - 0.7348845587775519698437916e-2 * c(m-3) - 0.8105784530576404277872603e-3 * c(m-2) + 0.9574633163221758060736592e-2 * c(m-1) - 0.1828896813877197352675410e-1 * c(m) + 0.1350326632905990039353503e-1 * c(m-6) - 0.1315967038382618382356495e-1 * c(m-5) 0.1241625568998496895352046e-1 * c(m-5) + 0.7403484645316174090533193e-1 * c(m-3) - 0.1023976547309387874453988e-2 * c(m-2) + 0.4440063948509876221050939e-1 * c(m-1) - 0.5224403464202056316702078e-1 * c(m) - 0.3213800979246298453953842e-1 * c(m-6) + 0.1178181682424363524005403e0 * c(m-4) -0.7684117160199014594442072e-1 * c(m-5) - 0.2928439026361256842196229e0 * c(m-4) + 0.2710530961648671297733465e-1 * c(m-2) - 0.1662748711097054895317080e0 * c(m-1) + 0.2251991532891353212689574e0 * c(m) + 0.1898341454096471754822498e-1 * c(m-6) 0.1582127073537215443965653e0 * c(m-5) + 0.3746489300753517635549495e0 * c(m-4) - 0.4364163147111892346990101e-1 * c(m-3) - 0.1845476106024151050283847e0 * c(m-1) + 0.7075642937243715046279337e-1 * c(m) + 0.2882271848190011329385407e-1 * c(m-6) -0.1195777325611201766551392e0 * c(m-5) - 0.2187310061229745694542609e0 * c(m-4) - 0.4128029838349298819825156e-1 * c(m-3) - 0.2845627370491611369031341e-1 * c(m-2) - 0.1016689339350338144430605e1 * c(m) - 0.3992391469197282238624438e-1 * c(m-6) 0.3167964748016105299646518e-1 * c(m-5) + 0.4976563420877041544013670e-1 * c(m-4) + 0.1632404042590951953384672e-1 * c(m-3) + 0.3185519088796429015220016e-2 * c(m-2) + 0.2968472090638000742888467e0 * c(m-1) + 0.7912667594695582093926295e0 * c(m) + 0.1332316557164627464149716e-1 * c(m-6);]; + + M(5,10)=M(10,5); + M(m-4,m-9)=M(m-9,m-4); + + M=M/h; + + D2=HI*(-M-diag(c)*e_1*S_1'+diag(c)*e_m*S_m'); + end + D2 = @D2_fun; + + S2_U=[0.35e2 / 0.12e2 -0.26e2 / 0.3e1 0.19e2 / 0.2e1 -0.14e2 / 0.3e1 0.11e2 / 0.12e2;]/h^2; + S2_1=zeros(1,m); + S2_1(1:5)=S2_U; + S2_m=zeros(1,m); + S2_m(m-4:m)=fliplr(S2_U); + S2_1 = S2_1'; + S2_m = S2_m'; + + + + + + % Fourth derivative, 1th order accurate at first 8 boundary points (still + % yield 5th order convergence if stable: for example u_tt=-u_xxxx + + m4=7/240;m3=-2/5;m2=169/60;m1=-122/15;m0=91/8; + M4=m4*(diag(ones(m-4,1),4)+diag(ones(m-4,1),-4))+m3*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3))+m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0); + + %M4=(-1/6*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3) ) + 2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2)) -13/2*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)) + 28/3*diag(ones(m,1),0)); + + M4_U=[0.1394226315049e13 / 0.367201486080e12 -0.1137054563243e13 / 0.114750464400e12 0.16614189027367e14 / 0.1836007430400e13 -0.1104821700277e13 / 0.306001238400e12 0.1355771086763e13 / 0.1836007430400e13 -0.27818686453e11 / 0.459001857600e12 -0.40671054239e11 / 0.1836007430400e13 0.5442887371e10 / 0.306001238400e12; -0.1137054563243e13 / 0.114750464400e12 0.70616795535409e14 / 0.2570410402560e13 -0.173266854731041e15 / 0.6426026006400e13 0.28938615291031e14 / 0.2570410402560e13 -0.146167361863e12 / 0.71400288960e11 0.2793470836571e13 / 0.12852052012800e14 0.6219558097e10 / 0.428401733760e12 -0.7313844559e10 / 0.166909766400e12; 0.16614189027367e14 / 0.1836007430400e13 -0.173266854731041e15 / 0.6426026006400e13 0.378613061504779e15 / 0.12852052012800e14 -0.9117069604217e13 / 0.642602600640e12 0.632177582849e12 / 0.233673672960e12 -0.1057776382577e13 / 0.6426026006400e13 0.443019868399e12 / 0.4284017337600e13 -0.3707981e7 / 0.2318191200e10; -0.1104821700277e13 / 0.306001238400e12 0.28938615291031e14 / 0.2570410402560e13 -0.9117069604217e13 / 0.642602600640e12 0.5029150721885e13 / 0.514082080512e12 -0.5209119714341e13 / 0.1285205201280e13 0.12235427457469e14 / 0.12852052012800e14 -0.13731270505e11 / 0.64260260064e11 0.2933596129e10 / 0.40800165120e11; 0.1355771086763e13 / 0.1836007430400e13 -0.146167361863e12 / 0.71400288960e11 0.632177582849e12 / 0.233673672960e12 -0.5209119714341e13 / 0.1285205201280e13 0.14871726798559e14 / 0.2570410402560e13 -0.7504337615347e13 / 0.1606506501600e13 0.310830296467e12 / 0.171360693504e12 -0.55284274391e11 / 0.183600743040e12; -0.27818686453e11 / 0.459001857600e12 0.2793470836571e13 / 0.12852052012800e14 -0.1057776382577e13 / 0.6426026006400e13 0.12235427457469e14 / 0.12852052012800e14 -0.7504337615347e13 / 0.1606506501600e13 0.106318657014853e15 / 0.12852052012800e14 -0.14432772918527e14 / 0.2142008668800e13 0.58102695589e11 / 0.22666758400e11; -0.40671054239e11 / 0.1836007430400e13 0.6219558097e10 / 0.428401733760e12 0.443019868399e12 / 0.4284017337600e13 -0.13731270505e11 / 0.64260260064e11 0.310830296467e12 / 0.171360693504e12 -0.14432772918527e14 / 0.2142008668800e13 0.27102479467823e14 / 0.2570410402560e13 -0.1216032192203e13 / 0.153000619200e12; 0.5442887371e10 / 0.306001238400e12 -0.7313844559e10 / 0.166909766400e12 -0.3707981e7 / 0.2318191200e10 0.2933596129e10 / 0.40800165120e11 -0.55284274391e11 / 0.183600743040e12 0.58102695589e11 / 0.22666758400e11 -0.1216032192203e13 / 0.153000619200e12 0.20799922829107e14 / 0.1836007430400e13;]; + + M4(1:8,1:8)=M4_U; + + M4(m-7:m,m-7:m)=flipud( fliplr( M4_U ) ); + M4=M4/h^3; + + S3_U=[-0.5e1 / 0.2e1 9 -12 7 -0.3e1 / 0.2e1;]/h^3; + S3_1=zeros(1,m); + S3_1(1:5)=S3_U; + S3_m=zeros(1,m); + S3_m(m-4:m)=fliplr(-S3_U); + S3_1 = S3_1'; + S3_m = S3_m'; + + D4=HI*(M4-e_1*S3_1'+e_m*S3_m' + S_1*S2_1'-S_m*S2_m'); + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/higher_variable6_2.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,108 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% 6:te ordn. SBP Finita differens %%% +%%% operatorer med diagonal norm %%% +%%% Extension to variable koeff %%% +%%% %%% +%%% H (Normen) %%% +%%% D1=H^(-1)Q (approx f?rsta derivatan) %%% +%%% D2 (approx andra derivatan) %%% +%%% D2=HI*(R+C*D*S %%% +%%% %%% +%%% R=-D1'*H*C*D1-RR %%% +%%% %%% +%%% RR ?r dissipation) %%% +%%% Dissipationen uppbyggd av D4: %%% +%%% DI=D4*B*H*D4 %%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% H?r med 6 RP ist?llet f?r 8 f?r D4 operatorn, dock samma randderivator +% Denna ?r noggrannare, och har 2a ordningens randdslutning och b?r ge 6te +% ordningens konvergens. Hade dock ingen fri parameter att optimera + +%m=10; %problemstorlek +%h=1/(m-1); + +% Variable koefficicients are stored in vector: c, size m, +% with the unknown stored as c(1), c(2), ..., c_m +% x=1:h:m*h;x=x'; +% c=x.^0; + + +H=diag(ones(m,1),0); +H(1:6,1:6)=[0.181e3 / 0.576e3 0 0 0 0 0; 0 0.1343e4 / 0.960e3 0 0 0 0; 0 0 0.293e3 / 0.480e3 0 0 0; 0 0 0 0.1811e4 / 0.1440e4 0 0; 0 0 0 0 0.289e3 / 0.320e3 0; 0 0 0 0 0 0.65e2 / 0.64e2;]; +H(m-5:m,m-5:m)=fliplr(flipud(H(1:6,1:6))); + + +e_1=zeros(m,1);e_1(1)=1; +e_m=zeros(m,1);e_m(m)=1; + +S_U=[-0.137e3 / 0.60e2 5 -5 0.10e2 / 0.3e1 -0.5e1 / 0.4e1 0.1e1 / 0.5e1;]/h; +S_1=zeros(1,m); +S_1(1:6)=S_U; +S_m=zeros(1,m); +S_m(m-5:m)=fliplr(-S_U); + + + +%DS=zeros(m,m); +%DS(1,1:5)=-[-25/12, 4, -3, 4/3, -1/4]; +%DS(m,m-4:m)=fliplr(-[-25/12, 4, -3, 4/3, -1/4]); +%DS=diag(c)*DS/h; + + +H=h*H; +HI=inv(H); + + +% M=zeros(m,m); +% +% for i=6:m-5 +% M(i,i-3:i+3)=[c(i-2) / 0.40e2 + c(i-1) / 0.40e2 - 0.11e2 / 0.360e3 * c(i-3) - 0.11e2 / 0.360e3 * c(i) c(i-3) / 0.20e2 - 0.3e1 / 0.10e2 * c(i-1) + c(i+1) / 0.20e2 + 0.7e1 / 0.40e2 * c(i) + 0.7e1 / 0.40e2 * c(i-2) -c(i-3) / 0.40e2 - 0.3e1 / 0.10e2 * c(i-2) - 0.3e1 / 0.10e2 * c(i+1) - c(i+2) / 0.40e2 - 0.17e2 / 0.40e2 * c(i) - 0.17e2 / 0.40e2 * c(i-1) c(i-3) / 0.180e3 + c(i-2) / 0.8e1 + 0.19e2 / 0.20e2 * c(i-1) + 0.19e2 / 0.20e2 * c(i+1) + c(i+2) / 0.8e1 + c(i+3) / 0.180e3 + 0.101e3 / 0.180e3 * c(i) -c(i-2) / 0.40e2 - 0.3e1 / 0.10e2 * c(i-1) - 0.3e1 / 0.10e2 * c(i+2) - c(i+3) / 0.40e2 - 0.17e2 / 0.40e2 * c(i) - 0.17e2 / 0.40e2 * c(i+1) c(i-1) / 0.20e2 - 0.3e1 / 0.10e2 * c(i+1) + c(i+3) / 0.20e2 + 0.7e1 / 0.40e2 * c(i) + 0.7e1 / 0.40e2 * c(i+2) c(i+1) / 0.40e2 + c(i+2) / 0.40e2 - 0.11e2 / 0.360e3 * c(i) - 0.11e2 / 0.360e3 * c(i+3);]; +% end +% +% +% M(1:9,1:9)=[0.7912667594695582093926295e0 * c(1) + 0.2968472090638000742888467e0 * c(2) + 0.3185519088796429015220016e-2 * c(3) + 0.1632404042590951953384672e-1 * c(4) + 0.3160302244094415087693968e-1 * c(5) + 0.3167964748016105299646518e-1 * c(6) + 0.3148577733947253920469418e-1 * c(7) -0.1016689339350338144430605e1 * c(1) - 0.2845627370491611369031341e-1 * c(3) - 0.4128029838349298819825156e-1 * c(4) - 0.1392281451620140507549866e0 * c(5) - 0.1195777325611201766551392e0 * c(6) - 0.1194267756529333410855186e0 * c(7) 0.7075642937243715046279337e-1 * c(1) - 0.1845476106024151050283847e0 * c(2) - 0.4364163147111892346990101e-1 * c(4) + 0.2432367907207732460874765e0 * c(5) + 0.1582127073537215443965653e0 * c(6) + 0.1602348578364786307613271e0 * c(7) 0.2251991532891353212689574e0 * c(1) - 0.1662748711097054895317080e0 * c(2) + 0.2710530961648671297733465e-1 * c(3) - 0.1916646185968439909125616e0 * c(5) - 0.7684117160199014594442072e-1 * c(6) - 0.8219586949831697575883635e-1 * c(7) -0.5224403464202056316702078e-1 * c(1) + 0.4440063948509876221050939e-1 * c(2) - 0.1023976547309387874453988e-2 * c(3) + 0.7403484645316174090533193e-1 * c(4) + 0.1241625568998496895352046e-1 * c(6) + 0.7188652847892601282652228e-1 * c(5) + 0.1379362997104735503447960e-1 * c(7) -0.1828896813877197352675410e-1 * c(1) + 0.9574633163221758060736592e-2 * c(2) - 0.8105784530576404277872603e-3 * c(3) - 0.7348845587775519698437916e-2 * c(4) + 0.1063601949723906997026904e-1 * c(5) - 0.1315967038382618382356495e-1 * c(6) - 0.2117936478838753524581943e-1 * c(7) 0.1911888563316170927411831e-2 * c(4) - 0.4068130355529149936100229e-1 * c(5) + 0.1319674981073749167009902e-1 * c(6) + 0.2557266518123783676349144e-1 * c(7) 0.1559652871136785763960685e-1 * c(5) - 0.6486184157331537899459796e-2 * c(6) - 0.9110344554036319740147054e-2 * c(7) 0.5593983696629863059347067e-3 * c(6) - 0.1384822535100796372263822e-2 * c(5) + 0.8254241654378100663291154e-3 * c(7); -0.1016689339350338144430605e1 * c(1) - 0.2845627370491611369031341e-1 * c(3) - 0.4128029838349298819825156e-1 * c(4) - 0.1392281451620140507549866e0 * c(5) - 0.1195777325611201766551392e0 * c(6) - 0.1194267756529333410855186e0 * c(7) 0.1306332157111667628555907e1 * c(1) + 0.2542001760457345743492403e0 * c(3) + 0.1043897828092562609502636e0 * c(4) + 0.6672328021032112950919876e0 * c(5) + 0.4681819359722749441073885e0 * c(6) + 0.4676415410195836920069412e0 * c(7) -0.9091410269992464604926176e-1 * c(1) + 0.1103611313171476425250639e0 * c(4) - 0.1290397544997518887000350e1 * c(5) - 0.6639605248735044787146222e0 * c(6) - 0.6615974464005206184151509e0 * c(7) -0.2893557395653431666593814e0 * c(1) - 0.2421320004064592721552708e0 * c(3) + 0.1187670255028031027693374e1 * c(5) + 0.3956598149904136332753521e0 * c(6) + 0.3860048921755800000681479e0 * c(7) 0.6712774475803763988977355e-1 * c(1) + 0.9147192682075630179962131e-2 * c(3) - 0.1872196143003808021730728e0 * c(4) - 0.1319358558853174530078498e0 * c(6) - 0.4871575736811911887376923e0 * c(5) - 0.1047516312275448138054418e0 * c(7) 0.2349927974590068869356781e-1 * c(1) + 0.7240905383565181316381731e-2 * c(3) + 0.1858378996391679448655070e-1 * c(4) - 0.9289616133938676174345208e-1 * c(5) + 0.1223513270418807666970488e0 * c(6) + 0.1113520320436295033894092e0 * c(7) -0.4834791406446907590553793e-2 * c(4) + 0.2310683832687820403062715e0 * c(5) - 0.1080774142196007991746827e0 * c(6) - 0.1181561776427343335410349e0 * c(7) -0.8368141434403455353724691e-1 * c(5) + 0.4093499466767054661591066e-1 * c(6) + 0.4274641967636400692133626e-1 * c(7) -0.3576545132696983143406173e-2 * c(6) + 0.7389399124121078682094445e-2 * c(5) - 0.3812853991424095538688273e-2 * c(7); 0.7075642937243715046279337e-1 * c(1) - 0.1845476106024151050283847e0 * c(2) - 0.4364163147111892346990101e-1 * c(4) + 0.2432367907207732460874765e0 * c(5) + 0.1582127073537215443965653e0 * c(6) + 0.1602348578364786307613271e0 * c(7) -0.9091410269992464604926176e-1 * c(1) + 0.1103611313171476425250639e0 * c(4) - 0.1290397544997518887000350e1 * c(5) - 0.6639605248735044787146222e0 * c(6) - 0.6615974464005206184151509e0 * c(7) 0.6327161147136873807796515e-2 * c(1) + 0.1147318200715868527529827e0 * c(2) + 0.1166740554279680007487795e0 * c(4) + 0.2766610808285444037240703e1 * c(5) + 0.1070920689960817104203947e1 * c(6) + 0.1013161391032973057171717e1 * c(7) 0.2013769413884797246646959e-1 * c(1) + 0.1033717994630886401730470e0 * c(2) - 0.2913221621151742724258117e1 * c(5) - 0.8755807343482262259774782e0 * c(6) - 0.6909957183488812426508351e0 * c(7) -0.4671751091575462868310238e-2 * c(1) - 0.2760353365637712827793337e-1 * c(2) - 0.1979290298620869974478871e0 * c(4) + 0.5402985338373433052255418e0 * c(6) + 0.1239177593031911077924537e1 * c(5) + 0.2628038050247358227280031e0 * c(7) -0.1635430866921887819487473e-2 * c(1) - 0.5952475275883259619711594e-2 * c(2) + 0.1964682777744275219350831e-1 * c(4) + 0.3236640012639046600590714e0 * c(5) - 0.4659516693228870973898560e0 * c(6) - 0.2217272720941736859420432e0 * c(7) -0.5111353189352474549563559e-2 * c(4) - 0.5355878163774754346032096e0 * c(5) + 0.3328335104489738933610597e0 * c(6) + 0.2078656591178540157917135e0 * c(7) 0.1824328174134289562208038e0 * c(5) - 0.1059816030196818445908057e0 * c(6) - 0.7645121439374711162999809e-1 * c(7) 0.9209089963443799485648361e-2 * c(6) - 0.1591502818872493167091475e-1 * c(5) + 0.6705938225281132185266388e-2 * c(7); 0.2251991532891353212689574e0 * c(1) - 0.1662748711097054895317080e0 * c(2) + 0.2710530961648671297733465e-1 * c(3) - 0.1916646185968439909125616e0 * c(5) - 0.7684117160199014594442072e-1 * c(6) - 0.8219586949831697575883635e-1 * c(7) -0.2893557395653431666593814e0 * c(1) - 0.2421320004064592721552708e0 * c(3) + 0.1187670255028031027693374e1 * c(5) + 0.3956598149904136332753521e0 * c(6) + 0.3860048921755800000681479e0 * c(7) 0.2013769413884797246646959e-1 * c(1) + 0.1033717994630886401730470e0 * c(2) - 0.2913221621151742724258117e1 * c(5) - 0.8755807343482262259774782e0 * c(6) - 0.6909957183488812426508351e0 * c(7) 0.6409299775987186986730499e-1 * c(1) + 0.9313657638804699948929701e-1 * c(2) + 0.2306367624634749229113646e0 * c(3) + 0.3689440308283716620260816e1 * c(5) + 0.1190550338687608873798462e1 * c(6) + 0.5912479546888856519443605e0 * c(7) -0.1486895819265604128572498e-1 * c(1) - 0.2487040599390160764166412e-1 * c(2) - 0.8712928907711754187084757e-2 * c(3) - 0.1263507837371824205693950e1 * c(6) - 0.3058317397843997326920898e0 * c(7) - 0.1470691926045802954795783e1 * c(5) -0.5205147429855955657625694e-2 * c(1) - 0.5363098747528542488971874e-2 * c(2) - 0.6897142765790609546343709e-2 * c(3) - 0.7857524521667450101721993e0 * c(5) + 0.2291148005423734600066709e0 * c(7) + 0.9977064356292750529201981e0 * c(6) 0.6697297488067662265210608e0 * c(5) - 0.5013247356072127938999311e0 * c(6) - 0.1795161243106645437322408e0 * c(7) -0.2022909060111751565150958e0 * c(5) + 0.1453421858063658498587377e0 * c(6) + 0.5694872020480930665635812e-1 * c(7) -0.1200429618441003833696998e-1 * c(6) - 0.4776915669385923841535432e-2 * c(7) + 0.1678121185379596217850541e-1 * c(5); -0.5224403464202056316702078e-1 * c(1) + 0.4440063948509876221050939e-1 * c(2) - 0.1023976547309387874453988e-2 * c(3) + 0.7403484645316174090533193e-1 * c(4) + 0.1241625568998496895352046e-1 * c(6) + 0.7188652847892601282652228e-1 * c(5) + 0.1379362997104735503447960e-1 * c(7) 0.6712774475803763988977355e-1 * c(1) + 0.9147192682075630179962131e-2 * c(3) - 0.1872196143003808021730728e0 * c(4) - 0.1319358558853174530078498e0 * c(6) - 0.4871575736811911887376923e0 * c(5) - 0.1047516312275448138054418e0 * c(7) -0.4671751091575462868310238e-2 * c(1) - 0.2760353365637712827793337e-1 * c(2) - 0.1979290298620869974478871e0 * c(4) + 0.5402985338373433052255418e0 * c(6) + 0.1239177593031911077924537e1 * c(5) + 0.2628038050247358227280031e0 * c(7) -0.1486895819265604128572498e-1 * c(1) - 0.2487040599390160764166412e-1 * c(2) - 0.8712928907711754187084757e-2 * c(3) - 0.1263507837371824205693950e1 * c(6) - 0.3058317397843997326920898e0 * c(7) - 0.1470691926045802954795783e1 * c(5) 0.3449455095910233625229891e-2 * c(1) + 0.6641183499427826101618457e-2 * c(2) + 0.3291545083271862858501887e-3 * c(3) + 0.3357721707576477199985656e0 * c(4) + 0.2096413329579026439044119e1 * c(6) + 0.2317323204183126854954203e0 * c(7) + 0.6107825764368264576481962e-2 * c(8) + 0.7109125850683376695640722e0 * c(5) 0.1207544072304193806052558e-2 * c(1) + 0.1432116665752147607469646e-2 * c(2) + 0.2605582646183255957264249e-3 * c(3) - 0.3332941113251635390801278e-1 * c(4) - 0.2808241697385532683612407e0 * c(7) - 0.2720908083525083608370563e-1 * c(8) + 0.1045865435682921987447929e0 * c(5) - 0.1348436986667115543203552e1 * c(6) 0.8671038084174692625075159e-2 * c(4) + 0.1736073411355428563685818e0 * c(6) + 0.5331362125287625412555844e-1 * c(8) - 0.2424935262404526301801157e0 * c(5) + 0.1569015257678588270609004e0 * c(7) -0.8631683980217122275970376e-1 * c(6) + 0.2698842360470999243492629e-1 * c(7) + 0.8098194147715651085292754e-1 * c(5) - 0.3276463639080639163926118e-1 * c(8) 0.7462059484530855073291365e-2 * c(6) - 0.8121640361668678949573496e-3 * c(7) + 0.5522702088127090209264064e-3 * c(8) - 0.7202165657176696199260422e-2 * c(5); -0.1828896813877197352675410e-1 * c(1) + 0.9574633163221758060736592e-2 * c(2) - 0.8105784530576404277872603e-3 * c(3) - 0.7348845587775519698437916e-2 * c(4) + 0.1063601949723906997026904e-1 * c(5) - 0.1315967038382618382356495e-1 * c(6) - 0.2117936478838753524581943e-1 * c(7) 0.2349927974590068869356781e-1 * c(1) + 0.7240905383565181316381731e-2 * c(3) + 0.1858378996391679448655070e-1 * c(4) - 0.9289616133938676174345208e-1 * c(5) + 0.1223513270418807666970488e0 * c(6) + 0.1113520320436295033894092e0 * c(7) -0.1635430866921887819487473e-2 * c(1) - 0.5952475275883259619711594e-2 * c(2) + 0.1964682777744275219350831e-1 * c(4) + 0.3236640012639046600590714e0 * c(5) - 0.4659516693228870973898560e0 * c(6) - 0.2217272720941736859420432e0 * c(7) -0.5205147429855955657625694e-2 * c(1) - 0.5363098747528542488971874e-2 * c(2) - 0.6897142765790609546343709e-2 * c(3) - 0.7857524521667450101721993e0 * c(5) + 0.2291148005423734600066709e0 * c(7) + 0.9977064356292750529201981e0 * c(6) 0.1207544072304193806052558e-2 * c(1) + 0.1432116665752147607469646e-2 * c(2) + 0.2605582646183255957264249e-3 * c(3) - 0.3332941113251635390801278e-1 * c(4) - 0.2808241697385532683612407e0 * c(7) - 0.2720908083525083608370563e-1 * c(8) + 0.1045865435682921987447929e0 * c(5) - 0.1348436986667115543203552e1 * c(6) 0.4227226173449345042468960e-3 * c(1) + 0.3088241944378964404772302e-3 * c(2) + 0.2062575706647430620228133e-3 * c(3) + 0.3308343404200968256656458e-2 * c(4) + 0.5828047016405001815804837e0 * c(5) + 0.8054174220366215473556835e0 * c(7) + 0.1338363233410033443348225e0 * c(8) + 0.5555555555555555555555556e-2 * c(9) + 0.1190362071861893051132274e1 * c(6) -0.8607044252686413302647675e-3 * c(4) - 0.1748074708673904989293256e0 * c(5) - 0.3132544850115050165022338e0 * c(8) - 0.2500000000000000000000000e-1 * c(9) - 0.3169166305310429271303167e0 * c(7) - 0.6691607091647929161078591e0 * c(6) 0.3354661791693352108660900e-1 * c(5) - 0.3343620022386971405018586e0 * c(7) + 0.5000000000000000000000000e-1 * c(9) + 0.2169790609807602750804271e0 * c(6) + 0.1838363233410033443348225e0 * c(8) 0.2912518476823004642951502e-1 * c(7) + 0.2279091916474916391629437e-1 * c(8) - 0.3068985997518740530511593e-1 * c(6) - 0.1781799513347360596249022e-2 * c(5) - 0.3055555555555555555555556e-1 * c(9); 0.1911888563316170927411831e-2 * c(4) - 0.4068130355529149936100229e-1 * c(5) + 0.1319674981073749167009902e-1 * c(6) + 0.2557266518123783676349144e-1 * c(7) -0.4834791406446907590553793e-2 * c(4) + 0.2310683832687820403062715e0 * c(5) - 0.1080774142196007991746827e0 * c(6) - 0.1181561776427343335410349e0 * c(7) -0.5111353189352474549563559e-2 * c(4) - 0.5355878163774754346032096e0 * c(5) + 0.3328335104489738933610597e0 * c(6) + 0.2078656591178540157917135e0 * c(7) 0.6697297488067662265210608e0 * c(5) - 0.5013247356072127938999311e0 * c(6) - 0.1795161243106645437322408e0 * c(7) 0.8671038084174692625075159e-2 * c(4) + 0.1736073411355428563685818e0 * c(6) + 0.5331362125287625412555844e-1 * c(8) - 0.2424935262404526301801157e0 * c(5) + 0.1569015257678588270609004e0 * c(7) -0.8607044252686413302647675e-3 * c(4) - 0.1748074708673904989293256e0 * c(5) - 0.3132544850115050165022338e0 * c(8) - 0.2500000000000000000000000e-1 * c(9) - 0.3169166305310429271303167e0 * c(7) - 0.6691607091647929161078591e0 * c(6) 0.2239223735771599178951297e-3 * c(4) + 0.1275437785430956673825710e0 * c(5) + 0.1011699483929608164601067e1 * c(6) + 0.9698817275172575247533506e0 * c(8) + 0.1250000000000000000000000e0 * c(9) + 0.5555555555555555555555556e-2 * c(10) + 0.4823177543031281500117826e0 * c(7) -0.3784113973033012949863031e-1 * c(5) - 0.2997556885134827361576001e0 * c(6) - 0.3000000000000000000000000e0 * c(9) - 0.2500000000000000000000000e-1 * c(10) - 0.3991486867446821178415359e0 * c(7) - 0.4382544850115050165022338e0 * c(8) 0.4698146218022683933926520e-1 * c(6) - 0.2966863787471237458744416e0 * c(8) + 0.5000000000000000000000000e-1 * c(10) + 0.1716355704146006481727960e0 * c(7) + 0.3069346152296258362380356e-2 * c(5) + 0.1750000000000000000000000e0 * c(9); 0.1559652871136785763960685e-1 * c(5) - 0.6486184157331537899459796e-2 * c(6) - 0.9110344554036319740147054e-2 * c(7) -0.8368141434403455353724691e-1 * c(5) + 0.4093499466767054661591066e-1 * c(6) + 0.4274641967636400692133626e-1 * c(7) 0.1824328174134289562208038e0 * c(5) - 0.1059816030196818445908057e0 * c(6) - 0.7645121439374711162999809e-1 * c(7) -0.2022909060111751565150958e0 * c(5) + 0.1453421858063658498587377e0 * c(6) + 0.5694872020480930665635812e-1 * c(7) -0.8631683980217122275970376e-1 * c(6) + 0.2698842360470999243492629e-1 * c(7) + 0.8098194147715651085292754e-1 * c(5) - 0.3276463639080639163926118e-1 * c(8) 0.3354661791693352108660900e-1 * c(5) - 0.3343620022386971405018586e0 * c(7) + 0.5000000000000000000000000e-1 * c(9) + 0.2169790609807602750804271e0 * c(6) + 0.1838363233410033443348225e0 * c(8) -0.3784113973033012949863031e-1 * c(5) - 0.2997556885134827361576001e0 * c(6) - 0.3000000000000000000000000e0 * c(9) - 0.2500000000000000000000000e-1 * c(10) - 0.3991486867446821178415359e0 * c(7) - 0.4382544850115050165022338e0 * c(8) 0.1230328942716804455358698e-1 * c(5) + 0.1183647529645898332481833e0 * c(6) + 0.9410511898227943334189628e0 * c(7) + 0.9500000000000000000000000e0 * c(9) + 0.1250000000000000000000000e0 * c(10) + 0.5555555555555555555555556e-2 * c(11) + 0.5699474344521144554459336e0 * c(8) -0.2308067892671916339568942e-1 * c(6) - 0.2986625053775149497180439e0 * c(7) - 0.3000000000000000000000000e0 * c(10) - 0.2500000000000000000000000e-1 * c(11) - 0.1047734860515050802561078e-2 * c(5) - 0.4272090808352508360837056e0 * c(8) - 0.4250000000000000000000000e0 * c(9); 0.5593983696629863059347067e-3 * c(6) - 0.1384822535100796372263822e-2 * c(5) + 0.8254241654378100663291154e-3 * c(7) -0.3576545132696983143406173e-2 * c(6) + 0.7389399124121078682094445e-2 * c(5) - 0.3812853991424095538688273e-2 * c(7) 0.9209089963443799485648361e-2 * c(6) - 0.1591502818872493167091475e-1 * c(5) + 0.6705938225281132185266388e-2 * c(7) -0.1200429618441003833696998e-1 * c(6) - 0.4776915669385923841535432e-2 * c(7) + 0.1678121185379596217850541e-1 * c(5) 0.7462059484530855073291365e-2 * c(6) - 0.8121640361668678949573496e-3 * c(7) + 0.5522702088127090209264064e-3 * c(8) - 0.7202165657176696199260422e-2 * c(5) 0.2912518476823004642951502e-1 * c(7) + 0.2279091916474916391629437e-1 * c(8) - 0.3068985997518740530511593e-1 * c(6) - 0.1781799513347360596249022e-2 * c(5) - 0.3055555555555555555555556e-1 * c(9) 0.4698146218022683933926520e-1 * c(6) - 0.2966863787471237458744416e0 * c(8) + 0.5000000000000000000000000e-1 * c(10) + 0.1716355704146006481727960e0 * c(7) + 0.3069346152296258362380356e-2 * c(5) + 0.1750000000000000000000000e0 * c(9) -0.2308067892671916339568942e-1 * c(6) - 0.2986625053775149497180439e0 * c(7) - 0.3000000000000000000000000e0 * c(10) - 0.2500000000000000000000000e-1 * c(11) - 0.1047734860515050802561078e-2 * c(5) - 0.4272090808352508360837056e0 * c(8) - 0.4250000000000000000000000e0 * c(9) 0.5139370221149109977041877e-2 * c(6) + 0.1247723215009422001393184e0 * c(7) + 0.9505522702088127090209264e0 * c(8) + 0.9500000000000000000000000e0 * c(10) + 0.1250000000000000000000000e0 * c(11) + 0.5555555555555555555555556e-2 * c(12) + 0.9159362465153641826887659e-4 * c(5) + 0.5611111111111111111111111e0 * c(9);]; +% +% M(m-8:m,m-8:m)=[0.5555555555555555555555556e-2 * c(m-11) + 0.1250000000000000000000000e0 * c(m-10) + 0.9500000000000000000000000e0 * c(m-9) + 0.9505522702088127090209264e0 * c(m-7) + 0.1247205076844361998744053e0 * c(m-6) + 0.5139370221149109977041877e-2 * c(m-5) + 0.5611111111111111111111111e0 * c(m-8) + 0.1434074411575366831819799e-3 * c(m-4) -0.2500000000000000000000000e-1 * c(m-10) - 0.3000000000000000000000000e0 * c(m-9) - 0.2980649679116425253322056e0 * c(m-6) - 0.2308067892671916339568942e-1 * c(m-5) - 0.4250000000000000000000000e0 * c(m-8) - 0.4272090808352508360837056e0 * c(m-7) - 0.1645272326387475188399322e-2 * c(m-4) 0.5000000000000000000000000e-1 * c(m-9) - 0.2966863787471237458744416e0 * c(m-7) + 0.4698146218022683933926520e-1 * c(m-5) + 0.1750000000000000000000000e0 * c(m-8) + 0.1700291833903489463825077e0 * c(m-6) + 0.4675733176547960152668626e-2 * c(m-4) 0.2279091916474916391629437e-1 * c(m-7) + 0.3097763128598982561225538e-1 * c(m-6) - 0.3055555555555555555555556e-1 * c(m-8) - 0.3068985997518740530511593e-1 * c(m-5) - 0.3634246031107139778989373e-2 * c(m-4) 0.5522702088127090209264064e-3 * c(m-7) - 0.3265435411305071914756373e-2 * c(m-6) + 0.7462059484530855073291365e-2 * c(m-5) - 0.4748894282038492179461399e-2 * c(m-4) 0.6272075574042975468177820e-3 * c(m-6) - 0.1200429618441003833696998e-1 * c(m-5) + 0.1137708862700574079015220e-1 * c(m-4) 0.9209089963443799485648361e-2 * c(m-5) - 0.3129629392354775191148163e-3 * c(m-6) - 0.8896127024208321966533544e-2 * c(m-4) -0.3576545132696983143406173e-2 * c(m-5) + 0.4335019854436220306755673e-3 * c(m-6) + 0.3143043147253361112730605e-2 * c(m-4) 0.5593983696629863059347067e-3 * c(m-5) - 0.1446656414398166805849327e-3 * c(m-6) - 0.4147327282231696253497740e-3 * c(m-4); -0.2500000000000000000000000e-1 * c(m-10) - 0.3000000000000000000000000e0 * c(m-9) - 0.2980649679116425253322056e0 * c(m-6) - 0.2308067892671916339568942e-1 * c(m-5) - 0.4250000000000000000000000e0 * c(m-8) - 0.4272090808352508360837056e0 * c(m-7) - 0.1645272326387475188399322e-2 * c(m-4) 0.5555555555555555555555556e-2 * c(m-10) + 0.1250000000000000000000000e0 * c(m-9) + 0.9500000000000000000000000e0 * c(m-8) + 0.9341601509609901526962449e0 * c(m-6) + 0.1183647529645898332481833e0 * c(m-5) + 0.1919432828897222527630486e-1 * c(m-4) + 0.5699474344521144554459336e0 * c(m-7) -0.2500000000000000000000000e-1 * c(m-9) - 0.3000000000000000000000000e0 * c(m-8) - 0.2997556885134827361576001e0 * c(m-5) - 0.5636663150858098975790317e-1 * c(m-4) - 0.4382544850115050165022338e0 * c(m-7) - 0.3806231949664312575822630e0 * c(m-6) 0.5000000000000000000000000e-1 * c(m-8) - 0.3557251496099816106154206e0 * c(m-6) + 0.5490976528821799120017102e-1 * c(m-4) + 0.1838363233410033443348225e0 * c(m-7) + 0.2169790609807602750804271e0 * c(m-5) 0.5528052133944605740009217e-1 * c(m-6) - 0.8631683980217122275970376e-1 * c(m-5) - 0.3276463639080639163926118e-1 * c(m-7) + 0.5268984374242044588776166e-1 * c(m-4) -0.5373770512016897565958305e-2 * c(m-6) + 0.1453421858063658498587377e0 * c(m-5) - 0.1399684152943489522927794e0 * c(m-4) -0.1059816030196818445908057e0 * c(m-5) + 0.1014880675788250237247178e0 * c(m-4) + 0.4493535440856820866087846e-2 * c(m-6) 0.4093499466767054661591066e-1 * c(m-5) - 0.3471075437892810033585296e-1 * c(m-4) - 0.6224240288742446280057699e-2 * c(m-6) -0.6486184157331537899459796e-2 * c(m-5) + 0.4409068609809831485979484e-2 * c(m-4) + 0.2077115547521706413480312e-2 * c(m-6); 0.5000000000000000000000000e-1 * c(m-9) - 0.2966863787471237458744416e0 * c(m-7) + 0.4698146218022683933926520e-1 * c(m-5) + 0.1750000000000000000000000e0 * c(m-8) + 0.1700291833903489463825077e0 * c(m-6) + 0.4675733176547960152668626e-2 * c(m-4) -0.2500000000000000000000000e-1 * c(m-9) - 0.3000000000000000000000000e0 * c(m-8) - 0.2997556885134827361576001e0 * c(m-5) - 0.5636663150858098975790317e-1 * c(m-4) - 0.4382544850115050165022338e0 * c(m-7) - 0.3806231949664312575822630e0 * c(m-6) 0.5555555555555555555555556e-2 * c(m-9) + 0.1250000000000000000000000e0 * c(m-8) + 0.9698817275172575247533506e0 * c(m-7) + 0.1011699483929608164601067e1 * c(m-5) + 0.1773466968705924819112984e0 * c(m-4) + 0.2239223735771599178951297e-3 * c(m-3) + 0.4325148359756313354830552e0 * c(m-6) -0.2500000000000000000000000e-1 * c(m-8) - 0.3132544850115050165022338e0 * c(m-7) - 0.2322389872063761557916742e0 * c(m-4) - 0.8607044252686413302647675e-3 * c(m-3) - 0.2594851141920572702679681e0 * c(m-6) - 0.6691607091647929161078591e0 * c(m-5) 0.5331362125287625412555844e-1 * c(m-7) + 0.1736073411355428563685818e0 * c(m-5) + 0.8671038084174692625075159e-2 * c(m-3) + 0.8084259844422177692569663e-1 * c(m-6) - 0.1664345989168155800449120e0 * c(m-4) -0.5013247356072127938999311e0 * c(m-5) + 0.5021853752328231128475915e0 * c(m-4) - 0.1197175073672143005877150e-1 * c(m-6) 0.3328335104489738933610597e0 * c(m-5) - 0.3179803804558436283847901e0 * c(m-4) - 0.5111353189352474549563559e-2 * c(m-3) - 0.9741776803777790426705996e-2 * c(m-6) -0.1080774142196007991746827e0 * c(m-5) + 0.9941834083648937298100811e-1 * c(m-4) - 0.4834791406446907590553793e-2 * c(m-3) + 0.1349386478955833378422842e-1 * c(m-6) 0.1319674981073749167009902e-1 * c(m-5) - 0.1060554802883657391328704e-1 * c(m-4) + 0.1911888563316170927411831e-2 * c(m-3) - 0.4503090345217088684223814e-2 * c(m-6); 0.2279091916474916391629437e-1 * c(m-7) + 0.3097763128598982561225538e-1 * c(m-6) - 0.3055555555555555555555556e-1 * c(m-8) - 0.3068985997518740530511593e-1 * c(m-5) - 0.3634246031107139778989373e-2 * c(m-4) 0.5000000000000000000000000e-1 * c(m-8) - 0.3557251496099816106154206e0 * c(m-6) + 0.5490976528821799120017102e-1 * c(m-4) + 0.1838363233410033443348225e0 * c(m-7) + 0.2169790609807602750804271e0 * c(m-5) -0.2500000000000000000000000e-1 * c(m-8) - 0.3132544850115050165022338e0 * c(m-7) - 0.2322389872063761557916742e0 * c(m-4) - 0.8607044252686413302647675e-3 * c(m-3) - 0.2594851141920572702679681e0 * c(m-6) - 0.6691607091647929161078591e0 * c(m-5) 0.5555555555555555555555556e-2 * c(m-8) + 0.1338363233410033443348225e0 * c(m-7) + 0.7391887916719206077121040e0 * c(m-6) + 0.6490333320052011212240632e0 * c(m-4) + 0.3308343404200968256656458e-2 * c(m-3) + 0.2062575706647430620228133e-3 * c(m-2) + 0.3088241944378964404772302e-3 * c(m-1) + 0.4227226173449345042468960e-3 * c(m) + 0.1190362071861893051132274e1 * c(m-5) -0.2720908083525083608370563e-1 * c(m-7) - 0.1931148612480615118957263e0 * c(m-6) - 0.3332941113251635390801278e-1 * c(m-3) + 0.2605582646183255957264249e-3 * c(m-2) + 0.1432116665752147607469646e-2 * c(m-1) + 0.1207544072304193806052558e-2 * c(m) - 0.1348436986667115543203552e1 * c(m-5) + 0.1687723507780044227927853e-1 * c(m-4) 0.3590669644811151307464697e-1 * c(m-6) - 0.5925443480724830632401754e0 * c(m-4) - 0.6897142765790609546343709e-2 * c(m-2) - 0.5363098747528542488971874e-2 * c(m-1) - 0.5205147429855955657625694e-2 * c(m) + 0.9977064356292750529201981e0 * c(m-5) 0.7272438906214475928744770e-1 * c(m-4) + 0.1964682777744275219350831e-1 * c(m-3) - 0.5952475275883259619711594e-2 * c(m-1) - 0.1635430866921887819487473e-2 * c(m) + 0.2921234010758621482958052e-1 * c(m-6) - 0.4659516693228870973898560e0 * c(m-5) 0.5891947149681041048896399e-1 * c(m-4) + 0.1858378996391679448655070e-1 * c(m-3) + 0.7240905383565181316381731e-2 * c(m-2) + 0.2349927974590068869356781e-1 * c(m) - 0.4046360079256766884300687e-1 * c(m-6) + 0.1223513270418807666970488e0 * c(m-5) -0.2404661162020836566908542e-1 * c(m-4) - 0.7348845587775519698437916e-2 * c(m-3) - 0.8105784530576404277872603e-3 * c(m-2) + 0.9574633163221758060736592e-2 * c(m-1) - 0.1828896813877197352675410e-1 * c(m) + 0.1350326632905990039353503e-1 * c(m-6) - 0.1315967038382618382356495e-1 * c(m-5); 0.5522702088127090209264064e-3 * c(m-7) - 0.3265435411305071914756373e-2 * c(m-6) + 0.7462059484530855073291365e-2 * c(m-5) - 0.4748894282038492179461399e-2 * c(m-4) 0.5528052133944605740009217e-1 * c(m-6) - 0.8631683980217122275970376e-1 * c(m-5) - 0.3276463639080639163926118e-1 * c(m-7) + 0.5268984374242044588776166e-1 * c(m-4) 0.5331362125287625412555844e-1 * c(m-7) + 0.1736073411355428563685818e0 * c(m-5) + 0.8671038084174692625075159e-2 * c(m-3) + 0.8084259844422177692569663e-1 * c(m-6) - 0.1664345989168155800449120e0 * c(m-4) -0.2720908083525083608370563e-1 * c(m-7) - 0.1931148612480615118957263e0 * c(m-6) - 0.3332941113251635390801278e-1 * c(m-3) + 0.2605582646183255957264249e-3 * c(m-2) + 0.1432116665752147607469646e-2 * c(m-1) + 0.1207544072304193806052558e-2 * c(m) - 0.1348436986667115543203552e1 * c(m-5) + 0.1687723507780044227927853e-1 * c(m-4) 0.6107825764368264576481962e-2 * c(m-7) + 0.1155752633643216628010304e0 * c(m-6) + 0.2096413329579026439044119e1 * c(m-5) + 0.3357721707576477199985656e0 * c(m-3) + 0.3291545083271862858501887e-3 * c(m-2) + 0.6641183499427826101618457e-2 * c(m-1) + 0.3449455095910233625229891e-2 * c(m) + 0.8270696421223286922584620e0 * c(m-4) -0.4995827370863505253765970e-1 * c(m-6) - 0.1263507837371824205693950e1 * c(m-5) - 0.8712928907711754187084757e-2 * c(m-2) - 0.2487040599390160764166412e-1 * c(m-1) - 0.1486895819265604128572498e-1 * c(m) - 0.1726565392121567634950213e1 * c(m-4) 0.5402985338373433052255418e0 * c(m-5) - 0.1979290298620869974478871e0 * c(m-3) - 0.2760353365637712827793337e-1 * c(m-1) - 0.4671751091575462868310238e-2 * c(m) - 0.6952587985456154591014641e-1 * c(m-6) + 0.1571507277911208446562686e1 * c(m-4) -0.1319358558853174530078498e0 * c(m-5) - 0.1872196143003808021730728e0 * c(m-3) + 0.9147192682075630179962131e-2 * c(m-2) + 0.6712774475803763988977355e-1 * c(m) + 0.9630407686703666967100804e-1 * c(m-6) - 0.6882132817757726722141421e0 * c(m-4) 0.1241625568998496895352046e-1 * c(m-5) + 0.7403484645316174090533193e-1 * c(m-3) - 0.1023976547309387874453988e-2 * c(m-2) + 0.4440063948509876221050939e-1 * c(m-1) - 0.5224403464202056316702078e-1 * c(m) - 0.3213800979246298453953842e-1 * c(m-6) + 0.1178181682424363524005403e0 * c(m-4); 0.6272075574042975468177820e-3 * c(m-6) - 0.1200429618441003833696998e-1 * c(m-5) + 0.1137708862700574079015220e-1 * c(m-4) -0.5373770512016897565958305e-2 * c(m-6) + 0.1453421858063658498587377e0 * c(m-5) - 0.1399684152943489522927794e0 * c(m-4) -0.5013247356072127938999311e0 * c(m-5) + 0.5021853752328231128475915e0 * c(m-4) - 0.1197175073672143005877150e-1 * c(m-6) 0.3590669644811151307464697e-1 * c(m-6) - 0.5925443480724830632401754e0 * c(m-4) - 0.6897142765790609546343709e-2 * c(m-2) - 0.5363098747528542488971874e-2 * c(m-1) - 0.5205147429855955657625694e-2 * c(m) + 0.9977064356292750529201981e0 * c(m-5) -0.4995827370863505253765970e-1 * c(m-6) - 0.1263507837371824205693950e1 * c(m-5) - 0.8712928907711754187084757e-2 * c(m-2) - 0.2487040599390160764166412e-1 * c(m-1) - 0.1486895819265604128572498e-1 * c(m) - 0.1726565392121567634950213e1 * c(m-4) 0.2760393423824887721078848e-1 * c(m-6) + 0.1190550338687608873798462e1 * c(m-5) + 0.4253084328734353394994388e1 * c(m-4) + 0.2306367624634749229113646e0 * c(m-2) + 0.9313657638804699948929701e-1 * c(m-1) + 0.6409299775987186986730499e-1 * c(m) -0.8755807343482262259774782e0 * c(m-5) - 0.3645285178085761821545207e1 * c(m-4) + 0.1033717994630886401730470e0 * c(m-1) + 0.2013769413884797246646959e-1 * c(m) + 0.4106783858513785463625543e-1 * c(m-6) 0.3956598149904136332753521e0 * c(m-5) + 0.1630560443616104907615866e1 * c(m-4) - 0.2421320004064592721552708e0 * c(m-2) - 0.2893557395653431666593814e0 * c(m) - 0.5688529641249387985434413e-1 * c(m-6) -0.7684117160199014594442072e-1 * c(m-5) - 0.2928439026361256842196229e0 * c(m-4) + 0.2710530961648671297733465e-1 * c(m-2) - 0.1662748711097054895317080e0 * c(m-1) + 0.2251991532891353212689574e0 * c(m) + 0.1898341454096471754822498e-1 * c(m-6); 0.9209089963443799485648361e-2 * c(m-5) - 0.3129629392354775191148163e-3 * c(m-6) - 0.8896127024208321966533544e-2 * c(m-4) -0.1059816030196818445908057e0 * c(m-5) + 0.1014880675788250237247178e0 * c(m-4) + 0.4493535440856820866087846e-2 * c(m-6) 0.3328335104489738933610597e0 * c(m-5) - 0.3179803804558436283847901e0 * c(m-4) - 0.5111353189352474549563559e-2 * c(m-3) - 0.9741776803777790426705996e-2 * c(m-6) 0.7272438906214475928744770e-1 * c(m-4) + 0.1964682777744275219350831e-1 * c(m-3) - 0.5952475275883259619711594e-2 * c(m-1) - 0.1635430866921887819487473e-2 * c(m) + 0.2921234010758621482958052e-1 * c(m-6) - 0.4659516693228870973898560e0 * c(m-5) 0.5402985338373433052255418e0 * c(m-5) - 0.1979290298620869974478871e0 * c(m-3) - 0.2760353365637712827793337e-1 * c(m-1) - 0.4671751091575462868310238e-2 * c(m) - 0.6952587985456154591014641e-1 * c(m-6) + 0.1571507277911208446562686e1 * c(m-4) -0.8755807343482262259774782e0 * c(m-5) - 0.3645285178085761821545207e1 * c(m-4) + 0.1033717994630886401730470e0 * c(m-1) + 0.2013769413884797246646959e-1 * c(m) + 0.4106783858513785463625543e-1 * c(m-6) 0.1070920689960817104203947e1 * c(m-5) + 0.3717418466925056542408153e1 * c(m-4) + 0.1166740554279680007487795e0 * c(m-3) + 0.1147318200715868527529827e0 * c(m-1) + 0.6327161147136873807796515e-2 * c(m) + 0.6235373239336055200426697e-1 * c(m-6) -0.6639605248735044787146222e0 * c(m-5) - 0.1865625445986772763641423e1 * c(m-4) + 0.1103611313171476425250639e0 * c(m-3) - 0.9091410269992464604926176e-1 * c(m) - 0.8636954541126674177407762e-1 * c(m-6) 0.1582127073537215443965653e0 * c(m-5) + 0.3746489300753517635549495e0 * c(m-4) - 0.4364163147111892346990101e-1 * c(m-3) - 0.1845476106024151050283847e0 * c(m-1) + 0.7075642937243715046279337e-1 * c(m) + 0.2882271848190011329385407e-1 * c(m-6); -0.3576545132696983143406173e-2 * c(m-5) + 0.4335019854436220306755673e-3 * c(m-6) + 0.3143043147253361112730605e-2 * c(m-4) 0.4093499466767054661591066e-1 * c(m-5) - 0.3471075437892810033585296e-1 * c(m-4) - 0.6224240288742446280057699e-2 * c(m-6) -0.1080774142196007991746827e0 * c(m-5) + 0.9941834083648937298100811e-1 * c(m-4) - 0.4834791406446907590553793e-2 * c(m-3) + 0.1349386478955833378422842e-1 * c(m-6) 0.5891947149681041048896399e-1 * c(m-4) + 0.1858378996391679448655070e-1 * c(m-3) + 0.7240905383565181316381731e-2 * c(m-2) + 0.2349927974590068869356781e-1 * c(m) - 0.4046360079256766884300687e-1 * c(m-6) + 0.1223513270418807666970488e0 * c(m-5) -0.1319358558853174530078498e0 * c(m-5) - 0.1872196143003808021730728e0 * c(m-3) + 0.9147192682075630179962131e-2 * c(m-2) + 0.6712774475803763988977355e-1 * c(m) + 0.9630407686703666967100804e-1 * c(m-6) - 0.6882132817757726722141421e0 * c(m-4) 0.3956598149904136332753521e0 * c(m-5) + 0.1630560443616104907615866e1 * c(m-4) - 0.2421320004064592721552708e0 * c(m-2) - 0.2893557395653431666593814e0 * c(m) - 0.5688529641249387985434413e-1 * c(m-6) -0.6639605248735044787146222e0 * c(m-5) - 0.1865625445986772763641423e1 * c(m-4) + 0.1103611313171476425250639e0 * c(m-3) - 0.9091410269992464604926176e-1 * c(m) - 0.8636954541126674177407762e-1 * c(m-6) 0.4681819359722749441073885e0 * c(m-5) + 0.1015239189167790053447110e1 * c(m-4) + 0.1043897828092562609502636e0 * c(m-3) + 0.2542001760457345743492403e0 * c(m-2) + 0.1306332157111667628555907e1 * c(m) + 0.1196351539550049336518187e0 * c(m-6) -0.1195777325611201766551392e0 * c(m-5) - 0.2187310061229745694542609e0 * c(m-4) - 0.4128029838349298819825156e-1 * c(m-3) - 0.2845627370491611369031341e-1 * c(m-2) - 0.1016689339350338144430605e1 * c(m) - 0.3992391469197282238624438e-1 * c(m-6); 0.5593983696629863059347067e-3 * c(m-5) - 0.1446656414398166805849327e-3 * c(m-6) - 0.4147327282231696253497740e-3 * c(m-4) -0.6486184157331537899459796e-2 * c(m-5) + 0.4409068609809831485979484e-2 * c(m-4) + 0.2077115547521706413480312e-2 * c(m-6) 0.1319674981073749167009902e-1 * c(m-5) - 0.1060554802883657391328704e-1 * c(m-4) + 0.1911888563316170927411831e-2 * c(m-3) - 0.4503090345217088684223814e-2 * c(m-6) -0.2404661162020836566908542e-1 * c(m-4) - 0.7348845587775519698437916e-2 * c(m-3) - 0.8105784530576404277872603e-3 * c(m-2) + 0.9574633163221758060736592e-2 * c(m-1) - 0.1828896813877197352675410e-1 * c(m) + 0.1350326632905990039353503e-1 * c(m-6) - 0.1315967038382618382356495e-1 * c(m-5) 0.1241625568998496895352046e-1 * c(m-5) + 0.7403484645316174090533193e-1 * c(m-3) - 0.1023976547309387874453988e-2 * c(m-2) + 0.4440063948509876221050939e-1 * c(m-1) - 0.5224403464202056316702078e-1 * c(m) - 0.3213800979246298453953842e-1 * c(m-6) + 0.1178181682424363524005403e0 * c(m-4) -0.7684117160199014594442072e-1 * c(m-5) - 0.2928439026361256842196229e0 * c(m-4) + 0.2710530961648671297733465e-1 * c(m-2) - 0.1662748711097054895317080e0 * c(m-1) + 0.2251991532891353212689574e0 * c(m) + 0.1898341454096471754822498e-1 * c(m-6) 0.1582127073537215443965653e0 * c(m-5) + 0.3746489300753517635549495e0 * c(m-4) - 0.4364163147111892346990101e-1 * c(m-3) - 0.1845476106024151050283847e0 * c(m-1) + 0.7075642937243715046279337e-1 * c(m) + 0.2882271848190011329385407e-1 * c(m-6) -0.1195777325611201766551392e0 * c(m-5) - 0.2187310061229745694542609e0 * c(m-4) - 0.4128029838349298819825156e-1 * c(m-3) - 0.2845627370491611369031341e-1 * c(m-2) - 0.1016689339350338144430605e1 * c(m) - 0.3992391469197282238624438e-1 * c(m-6) 0.3167964748016105299646518e-1 * c(m-5) + 0.4976563420877041544013670e-1 * c(m-4) + 0.1632404042590951953384672e-1 * c(m-3) + 0.3185519088796429015220016e-2 * c(m-2) + 0.2968472090638000742888467e0 * c(m-1) + 0.7912667594695582093926295e0 * c(m) + 0.1332316557164627464149716e-1 * c(m-6);]; +% +% M(5,10)=M(10,5); +% M(m-4,m-9)=M(m-9,m-4); +% +% M=M/h; +% +% D2=HI*(-M-diag(c)*e_1*S_1+diag(c)*e_m*S_m); + +S2_U=[0.15e2 / 0.4e1 -0.77e2 / 0.6e1 0.107e3 / 0.6e1 -13 0.61e2 / 0.12e2 -0.5e1 / 0.6e1;]/h^2; +S2_1=zeros(1,m); +S2_1(1:6)=S2_U; +S2_m=zeros(1,m); +S2_m(m-5:m)=fliplr(S2_U); + + + + + +% Fourth derivative, 1th order accurate at first 8 boundary points (still +% yield 5th order convergence if stable: for example u_tt=-u_xxxx + +m4=7/240;m3=-2/5;m2=169/60;m1=-122/15;m0=91/8; +M4=m4*(diag(ones(m-4,1),4)+diag(ones(m-4,1),-4))+m3*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3))+m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0); + +%M4=(-1/6*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3) ) + 2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2)) -13/2*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)) + 28/3*diag(ones(m,1),0)); + +M4_U=[0.1009e4 / 0.192e3 -0.7657e4 / 0.480e3 0.9307e4 / 0.480e3 -0.509e3 / 0.40e2 0.4621e4 / 0.960e3 -0.25e2 / 0.32e2; -0.7657e4 / 0.480e3 0.49513e5 / 0.960e3 -0.4007e4 / 0.60e2 0.21799e5 / 0.480e3 -0.8171e4 / 0.480e3 0.2657e4 / 0.960e3; 0.9307e4 / 0.480e3 -0.4007e4 / 0.60e2 0.1399e4 / 0.15e2 -0.2721e4 / 0.40e2 0.12703e5 / 0.480e3 -0.521e3 / 0.120e3; -0.509e3 / 0.40e2 0.21799e5 / 0.480e3 -0.2721e4 / 0.40e2 0.3349e4 / 0.60e2 -0.389e3 / 0.15e2 0.559e3 / 0.96e2; 0.4621e4 / 0.960e3 -0.8171e4 / 0.480e3 0.12703e5 / 0.480e3 -0.389e3 / 0.15e2 0.17857e5 / 0.960e3 -0.1499e4 / 0.160e3; -0.25e2 / 0.32e2 0.2657e4 / 0.960e3 -0.521e3 / 0.120e3 0.559e3 / 0.96e2 -0.1499e4 / 0.160e3 0.2225e4 / 0.192e3;]; + +M4(1:6,1:6)=M4_U; + +M4(m-5:m,m-5:m)=flipud( fliplr( M4_U ) ); +M4=M4/h^3; + +S3_U=[-0.17e2 / 0.4e1 0.71e2 / 0.4e1 -0.59e2 / 0.2e1 0.49e2 / 0.2e1 -0.41e2 / 0.4e1 0.7e1 / 0.4e1;]/h^3; +S3_1=zeros(1,m); +S3_1(1:6)=S3_U; +S3_m=zeros(1,m); +S3_m(m-5:m)=fliplr(-S3_U); + +D4=HI*(M4-e_1*S3_1+e_m*S3_m + S_1'*S2_1-S_m'*S2_m); + +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/higher_variable6_3.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,109 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% 6:te ordn. SBP Finita differens %%% +%%% operatorer med diagonal norm %%% +%%% Extension to variable koeff %%% +%%% %%% +%%% H (Normen) %%% +%%% D1=H^(-1)Q (approx f?rsta derivatan) %%% +%%% D2 (approx andra derivatan) %%% +%%% D2=HI*(R+C*D*S %%% +%%% %%% +%%% R=-D1'*H*C*D1-RR %%% +%%% %%% +%%% RR ?r dissipation) %%% +%%% Dissipationen uppbyggd av D4: %%% +%%% DI=D4*B*H*D4 %%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% H?r med 7 RP ist?llet f?r 8 f?r D4 operatorn, dock samma randderivator +% Denna ?r noggrannare, och har 2a ordningens randdslutning och b?r ge 6te +% ordningens konvergens. Hade 2 fria parametrar att optimera + +%m=10; %problemstorlek +%h=1/(m-1); + +% Variable koefficicients are stored in vector: c, size m, +% with the unknown stored as c(1), c(2), ..., c_m +% x=1:h:m*h;x=x'; +% c=x.^0; + + +H=diag(ones(m,1),0); +H(1:7,1:7)=[0.414837907e9 / 0.1191965760e10 0 0 0 0 0 0; 0 0.475278367e9 / 0.397321920e9 0 0 0 0 0; 0 0 0.13872751e8 / 0.12416310e8 0 0 0 0; 0 0 0 0.346739027e9 / 0.595982880e9 0 0 0; 0 0 0 0 0.560227469e9 / 0.397321920e9 0 0; 0 0 0 0 0 0.322971631e9 / 0.397321920e9 0; 0 0 0 0 0 0 0.616122491e9 / 0.595982880e9;]; + +H(m-6:m,m-6:m)=fliplr(flipud(H(1:7,1:7))); + + +e_1=zeros(m,1);e_1(1)=1; +e_m=zeros(m,1);e_m(m)=1; + +S_U=[-0.137e3 / 0.60e2 5 -5 0.10e2 / 0.3e1 -0.5e1 / 0.4e1 0.1e1 / 0.5e1;]/h; +S_1=zeros(1,m); +S_1(1:6)=S_U; +S_m=zeros(1,m); +S_m(m-5:m)=fliplr(-S_U); + + + +%DS=zeros(m,m); +%DS(1,1:5)=-[-25/12, 4, -3, 4/3, -1/4]; +%DS(m,m-4:m)=fliplr(-[-25/12, 4, -3, 4/3, -1/4]); +%DS=diag(c)*DS/h; + + +H=h*H; +HI=inv(H); + + +% M=zeros(m,m); +% +% for i=6:m-5 +% M(i,i-3:i+3)=[c(i-2) / 0.40e2 + c(i-1) / 0.40e2 - 0.11e2 / 0.360e3 * c(i-3) - 0.11e2 / 0.360e3 * c(i) c(i-3) / 0.20e2 - 0.3e1 / 0.10e2 * c(i-1) + c(i+1) / 0.20e2 + 0.7e1 / 0.40e2 * c(i) + 0.7e1 / 0.40e2 * c(i-2) -c(i-3) / 0.40e2 - 0.3e1 / 0.10e2 * c(i-2) - 0.3e1 / 0.10e2 * c(i+1) - c(i+2) / 0.40e2 - 0.17e2 / 0.40e2 * c(i) - 0.17e2 / 0.40e2 * c(i-1) c(i-3) / 0.180e3 + c(i-2) / 0.8e1 + 0.19e2 / 0.20e2 * c(i-1) + 0.19e2 / 0.20e2 * c(i+1) + c(i+2) / 0.8e1 + c(i+3) / 0.180e3 + 0.101e3 / 0.180e3 * c(i) -c(i-2) / 0.40e2 - 0.3e1 / 0.10e2 * c(i-1) - 0.3e1 / 0.10e2 * c(i+2) - c(i+3) / 0.40e2 - 0.17e2 / 0.40e2 * c(i) - 0.17e2 / 0.40e2 * c(i+1) c(i-1) / 0.20e2 - 0.3e1 / 0.10e2 * c(i+1) + c(i+3) / 0.20e2 + 0.7e1 / 0.40e2 * c(i) + 0.7e1 / 0.40e2 * c(i+2) c(i+1) / 0.40e2 + c(i+2) / 0.40e2 - 0.11e2 / 0.360e3 * c(i) - 0.11e2 / 0.360e3 * c(i+3);]; +% end +% +% +% M(1:9,1:9)=[0.7912667594695582093926295e0 * c(1) + 0.2968472090638000742888467e0 * c(2) + 0.3185519088796429015220016e-2 * c(3) + 0.1632404042590951953384672e-1 * c(4) + 0.3160302244094415087693968e-1 * c(5) + 0.3167964748016105299646518e-1 * c(6) + 0.3148577733947253920469418e-1 * c(7) -0.1016689339350338144430605e1 * c(1) - 0.2845627370491611369031341e-1 * c(3) - 0.4128029838349298819825156e-1 * c(4) - 0.1392281451620140507549866e0 * c(5) - 0.1195777325611201766551392e0 * c(6) - 0.1194267756529333410855186e0 * c(7) 0.7075642937243715046279337e-1 * c(1) - 0.1845476106024151050283847e0 * c(2) - 0.4364163147111892346990101e-1 * c(4) + 0.2432367907207732460874765e0 * c(5) + 0.1582127073537215443965653e0 * c(6) + 0.1602348578364786307613271e0 * c(7) 0.2251991532891353212689574e0 * c(1) - 0.1662748711097054895317080e0 * c(2) + 0.2710530961648671297733465e-1 * c(3) - 0.1916646185968439909125616e0 * c(5) - 0.7684117160199014594442072e-1 * c(6) - 0.8219586949831697575883635e-1 * c(7) -0.5224403464202056316702078e-1 * c(1) + 0.4440063948509876221050939e-1 * c(2) - 0.1023976547309387874453988e-2 * c(3) + 0.7403484645316174090533193e-1 * c(4) + 0.1241625568998496895352046e-1 * c(6) + 0.7188652847892601282652228e-1 * c(5) + 0.1379362997104735503447960e-1 * c(7) -0.1828896813877197352675410e-1 * c(1) + 0.9574633163221758060736592e-2 * c(2) - 0.8105784530576404277872603e-3 * c(3) - 0.7348845587775519698437916e-2 * c(4) + 0.1063601949723906997026904e-1 * c(5) - 0.1315967038382618382356495e-1 * c(6) - 0.2117936478838753524581943e-1 * c(7) 0.1911888563316170927411831e-2 * c(4) - 0.4068130355529149936100229e-1 * c(5) + 0.1319674981073749167009902e-1 * c(6) + 0.2557266518123783676349144e-1 * c(7) 0.1559652871136785763960685e-1 * c(5) - 0.6486184157331537899459796e-2 * c(6) - 0.9110344554036319740147054e-2 * c(7) 0.5593983696629863059347067e-3 * c(6) - 0.1384822535100796372263822e-2 * c(5) + 0.8254241654378100663291154e-3 * c(7); -0.1016689339350338144430605e1 * c(1) - 0.2845627370491611369031341e-1 * c(3) - 0.4128029838349298819825156e-1 * c(4) - 0.1392281451620140507549866e0 * c(5) - 0.1195777325611201766551392e0 * c(6) - 0.1194267756529333410855186e0 * c(7) 0.1306332157111667628555907e1 * c(1) + 0.2542001760457345743492403e0 * c(3) + 0.1043897828092562609502636e0 * c(4) + 0.6672328021032112950919876e0 * c(5) + 0.4681819359722749441073885e0 * c(6) + 0.4676415410195836920069412e0 * c(7) -0.9091410269992464604926176e-1 * c(1) + 0.1103611313171476425250639e0 * c(4) - 0.1290397544997518887000350e1 * c(5) - 0.6639605248735044787146222e0 * c(6) - 0.6615974464005206184151509e0 * c(7) -0.2893557395653431666593814e0 * c(1) - 0.2421320004064592721552708e0 * c(3) + 0.1187670255028031027693374e1 * c(5) + 0.3956598149904136332753521e0 * c(6) + 0.3860048921755800000681479e0 * c(7) 0.6712774475803763988977355e-1 * c(1) + 0.9147192682075630179962131e-2 * c(3) - 0.1872196143003808021730728e0 * c(4) - 0.1319358558853174530078498e0 * c(6) - 0.4871575736811911887376923e0 * c(5) - 0.1047516312275448138054418e0 * c(7) 0.2349927974590068869356781e-1 * c(1) + 0.7240905383565181316381731e-2 * c(3) + 0.1858378996391679448655070e-1 * c(4) - 0.9289616133938676174345208e-1 * c(5) + 0.1223513270418807666970488e0 * c(6) + 0.1113520320436295033894092e0 * c(7) -0.4834791406446907590553793e-2 * c(4) + 0.2310683832687820403062715e0 * c(5) - 0.1080774142196007991746827e0 * c(6) - 0.1181561776427343335410349e0 * c(7) -0.8368141434403455353724691e-1 * c(5) + 0.4093499466767054661591066e-1 * c(6) + 0.4274641967636400692133626e-1 * c(7) -0.3576545132696983143406173e-2 * c(6) + 0.7389399124121078682094445e-2 * c(5) - 0.3812853991424095538688273e-2 * c(7); 0.7075642937243715046279337e-1 * c(1) - 0.1845476106024151050283847e0 * c(2) - 0.4364163147111892346990101e-1 * c(4) + 0.2432367907207732460874765e0 * c(5) + 0.1582127073537215443965653e0 * c(6) + 0.1602348578364786307613271e0 * c(7) -0.9091410269992464604926176e-1 * c(1) + 0.1103611313171476425250639e0 * c(4) - 0.1290397544997518887000350e1 * c(5) - 0.6639605248735044787146222e0 * c(6) - 0.6615974464005206184151509e0 * c(7) 0.6327161147136873807796515e-2 * c(1) + 0.1147318200715868527529827e0 * c(2) + 0.1166740554279680007487795e0 * c(4) + 0.2766610808285444037240703e1 * c(5) + 0.1070920689960817104203947e1 * c(6) + 0.1013161391032973057171717e1 * c(7) 0.2013769413884797246646959e-1 * c(1) + 0.1033717994630886401730470e0 * c(2) - 0.2913221621151742724258117e1 * c(5) - 0.8755807343482262259774782e0 * c(6) - 0.6909957183488812426508351e0 * c(7) -0.4671751091575462868310238e-2 * c(1) - 0.2760353365637712827793337e-1 * c(2) - 0.1979290298620869974478871e0 * c(4) + 0.5402985338373433052255418e0 * c(6) + 0.1239177593031911077924537e1 * c(5) + 0.2628038050247358227280031e0 * c(7) -0.1635430866921887819487473e-2 * c(1) - 0.5952475275883259619711594e-2 * c(2) + 0.1964682777744275219350831e-1 * c(4) + 0.3236640012639046600590714e0 * c(5) - 0.4659516693228870973898560e0 * c(6) - 0.2217272720941736859420432e0 * c(7) -0.5111353189352474549563559e-2 * c(4) - 0.5355878163774754346032096e0 * c(5) + 0.3328335104489738933610597e0 * c(6) + 0.2078656591178540157917135e0 * c(7) 0.1824328174134289562208038e0 * c(5) - 0.1059816030196818445908057e0 * c(6) - 0.7645121439374711162999809e-1 * c(7) 0.9209089963443799485648361e-2 * c(6) - 0.1591502818872493167091475e-1 * c(5) + 0.6705938225281132185266388e-2 * c(7); 0.2251991532891353212689574e0 * c(1) - 0.1662748711097054895317080e0 * c(2) + 0.2710530961648671297733465e-1 * c(3) - 0.1916646185968439909125616e0 * c(5) - 0.7684117160199014594442072e-1 * c(6) - 0.8219586949831697575883635e-1 * c(7) -0.2893557395653431666593814e0 * c(1) - 0.2421320004064592721552708e0 * c(3) + 0.1187670255028031027693374e1 * c(5) + 0.3956598149904136332753521e0 * c(6) + 0.3860048921755800000681479e0 * c(7) 0.2013769413884797246646959e-1 * c(1) + 0.1033717994630886401730470e0 * c(2) - 0.2913221621151742724258117e1 * c(5) - 0.8755807343482262259774782e0 * c(6) - 0.6909957183488812426508351e0 * c(7) 0.6409299775987186986730499e-1 * c(1) + 0.9313657638804699948929701e-1 * c(2) + 0.2306367624634749229113646e0 * c(3) + 0.3689440308283716620260816e1 * c(5) + 0.1190550338687608873798462e1 * c(6) + 0.5912479546888856519443605e0 * c(7) -0.1486895819265604128572498e-1 * c(1) - 0.2487040599390160764166412e-1 * c(2) - 0.8712928907711754187084757e-2 * c(3) - 0.1263507837371824205693950e1 * c(6) - 0.3058317397843997326920898e0 * c(7) - 0.1470691926045802954795783e1 * c(5) -0.5205147429855955657625694e-2 * c(1) - 0.5363098747528542488971874e-2 * c(2) - 0.6897142765790609546343709e-2 * c(3) - 0.7857524521667450101721993e0 * c(5) + 0.2291148005423734600066709e0 * c(7) + 0.9977064356292750529201981e0 * c(6) 0.6697297488067662265210608e0 * c(5) - 0.5013247356072127938999311e0 * c(6) - 0.1795161243106645437322408e0 * c(7) -0.2022909060111751565150958e0 * c(5) + 0.1453421858063658498587377e0 * c(6) + 0.5694872020480930665635812e-1 * c(7) -0.1200429618441003833696998e-1 * c(6) - 0.4776915669385923841535432e-2 * c(7) + 0.1678121185379596217850541e-1 * c(5); -0.5224403464202056316702078e-1 * c(1) + 0.4440063948509876221050939e-1 * c(2) - 0.1023976547309387874453988e-2 * c(3) + 0.7403484645316174090533193e-1 * c(4) + 0.1241625568998496895352046e-1 * c(6) + 0.7188652847892601282652228e-1 * c(5) + 0.1379362997104735503447960e-1 * c(7) 0.6712774475803763988977355e-1 * c(1) + 0.9147192682075630179962131e-2 * c(3) - 0.1872196143003808021730728e0 * c(4) - 0.1319358558853174530078498e0 * c(6) - 0.4871575736811911887376923e0 * c(5) - 0.1047516312275448138054418e0 * c(7) -0.4671751091575462868310238e-2 * c(1) - 0.2760353365637712827793337e-1 * c(2) - 0.1979290298620869974478871e0 * c(4) + 0.5402985338373433052255418e0 * c(6) + 0.1239177593031911077924537e1 * c(5) + 0.2628038050247358227280031e0 * c(7) -0.1486895819265604128572498e-1 * c(1) - 0.2487040599390160764166412e-1 * c(2) - 0.8712928907711754187084757e-2 * c(3) - 0.1263507837371824205693950e1 * c(6) - 0.3058317397843997326920898e0 * c(7) - 0.1470691926045802954795783e1 * c(5) 0.3449455095910233625229891e-2 * c(1) + 0.6641183499427826101618457e-2 * c(2) + 0.3291545083271862858501887e-3 * c(3) + 0.3357721707576477199985656e0 * c(4) + 0.2096413329579026439044119e1 * c(6) + 0.2317323204183126854954203e0 * c(7) + 0.6107825764368264576481962e-2 * c(8) + 0.7109125850683376695640722e0 * c(5) 0.1207544072304193806052558e-2 * c(1) + 0.1432116665752147607469646e-2 * c(2) + 0.2605582646183255957264249e-3 * c(3) - 0.3332941113251635390801278e-1 * c(4) - 0.2808241697385532683612407e0 * c(7) - 0.2720908083525083608370563e-1 * c(8) + 0.1045865435682921987447929e0 * c(5) - 0.1348436986667115543203552e1 * c(6) 0.8671038084174692625075159e-2 * c(4) + 0.1736073411355428563685818e0 * c(6) + 0.5331362125287625412555844e-1 * c(8) - 0.2424935262404526301801157e0 * c(5) + 0.1569015257678588270609004e0 * c(7) -0.8631683980217122275970376e-1 * c(6) + 0.2698842360470999243492629e-1 * c(7) + 0.8098194147715651085292754e-1 * c(5) - 0.3276463639080639163926118e-1 * c(8) 0.7462059484530855073291365e-2 * c(6) - 0.8121640361668678949573496e-3 * c(7) + 0.5522702088127090209264064e-3 * c(8) - 0.7202165657176696199260422e-2 * c(5); -0.1828896813877197352675410e-1 * c(1) + 0.9574633163221758060736592e-2 * c(2) - 0.8105784530576404277872603e-3 * c(3) - 0.7348845587775519698437916e-2 * c(4) + 0.1063601949723906997026904e-1 * c(5) - 0.1315967038382618382356495e-1 * c(6) - 0.2117936478838753524581943e-1 * c(7) 0.2349927974590068869356781e-1 * c(1) + 0.7240905383565181316381731e-2 * c(3) + 0.1858378996391679448655070e-1 * c(4) - 0.9289616133938676174345208e-1 * c(5) + 0.1223513270418807666970488e0 * c(6) + 0.1113520320436295033894092e0 * c(7) -0.1635430866921887819487473e-2 * c(1) - 0.5952475275883259619711594e-2 * c(2) + 0.1964682777744275219350831e-1 * c(4) + 0.3236640012639046600590714e0 * c(5) - 0.4659516693228870973898560e0 * c(6) - 0.2217272720941736859420432e0 * c(7) -0.5205147429855955657625694e-2 * c(1) - 0.5363098747528542488971874e-2 * c(2) - 0.6897142765790609546343709e-2 * c(3) - 0.7857524521667450101721993e0 * c(5) + 0.2291148005423734600066709e0 * c(7) + 0.9977064356292750529201981e0 * c(6) 0.1207544072304193806052558e-2 * c(1) + 0.1432116665752147607469646e-2 * c(2) + 0.2605582646183255957264249e-3 * c(3) - 0.3332941113251635390801278e-1 * c(4) - 0.2808241697385532683612407e0 * c(7) - 0.2720908083525083608370563e-1 * c(8) + 0.1045865435682921987447929e0 * c(5) - 0.1348436986667115543203552e1 * c(6) 0.4227226173449345042468960e-3 * c(1) + 0.3088241944378964404772302e-3 * c(2) + 0.2062575706647430620228133e-3 * c(3) + 0.3308343404200968256656458e-2 * c(4) + 0.5828047016405001815804837e0 * c(5) + 0.8054174220366215473556835e0 * c(7) + 0.1338363233410033443348225e0 * c(8) + 0.5555555555555555555555556e-2 * c(9) + 0.1190362071861893051132274e1 * c(6) -0.8607044252686413302647675e-3 * c(4) - 0.1748074708673904989293256e0 * c(5) - 0.3132544850115050165022338e0 * c(8) - 0.2500000000000000000000000e-1 * c(9) - 0.3169166305310429271303167e0 * c(7) - 0.6691607091647929161078591e0 * c(6) 0.3354661791693352108660900e-1 * c(5) - 0.3343620022386971405018586e0 * c(7) + 0.5000000000000000000000000e-1 * c(9) + 0.2169790609807602750804271e0 * c(6) + 0.1838363233410033443348225e0 * c(8) 0.2912518476823004642951502e-1 * c(7) + 0.2279091916474916391629437e-1 * c(8) - 0.3068985997518740530511593e-1 * c(6) - 0.1781799513347360596249022e-2 * c(5) - 0.3055555555555555555555556e-1 * c(9); 0.1911888563316170927411831e-2 * c(4) - 0.4068130355529149936100229e-1 * c(5) + 0.1319674981073749167009902e-1 * c(6) + 0.2557266518123783676349144e-1 * c(7) -0.4834791406446907590553793e-2 * c(4) + 0.2310683832687820403062715e0 * c(5) - 0.1080774142196007991746827e0 * c(6) - 0.1181561776427343335410349e0 * c(7) -0.5111353189352474549563559e-2 * c(4) - 0.5355878163774754346032096e0 * c(5) + 0.3328335104489738933610597e0 * c(6) + 0.2078656591178540157917135e0 * c(7) 0.6697297488067662265210608e0 * c(5) - 0.5013247356072127938999311e0 * c(6) - 0.1795161243106645437322408e0 * c(7) 0.8671038084174692625075159e-2 * c(4) + 0.1736073411355428563685818e0 * c(6) + 0.5331362125287625412555844e-1 * c(8) - 0.2424935262404526301801157e0 * c(5) + 0.1569015257678588270609004e0 * c(7) -0.8607044252686413302647675e-3 * c(4) - 0.1748074708673904989293256e0 * c(5) - 0.3132544850115050165022338e0 * c(8) - 0.2500000000000000000000000e-1 * c(9) - 0.3169166305310429271303167e0 * c(7) - 0.6691607091647929161078591e0 * c(6) 0.2239223735771599178951297e-3 * c(4) + 0.1275437785430956673825710e0 * c(5) + 0.1011699483929608164601067e1 * c(6) + 0.9698817275172575247533506e0 * c(8) + 0.1250000000000000000000000e0 * c(9) + 0.5555555555555555555555556e-2 * c(10) + 0.4823177543031281500117826e0 * c(7) -0.3784113973033012949863031e-1 * c(5) - 0.2997556885134827361576001e0 * c(6) - 0.3000000000000000000000000e0 * c(9) - 0.2500000000000000000000000e-1 * c(10) - 0.3991486867446821178415359e0 * c(7) - 0.4382544850115050165022338e0 * c(8) 0.4698146218022683933926520e-1 * c(6) - 0.2966863787471237458744416e0 * c(8) + 0.5000000000000000000000000e-1 * c(10) + 0.1716355704146006481727960e0 * c(7) + 0.3069346152296258362380356e-2 * c(5) + 0.1750000000000000000000000e0 * c(9); 0.1559652871136785763960685e-1 * c(5) - 0.6486184157331537899459796e-2 * c(6) - 0.9110344554036319740147054e-2 * c(7) -0.8368141434403455353724691e-1 * c(5) + 0.4093499466767054661591066e-1 * c(6) + 0.4274641967636400692133626e-1 * c(7) 0.1824328174134289562208038e0 * c(5) - 0.1059816030196818445908057e0 * c(6) - 0.7645121439374711162999809e-1 * c(7) -0.2022909060111751565150958e0 * c(5) + 0.1453421858063658498587377e0 * c(6) + 0.5694872020480930665635812e-1 * c(7) -0.8631683980217122275970376e-1 * c(6) + 0.2698842360470999243492629e-1 * c(7) + 0.8098194147715651085292754e-1 * c(5) - 0.3276463639080639163926118e-1 * c(8) 0.3354661791693352108660900e-1 * c(5) - 0.3343620022386971405018586e0 * c(7) + 0.5000000000000000000000000e-1 * c(9) + 0.2169790609807602750804271e0 * c(6) + 0.1838363233410033443348225e0 * c(8) -0.3784113973033012949863031e-1 * c(5) - 0.2997556885134827361576001e0 * c(6) - 0.3000000000000000000000000e0 * c(9) - 0.2500000000000000000000000e-1 * c(10) - 0.3991486867446821178415359e0 * c(7) - 0.4382544850115050165022338e0 * c(8) 0.1230328942716804455358698e-1 * c(5) + 0.1183647529645898332481833e0 * c(6) + 0.9410511898227943334189628e0 * c(7) + 0.9500000000000000000000000e0 * c(9) + 0.1250000000000000000000000e0 * c(10) + 0.5555555555555555555555556e-2 * c(11) + 0.5699474344521144554459336e0 * c(8) -0.2308067892671916339568942e-1 * c(6) - 0.2986625053775149497180439e0 * c(7) - 0.3000000000000000000000000e0 * c(10) - 0.2500000000000000000000000e-1 * c(11) - 0.1047734860515050802561078e-2 * c(5) - 0.4272090808352508360837056e0 * c(8) - 0.4250000000000000000000000e0 * c(9); 0.5593983696629863059347067e-3 * c(6) - 0.1384822535100796372263822e-2 * c(5) + 0.8254241654378100663291154e-3 * c(7) -0.3576545132696983143406173e-2 * c(6) + 0.7389399124121078682094445e-2 * c(5) - 0.3812853991424095538688273e-2 * c(7) 0.9209089963443799485648361e-2 * c(6) - 0.1591502818872493167091475e-1 * c(5) + 0.6705938225281132185266388e-2 * c(7) -0.1200429618441003833696998e-1 * c(6) - 0.4776915669385923841535432e-2 * c(7) + 0.1678121185379596217850541e-1 * c(5) 0.7462059484530855073291365e-2 * c(6) - 0.8121640361668678949573496e-3 * c(7) + 0.5522702088127090209264064e-3 * c(8) - 0.7202165657176696199260422e-2 * c(5) 0.2912518476823004642951502e-1 * c(7) + 0.2279091916474916391629437e-1 * c(8) - 0.3068985997518740530511593e-1 * c(6) - 0.1781799513347360596249022e-2 * c(5) - 0.3055555555555555555555556e-1 * c(9) 0.4698146218022683933926520e-1 * c(6) - 0.2966863787471237458744416e0 * c(8) + 0.5000000000000000000000000e-1 * c(10) + 0.1716355704146006481727960e0 * c(7) + 0.3069346152296258362380356e-2 * c(5) + 0.1750000000000000000000000e0 * c(9) -0.2308067892671916339568942e-1 * c(6) - 0.2986625053775149497180439e0 * c(7) - 0.3000000000000000000000000e0 * c(10) - 0.2500000000000000000000000e-1 * c(11) - 0.1047734860515050802561078e-2 * c(5) - 0.4272090808352508360837056e0 * c(8) - 0.4250000000000000000000000e0 * c(9) 0.5139370221149109977041877e-2 * c(6) + 0.1247723215009422001393184e0 * c(7) + 0.9505522702088127090209264e0 * c(8) + 0.9500000000000000000000000e0 * c(10) + 0.1250000000000000000000000e0 * c(11) + 0.5555555555555555555555556e-2 * c(12) + 0.9159362465153641826887659e-4 * c(5) + 0.5611111111111111111111111e0 * c(9);]; +% +% M(m-8:m,m-8:m)=[0.5555555555555555555555556e-2 * c(m-11) + 0.1250000000000000000000000e0 * c(m-10) + 0.9500000000000000000000000e0 * c(m-9) + 0.9505522702088127090209264e0 * c(m-7) + 0.1247205076844361998744053e0 * c(m-6) + 0.5139370221149109977041877e-2 * c(m-5) + 0.5611111111111111111111111e0 * c(m-8) + 0.1434074411575366831819799e-3 * c(m-4) -0.2500000000000000000000000e-1 * c(m-10) - 0.3000000000000000000000000e0 * c(m-9) - 0.2980649679116425253322056e0 * c(m-6) - 0.2308067892671916339568942e-1 * c(m-5) - 0.4250000000000000000000000e0 * c(m-8) - 0.4272090808352508360837056e0 * c(m-7) - 0.1645272326387475188399322e-2 * c(m-4) 0.5000000000000000000000000e-1 * c(m-9) - 0.2966863787471237458744416e0 * c(m-7) + 0.4698146218022683933926520e-1 * c(m-5) + 0.1750000000000000000000000e0 * c(m-8) + 0.1700291833903489463825077e0 * c(m-6) + 0.4675733176547960152668626e-2 * c(m-4) 0.2279091916474916391629437e-1 * c(m-7) + 0.3097763128598982561225538e-1 * c(m-6) - 0.3055555555555555555555556e-1 * c(m-8) - 0.3068985997518740530511593e-1 * c(m-5) - 0.3634246031107139778989373e-2 * c(m-4) 0.5522702088127090209264064e-3 * c(m-7) - 0.3265435411305071914756373e-2 * c(m-6) + 0.7462059484530855073291365e-2 * c(m-5) - 0.4748894282038492179461399e-2 * c(m-4) 0.6272075574042975468177820e-3 * c(m-6) - 0.1200429618441003833696998e-1 * c(m-5) + 0.1137708862700574079015220e-1 * c(m-4) 0.9209089963443799485648361e-2 * c(m-5) - 0.3129629392354775191148163e-3 * c(m-6) - 0.8896127024208321966533544e-2 * c(m-4) -0.3576545132696983143406173e-2 * c(m-5) + 0.4335019854436220306755673e-3 * c(m-6) + 0.3143043147253361112730605e-2 * c(m-4) 0.5593983696629863059347067e-3 * c(m-5) - 0.1446656414398166805849327e-3 * c(m-6) - 0.4147327282231696253497740e-3 * c(m-4); -0.2500000000000000000000000e-1 * c(m-10) - 0.3000000000000000000000000e0 * c(m-9) - 0.2980649679116425253322056e0 * c(m-6) - 0.2308067892671916339568942e-1 * c(m-5) - 0.4250000000000000000000000e0 * c(m-8) - 0.4272090808352508360837056e0 * c(m-7) - 0.1645272326387475188399322e-2 * c(m-4) 0.5555555555555555555555556e-2 * c(m-10) + 0.1250000000000000000000000e0 * c(m-9) + 0.9500000000000000000000000e0 * c(m-8) + 0.9341601509609901526962449e0 * c(m-6) + 0.1183647529645898332481833e0 * c(m-5) + 0.1919432828897222527630486e-1 * c(m-4) + 0.5699474344521144554459336e0 * c(m-7) -0.2500000000000000000000000e-1 * c(m-9) - 0.3000000000000000000000000e0 * c(m-8) - 0.2997556885134827361576001e0 * c(m-5) - 0.5636663150858098975790317e-1 * c(m-4) - 0.4382544850115050165022338e0 * c(m-7) - 0.3806231949664312575822630e0 * c(m-6) 0.5000000000000000000000000e-1 * c(m-8) - 0.3557251496099816106154206e0 * c(m-6) + 0.5490976528821799120017102e-1 * c(m-4) + 0.1838363233410033443348225e0 * c(m-7) + 0.2169790609807602750804271e0 * c(m-5) 0.5528052133944605740009217e-1 * c(m-6) - 0.8631683980217122275970376e-1 * c(m-5) - 0.3276463639080639163926118e-1 * c(m-7) + 0.5268984374242044588776166e-1 * c(m-4) -0.5373770512016897565958305e-2 * c(m-6) + 0.1453421858063658498587377e0 * c(m-5) - 0.1399684152943489522927794e0 * c(m-4) -0.1059816030196818445908057e0 * c(m-5) + 0.1014880675788250237247178e0 * c(m-4) + 0.4493535440856820866087846e-2 * c(m-6) 0.4093499466767054661591066e-1 * c(m-5) - 0.3471075437892810033585296e-1 * c(m-4) - 0.6224240288742446280057699e-2 * c(m-6) -0.6486184157331537899459796e-2 * c(m-5) + 0.4409068609809831485979484e-2 * c(m-4) + 0.2077115547521706413480312e-2 * c(m-6); 0.5000000000000000000000000e-1 * c(m-9) - 0.2966863787471237458744416e0 * c(m-7) + 0.4698146218022683933926520e-1 * c(m-5) + 0.1750000000000000000000000e0 * c(m-8) + 0.1700291833903489463825077e0 * c(m-6) + 0.4675733176547960152668626e-2 * c(m-4) -0.2500000000000000000000000e-1 * c(m-9) - 0.3000000000000000000000000e0 * c(m-8) - 0.2997556885134827361576001e0 * c(m-5) - 0.5636663150858098975790317e-1 * c(m-4) - 0.4382544850115050165022338e0 * c(m-7) - 0.3806231949664312575822630e0 * c(m-6) 0.5555555555555555555555556e-2 * c(m-9) + 0.1250000000000000000000000e0 * c(m-8) + 0.9698817275172575247533506e0 * c(m-7) + 0.1011699483929608164601067e1 * c(m-5) + 0.1773466968705924819112984e0 * c(m-4) + 0.2239223735771599178951297e-3 * c(m-3) + 0.4325148359756313354830552e0 * c(m-6) -0.2500000000000000000000000e-1 * c(m-8) - 0.3132544850115050165022338e0 * c(m-7) - 0.2322389872063761557916742e0 * c(m-4) - 0.8607044252686413302647675e-3 * c(m-3) - 0.2594851141920572702679681e0 * c(m-6) - 0.6691607091647929161078591e0 * c(m-5) 0.5331362125287625412555844e-1 * c(m-7) + 0.1736073411355428563685818e0 * c(m-5) + 0.8671038084174692625075159e-2 * c(m-3) + 0.8084259844422177692569663e-1 * c(m-6) - 0.1664345989168155800449120e0 * c(m-4) -0.5013247356072127938999311e0 * c(m-5) + 0.5021853752328231128475915e0 * c(m-4) - 0.1197175073672143005877150e-1 * c(m-6) 0.3328335104489738933610597e0 * c(m-5) - 0.3179803804558436283847901e0 * c(m-4) - 0.5111353189352474549563559e-2 * c(m-3) - 0.9741776803777790426705996e-2 * c(m-6) -0.1080774142196007991746827e0 * c(m-5) + 0.9941834083648937298100811e-1 * c(m-4) - 0.4834791406446907590553793e-2 * c(m-3) + 0.1349386478955833378422842e-1 * c(m-6) 0.1319674981073749167009902e-1 * c(m-5) - 0.1060554802883657391328704e-1 * c(m-4) + 0.1911888563316170927411831e-2 * c(m-3) - 0.4503090345217088684223814e-2 * c(m-6); 0.2279091916474916391629437e-1 * c(m-7) + 0.3097763128598982561225538e-1 * c(m-6) - 0.3055555555555555555555556e-1 * c(m-8) - 0.3068985997518740530511593e-1 * c(m-5) - 0.3634246031107139778989373e-2 * c(m-4) 0.5000000000000000000000000e-1 * c(m-8) - 0.3557251496099816106154206e0 * c(m-6) + 0.5490976528821799120017102e-1 * c(m-4) + 0.1838363233410033443348225e0 * c(m-7) + 0.2169790609807602750804271e0 * c(m-5) -0.2500000000000000000000000e-1 * c(m-8) - 0.3132544850115050165022338e0 * c(m-7) - 0.2322389872063761557916742e0 * c(m-4) - 0.8607044252686413302647675e-3 * c(m-3) - 0.2594851141920572702679681e0 * c(m-6) - 0.6691607091647929161078591e0 * c(m-5) 0.5555555555555555555555556e-2 * c(m-8) + 0.1338363233410033443348225e0 * c(m-7) + 0.7391887916719206077121040e0 * c(m-6) + 0.6490333320052011212240632e0 * c(m-4) + 0.3308343404200968256656458e-2 * c(m-3) + 0.2062575706647430620228133e-3 * c(m-2) + 0.3088241944378964404772302e-3 * c(m-1) + 0.4227226173449345042468960e-3 * c(m) + 0.1190362071861893051132274e1 * c(m-5) -0.2720908083525083608370563e-1 * c(m-7) - 0.1931148612480615118957263e0 * c(m-6) - 0.3332941113251635390801278e-1 * c(m-3) + 0.2605582646183255957264249e-3 * c(m-2) + 0.1432116665752147607469646e-2 * c(m-1) + 0.1207544072304193806052558e-2 * c(m) - 0.1348436986667115543203552e1 * c(m-5) + 0.1687723507780044227927853e-1 * c(m-4) 0.3590669644811151307464697e-1 * c(m-6) - 0.5925443480724830632401754e0 * c(m-4) - 0.6897142765790609546343709e-2 * c(m-2) - 0.5363098747528542488971874e-2 * c(m-1) - 0.5205147429855955657625694e-2 * c(m) + 0.9977064356292750529201981e0 * c(m-5) 0.7272438906214475928744770e-1 * c(m-4) + 0.1964682777744275219350831e-1 * c(m-3) - 0.5952475275883259619711594e-2 * c(m-1) - 0.1635430866921887819487473e-2 * c(m) + 0.2921234010758621482958052e-1 * c(m-6) - 0.4659516693228870973898560e0 * c(m-5) 0.5891947149681041048896399e-1 * c(m-4) + 0.1858378996391679448655070e-1 * c(m-3) + 0.7240905383565181316381731e-2 * c(m-2) + 0.2349927974590068869356781e-1 * c(m) - 0.4046360079256766884300687e-1 * c(m-6) + 0.1223513270418807666970488e0 * c(m-5) -0.2404661162020836566908542e-1 * c(m-4) - 0.7348845587775519698437916e-2 * c(m-3) - 0.8105784530576404277872603e-3 * c(m-2) + 0.9574633163221758060736592e-2 * c(m-1) - 0.1828896813877197352675410e-1 * c(m) + 0.1350326632905990039353503e-1 * c(m-6) - 0.1315967038382618382356495e-1 * c(m-5); 0.5522702088127090209264064e-3 * c(m-7) - 0.3265435411305071914756373e-2 * c(m-6) + 0.7462059484530855073291365e-2 * c(m-5) - 0.4748894282038492179461399e-2 * c(m-4) 0.5528052133944605740009217e-1 * c(m-6) - 0.8631683980217122275970376e-1 * c(m-5) - 0.3276463639080639163926118e-1 * c(m-7) + 0.5268984374242044588776166e-1 * c(m-4) 0.5331362125287625412555844e-1 * c(m-7) + 0.1736073411355428563685818e0 * c(m-5) + 0.8671038084174692625075159e-2 * c(m-3) + 0.8084259844422177692569663e-1 * c(m-6) - 0.1664345989168155800449120e0 * c(m-4) -0.2720908083525083608370563e-1 * c(m-7) - 0.1931148612480615118957263e0 * c(m-6) - 0.3332941113251635390801278e-1 * c(m-3) + 0.2605582646183255957264249e-3 * c(m-2) + 0.1432116665752147607469646e-2 * c(m-1) + 0.1207544072304193806052558e-2 * c(m) - 0.1348436986667115543203552e1 * c(m-5) + 0.1687723507780044227927853e-1 * c(m-4) 0.6107825764368264576481962e-2 * c(m-7) + 0.1155752633643216628010304e0 * c(m-6) + 0.2096413329579026439044119e1 * c(m-5) + 0.3357721707576477199985656e0 * c(m-3) + 0.3291545083271862858501887e-3 * c(m-2) + 0.6641183499427826101618457e-2 * c(m-1) + 0.3449455095910233625229891e-2 * c(m) + 0.8270696421223286922584620e0 * c(m-4) -0.4995827370863505253765970e-1 * c(m-6) - 0.1263507837371824205693950e1 * c(m-5) - 0.8712928907711754187084757e-2 * c(m-2) - 0.2487040599390160764166412e-1 * c(m-1) - 0.1486895819265604128572498e-1 * c(m) - 0.1726565392121567634950213e1 * c(m-4) 0.5402985338373433052255418e0 * c(m-5) - 0.1979290298620869974478871e0 * c(m-3) - 0.2760353365637712827793337e-1 * c(m-1) - 0.4671751091575462868310238e-2 * c(m) - 0.6952587985456154591014641e-1 * c(m-6) + 0.1571507277911208446562686e1 * c(m-4) -0.1319358558853174530078498e0 * c(m-5) - 0.1872196143003808021730728e0 * c(m-3) + 0.9147192682075630179962131e-2 * c(m-2) + 0.6712774475803763988977355e-1 * c(m) + 0.9630407686703666967100804e-1 * c(m-6) - 0.6882132817757726722141421e0 * c(m-4) 0.1241625568998496895352046e-1 * c(m-5) + 0.7403484645316174090533193e-1 * c(m-3) - 0.1023976547309387874453988e-2 * c(m-2) + 0.4440063948509876221050939e-1 * c(m-1) - 0.5224403464202056316702078e-1 * c(m) - 0.3213800979246298453953842e-1 * c(m-6) + 0.1178181682424363524005403e0 * c(m-4); 0.6272075574042975468177820e-3 * c(m-6) - 0.1200429618441003833696998e-1 * c(m-5) + 0.1137708862700574079015220e-1 * c(m-4) -0.5373770512016897565958305e-2 * c(m-6) + 0.1453421858063658498587377e0 * c(m-5) - 0.1399684152943489522927794e0 * c(m-4) -0.5013247356072127938999311e0 * c(m-5) + 0.5021853752328231128475915e0 * c(m-4) - 0.1197175073672143005877150e-1 * c(m-6) 0.3590669644811151307464697e-1 * c(m-6) - 0.5925443480724830632401754e0 * c(m-4) - 0.6897142765790609546343709e-2 * c(m-2) - 0.5363098747528542488971874e-2 * c(m-1) - 0.5205147429855955657625694e-2 * c(m) + 0.9977064356292750529201981e0 * c(m-5) -0.4995827370863505253765970e-1 * c(m-6) - 0.1263507837371824205693950e1 * c(m-5) - 0.8712928907711754187084757e-2 * c(m-2) - 0.2487040599390160764166412e-1 * c(m-1) - 0.1486895819265604128572498e-1 * c(m) - 0.1726565392121567634950213e1 * c(m-4) 0.2760393423824887721078848e-1 * c(m-6) + 0.1190550338687608873798462e1 * c(m-5) + 0.4253084328734353394994388e1 * c(m-4) + 0.2306367624634749229113646e0 * c(m-2) + 0.9313657638804699948929701e-1 * c(m-1) + 0.6409299775987186986730499e-1 * c(m) -0.8755807343482262259774782e0 * c(m-5) - 0.3645285178085761821545207e1 * c(m-4) + 0.1033717994630886401730470e0 * c(m-1) + 0.2013769413884797246646959e-1 * c(m) + 0.4106783858513785463625543e-1 * c(m-6) 0.3956598149904136332753521e0 * c(m-5) + 0.1630560443616104907615866e1 * c(m-4) - 0.2421320004064592721552708e0 * c(m-2) - 0.2893557395653431666593814e0 * c(m) - 0.5688529641249387985434413e-1 * c(m-6) -0.7684117160199014594442072e-1 * c(m-5) - 0.2928439026361256842196229e0 * c(m-4) + 0.2710530961648671297733465e-1 * c(m-2) - 0.1662748711097054895317080e0 * c(m-1) + 0.2251991532891353212689574e0 * c(m) + 0.1898341454096471754822498e-1 * c(m-6); 0.9209089963443799485648361e-2 * c(m-5) - 0.3129629392354775191148163e-3 * c(m-6) - 0.8896127024208321966533544e-2 * c(m-4) -0.1059816030196818445908057e0 * c(m-5) + 0.1014880675788250237247178e0 * c(m-4) + 0.4493535440856820866087846e-2 * c(m-6) 0.3328335104489738933610597e0 * c(m-5) - 0.3179803804558436283847901e0 * c(m-4) - 0.5111353189352474549563559e-2 * c(m-3) - 0.9741776803777790426705996e-2 * c(m-6) 0.7272438906214475928744770e-1 * c(m-4) + 0.1964682777744275219350831e-1 * c(m-3) - 0.5952475275883259619711594e-2 * c(m-1) - 0.1635430866921887819487473e-2 * c(m) + 0.2921234010758621482958052e-1 * c(m-6) - 0.4659516693228870973898560e0 * c(m-5) 0.5402985338373433052255418e0 * c(m-5) - 0.1979290298620869974478871e0 * c(m-3) - 0.2760353365637712827793337e-1 * c(m-1) - 0.4671751091575462868310238e-2 * c(m) - 0.6952587985456154591014641e-1 * c(m-6) + 0.1571507277911208446562686e1 * c(m-4) -0.8755807343482262259774782e0 * c(m-5) - 0.3645285178085761821545207e1 * c(m-4) + 0.1033717994630886401730470e0 * c(m-1) + 0.2013769413884797246646959e-1 * c(m) + 0.4106783858513785463625543e-1 * c(m-6) 0.1070920689960817104203947e1 * c(m-5) + 0.3717418466925056542408153e1 * c(m-4) + 0.1166740554279680007487795e0 * c(m-3) + 0.1147318200715868527529827e0 * c(m-1) + 0.6327161147136873807796515e-2 * c(m) + 0.6235373239336055200426697e-1 * c(m-6) -0.6639605248735044787146222e0 * c(m-5) - 0.1865625445986772763641423e1 * c(m-4) + 0.1103611313171476425250639e0 * c(m-3) - 0.9091410269992464604926176e-1 * c(m) - 0.8636954541126674177407762e-1 * c(m-6) 0.1582127073537215443965653e0 * c(m-5) + 0.3746489300753517635549495e0 * c(m-4) - 0.4364163147111892346990101e-1 * c(m-3) - 0.1845476106024151050283847e0 * c(m-1) + 0.7075642937243715046279337e-1 * c(m) + 0.2882271848190011329385407e-1 * c(m-6); -0.3576545132696983143406173e-2 * c(m-5) + 0.4335019854436220306755673e-3 * c(m-6) + 0.3143043147253361112730605e-2 * c(m-4) 0.4093499466767054661591066e-1 * c(m-5) - 0.3471075437892810033585296e-1 * c(m-4) - 0.6224240288742446280057699e-2 * c(m-6) -0.1080774142196007991746827e0 * c(m-5) + 0.9941834083648937298100811e-1 * c(m-4) - 0.4834791406446907590553793e-2 * c(m-3) + 0.1349386478955833378422842e-1 * c(m-6) 0.5891947149681041048896399e-1 * c(m-4) + 0.1858378996391679448655070e-1 * c(m-3) + 0.7240905383565181316381731e-2 * c(m-2) + 0.2349927974590068869356781e-1 * c(m) - 0.4046360079256766884300687e-1 * c(m-6) + 0.1223513270418807666970488e0 * c(m-5) -0.1319358558853174530078498e0 * c(m-5) - 0.1872196143003808021730728e0 * c(m-3) + 0.9147192682075630179962131e-2 * c(m-2) + 0.6712774475803763988977355e-1 * c(m) + 0.9630407686703666967100804e-1 * c(m-6) - 0.6882132817757726722141421e0 * c(m-4) 0.3956598149904136332753521e0 * c(m-5) + 0.1630560443616104907615866e1 * c(m-4) - 0.2421320004064592721552708e0 * c(m-2) - 0.2893557395653431666593814e0 * c(m) - 0.5688529641249387985434413e-1 * c(m-6) -0.6639605248735044787146222e0 * c(m-5) - 0.1865625445986772763641423e1 * c(m-4) + 0.1103611313171476425250639e0 * c(m-3) - 0.9091410269992464604926176e-1 * c(m) - 0.8636954541126674177407762e-1 * c(m-6) 0.4681819359722749441073885e0 * c(m-5) + 0.1015239189167790053447110e1 * c(m-4) + 0.1043897828092562609502636e0 * c(m-3) + 0.2542001760457345743492403e0 * c(m-2) + 0.1306332157111667628555907e1 * c(m) + 0.1196351539550049336518187e0 * c(m-6) -0.1195777325611201766551392e0 * c(m-5) - 0.2187310061229745694542609e0 * c(m-4) - 0.4128029838349298819825156e-1 * c(m-3) - 0.2845627370491611369031341e-1 * c(m-2) - 0.1016689339350338144430605e1 * c(m) - 0.3992391469197282238624438e-1 * c(m-6); 0.5593983696629863059347067e-3 * c(m-5) - 0.1446656414398166805849327e-3 * c(m-6) - 0.4147327282231696253497740e-3 * c(m-4) -0.6486184157331537899459796e-2 * c(m-5) + 0.4409068609809831485979484e-2 * c(m-4) + 0.2077115547521706413480312e-2 * c(m-6) 0.1319674981073749167009902e-1 * c(m-5) - 0.1060554802883657391328704e-1 * c(m-4) + 0.1911888563316170927411831e-2 * c(m-3) - 0.4503090345217088684223814e-2 * c(m-6) -0.2404661162020836566908542e-1 * c(m-4) - 0.7348845587775519698437916e-2 * c(m-3) - 0.8105784530576404277872603e-3 * c(m-2) + 0.9574633163221758060736592e-2 * c(m-1) - 0.1828896813877197352675410e-1 * c(m) + 0.1350326632905990039353503e-1 * c(m-6) - 0.1315967038382618382356495e-1 * c(m-5) 0.1241625568998496895352046e-1 * c(m-5) + 0.7403484645316174090533193e-1 * c(m-3) - 0.1023976547309387874453988e-2 * c(m-2) + 0.4440063948509876221050939e-1 * c(m-1) - 0.5224403464202056316702078e-1 * c(m) - 0.3213800979246298453953842e-1 * c(m-6) + 0.1178181682424363524005403e0 * c(m-4) -0.7684117160199014594442072e-1 * c(m-5) - 0.2928439026361256842196229e0 * c(m-4) + 0.2710530961648671297733465e-1 * c(m-2) - 0.1662748711097054895317080e0 * c(m-1) + 0.2251991532891353212689574e0 * c(m) + 0.1898341454096471754822498e-1 * c(m-6) 0.1582127073537215443965653e0 * c(m-5) + 0.3746489300753517635549495e0 * c(m-4) - 0.4364163147111892346990101e-1 * c(m-3) - 0.1845476106024151050283847e0 * c(m-1) + 0.7075642937243715046279337e-1 * c(m) + 0.2882271848190011329385407e-1 * c(m-6) -0.1195777325611201766551392e0 * c(m-5) - 0.2187310061229745694542609e0 * c(m-4) - 0.4128029838349298819825156e-1 * c(m-3) - 0.2845627370491611369031341e-1 * c(m-2) - 0.1016689339350338144430605e1 * c(m) - 0.3992391469197282238624438e-1 * c(m-6) 0.3167964748016105299646518e-1 * c(m-5) + 0.4976563420877041544013670e-1 * c(m-4) + 0.1632404042590951953384672e-1 * c(m-3) + 0.3185519088796429015220016e-2 * c(m-2) + 0.2968472090638000742888467e0 * c(m-1) + 0.7912667594695582093926295e0 * c(m) + 0.1332316557164627464149716e-1 * c(m-6);]; +% +% M(5,10)=M(10,5); +% M(m-4,m-9)=M(m-9,m-4); +% +% M=M/h; +% +% D2=HI*(-M-diag(c)*e_1*S_1+diag(c)*e_m*S_m); + +S2_U=[0.15e2 / 0.4e1 -0.77e2 / 0.6e1 0.107e3 / 0.6e1 -13 0.61e2 / 0.12e2 -0.5e1 / 0.6e1;]/h^2; +S2_1=zeros(1,m); +S2_1(1:6)=S2_U; +S2_m=zeros(1,m); +S2_m(m-5:m)=fliplr(S2_U); + + + + + +% Fourth derivative, 1th order accurate at first 8 boundary points (still +% yield 5th order convergence if stable: for example u_tt=-u_xxxx + +m4=7/240;m3=-2/5;m2=169/60;m1=-122/15;m0=91/8; +M4=m4*(diag(ones(m-4,1),4)+diag(ones(m-4,1),-4))+m3*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3))+m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0); + +%M4=(-1/6*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3) ) + 2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2)) -13/2*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)) + 28/3*diag(ones(m,1),0)); + +M4_U=[0.1399708478939e13 / 0.263487168000e12 -0.13482796013041e14 / 0.834376032000e12 0.344344095859e12 / 0.17565811200e11 -0.3166261424681e13 / 0.250312809600e12 0.1508605165681e13 / 0.333750412800e12 -0.486270829441e12 / 0.834376032000e12 -0.221976356359e12 / 0.5006256192000e13; -0.13482796013041e14 / 0.834376032000e12 0.7260475818391e13 / 0.139062672000e12 -0.27224036353e11 / 0.406022400e9 0.1847477458951e13 / 0.41718801600e11 -0.848984558161e12 / 0.55625068800e11 0.247494925991e12 / 0.139062672000e12 0.165585445559e12 / 0.834376032000e12; 0.344344095859e12 / 0.17565811200e11 -0.27224036353e11 / 0.406022400e9 0.2044938640393e13 / 0.22250027520e11 -0.1071086785417e13 / 0.16687520640e11 0.502199537033e12 / 0.22250027520e11 -0.143589154441e12 / 0.55625068800e11 -0.88181965559e11 / 0.333750412800e12; -0.3166261424681e13 / 0.250312809600e12 0.1847477458951e13 / 0.41718801600e11 -0.1071086785417e13 / 0.16687520640e11 0.628860435593e12 / 0.12515640480e11 -0.73736245829e11 / 0.3337504128e10 0.195760572271e12 / 0.41718801600e11 -0.81156046361e11 / 0.250312809600e12; 0.1508605165681e13 / 0.333750412800e12 -0.848984558161e12 / 0.55625068800e11 0.502199537033e12 / 0.22250027520e11 -0.73736245829e11 / 0.3337504128e10 0.76725285869e11 / 0.4450005504e10 -0.3912429433e10 / 0.406022400e9 0.53227370659e11 / 0.17565811200e11; -0.486270829441e12 / 0.834376032000e12 0.247494925991e12 / 0.139062672000e12 -0.143589154441e12 / 0.55625068800e11 0.195760572271e12 / 0.41718801600e11 -0.3912429433e10 / 0.406022400e9 0.1699707221791e13 / 0.139062672000e12 -0.6959018412841e13 / 0.834376032000e12; -0.221976356359e12 / 0.5006256192000e13 0.165585445559e12 / 0.834376032000e12 -0.88181965559e11 / 0.333750412800e12 -0.81156046361e11 / 0.250312809600e12 0.53227370659e11 / 0.17565811200e11 -0.6959018412841e13 / 0.834376032000e12 0.3012195053939e13 / 0.263487168000e12;]; + +M4(1:7,1:7)=M4_U; + +M4(m-6:m,m-6:m)=flipud( fliplr( M4_U ) ); +M4=M4/h^3; + +S3_U=[-0.17e2 / 0.4e1 0.71e2 / 0.4e1 -0.59e2 / 0.2e1 0.49e2 / 0.2e1 -0.41e2 / 0.4e1 0.7e1 / 0.4e1;]/h^3; +S3_1=zeros(1,m); +S3_1(1:6)=S3_U; +S3_m=zeros(1,m); +S3_m(m-5:m)=fliplr(-S3_U); + +D4=HI*(M4-e_1*S3_1+e_m*S3_m + S_1'*S2_1-S_m'*S2_m); + +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/higher_variable6_min_boundary_points.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,136 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% 6:te ordn. SBP Finita differens %%% +%%% operatorer med diagonal norm %%% +%%% Extension to variable koeff %%% +%%% %%% +%%% H (Normen) %%% +%%% D1=H^(-1)Q (approx f?rsta derivatan) %%% +%%% D2 (approx andra derivatan) %%% +%%% D2=HI*(R+C*D*S %%% +%%% %%% +%%% R=-D1'*H*C*D1-RR %%% +%%% %%% +%%% RR ?r dissipation) %%% +%%% Dissipationen uppbyggd av D4: %%% +%%% DI=D4*B*H*D4 %%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% H?r med 6 RP ist?llet f?r 8 f?r D4 operatorn, dock samma randderivator + +%m=10; %problemstorlek +%h=1/(m-1); + +% Variable koefficicients are stored in vector: c, size m, +% with the unknown stored as c(1), c(2), ..., c_m +% x=1:h:m*h;x=x'; +% c=x.^0; + + +H=diag(ones(m,1),0); +H(1:6,1:6)=diag([13649/43200,12013/8640,2711/4320,5359/4320,7877/8640, ... + 43801/43200]); +H(m-5:m,m-5:m)=fliplr(flipud(diag([13649/43200,12013/8640, ... + 2711/4320,5359/4320,7877/8640,43801/43200]))); + + +x1=0.70127127127127; + + +D1=(1/60*diag(ones(m-3,1),3)-9/60*diag(ones(m-2,1),2)+45/60*diag(ones(m-1,1),1)-45/60*diag(ones(m-1,1),-1)+9/60*diag(ones(m-2,1),-2)-1/60*diag(ones(m-3,1),-3)); + + + +D1(1:6,1:9)=[-21600/13649, 43200/13649*x1-7624/40947, -172800/13649*x1+ ... + 715489/81894, 259200/13649*x1-187917/13649, -172800/13649* ... + x1+735635/81894, 43200/13649*x1-89387/40947, 0, 0, 0; ... + -8640/12013*x1+7624/180195, 0, 86400/12013*x1-57139/12013, ... + -172800/12013*x1+745733/72078, 129600/12013*x1-91715/12013, ... + -34560/12013*x1+240569/120130, 0, 0, 0; ... + 17280/2711*x1-715489/162660, -43200/2711*x1+57139/5422, 0, ... + 86400/2711*x1-176839/8133, -86400/2711*x1+242111/10844, ... + 25920/2711*x1-182261/27110, 0, 0, 0; ... + -25920/5359*x1+187917/53590, 86400/5359*x1-745733/64308, ... + -86400/5359*x1+176839/16077, 0, 43200/5359*x1-165041/32154, ... + -17280/5359*x1+710473/321540, 72/5359, 0, 0; ... + 34560/7877*x1-147127/47262, -129600/7877*x1+91715/7877, ... + 172800/7877*x1-242111/15754, -86400/7877*x1+165041/23631, ... + 0, 8640/7877*x1, -1296/7877, 144/7877, 0; ... + -43200/43801*x1+89387/131403, 172800/43801*x1-240569/87602,... + -259200/43801*x1+182261/43801, 172800/43801*x1-710473/262806, ... + -43200/43801*x1, 0, 32400/43801, -6480/43801, 720/43801]; +D1(m-5:m,m-8:m)=flipud( fliplr(-D1(1:6,1:9))); +D1=D1/h; + +e_1=zeros(m,1);e_1(1)=1; +e_m=zeros(m,1);e_m(m)=1; + +S_U=[-25/12, 4, -3, 4/3, -1/4]/h; +S_1=zeros(1,m); +S_1(1:5)=S_U; +S_m=zeros(1,m); +S_m(m-4:m)=fliplr(-S_U); + + + +%DS=zeros(m,m); +%DS(1,1:5)=-[-25/12, 4, -3, 4/3, -1/4]; +%DS(m,m-4:m)=fliplr(-[-25/12, 4, -3, 4/3, -1/4]); +%DS=diag(c)*DS/h; + + +H=h*H; +HI=inv(H); + + +M=zeros(m,m); + +for i=6:m-5 +M(i,i-3:i+3)=[c(i-2) / 0.40e2 + c(i-1) / 0.40e2 - 0.11e2 / 0.360e3 * c(i-3) - 0.11e2 / 0.360e3 * c(i) c(i-3) / 0.20e2 - 0.3e1 / 0.10e2 * c(i-1) + c(i+1) / 0.20e2 + 0.7e1 / 0.40e2 * c(i) + 0.7e1 / 0.40e2 * c(i-2) -c(i-3) / 0.40e2 - 0.3e1 / 0.10e2 * c(i-2) - 0.3e1 / 0.10e2 * c(i+1) - c(i+2) / 0.40e2 - 0.17e2 / 0.40e2 * c(i) - 0.17e2 / 0.40e2 * c(i-1) c(i-3) / 0.180e3 + c(i-2) / 0.8e1 + 0.19e2 / 0.20e2 * c(i-1) + 0.19e2 / 0.20e2 * c(i+1) + c(i+2) / 0.8e1 + c(i+3) / 0.180e3 + 0.101e3 / 0.180e3 * c(i) -c(i-2) / 0.40e2 - 0.3e1 / 0.10e2 * c(i-1) - 0.3e1 / 0.10e2 * c(i+2) - c(i+3) / 0.40e2 - 0.17e2 / 0.40e2 * c(i) - 0.17e2 / 0.40e2 * c(i+1) c(i-1) / 0.20e2 - 0.3e1 / 0.10e2 * c(i+1) + c(i+3) / 0.20e2 + 0.7e1 / 0.40e2 * c(i) + 0.7e1 / 0.40e2 * c(i+2) c(i+1) / 0.40e2 + c(i+2) / 0.40e2 - 0.11e2 / 0.360e3 * c(i) - 0.11e2 / 0.360e3 * c(i+3);]; +end + + +M(1:9,1:9)=[0.7912667594695582093926295e0 * c(1) + 0.2968472090638000742888467e0 * c(2) + 0.3185519088796429015220016e-2 * c(3) + 0.1632404042590951953384672e-1 * c(4) + 0.3160302244094415087693968e-1 * c(5) + 0.3167964748016105299646518e-1 * c(6) + 0.3148577733947253920469418e-1 * c(7) -0.1016689339350338144430605e1 * c(1) - 0.2845627370491611369031341e-1 * c(3) - 0.4128029838349298819825156e-1 * c(4) - 0.1392281451620140507549866e0 * c(5) - 0.1195777325611201766551392e0 * c(6) - 0.1194267756529333410855186e0 * c(7) 0.7075642937243715046279337e-1 * c(1) - 0.1845476106024151050283847e0 * c(2) - 0.4364163147111892346990101e-1 * c(4) + 0.2432367907207732460874765e0 * c(5) + 0.1582127073537215443965653e0 * c(6) + 0.1602348578364786307613271e0 * c(7) 0.2251991532891353212689574e0 * c(1) - 0.1662748711097054895317080e0 * c(2) + 0.2710530961648671297733465e-1 * c(3) - 0.1916646185968439909125616e0 * c(5) - 0.7684117160199014594442072e-1 * c(6) - 0.8219586949831697575883635e-1 * c(7) -0.5224403464202056316702078e-1 * c(1) + 0.4440063948509876221050939e-1 * c(2) - 0.1023976547309387874453988e-2 * c(3) + 0.7403484645316174090533193e-1 * c(4) + 0.1241625568998496895352046e-1 * c(6) + 0.7188652847892601282652228e-1 * c(5) + 0.1379362997104735503447960e-1 * c(7) -0.1828896813877197352675410e-1 * c(1) + 0.9574633163221758060736592e-2 * c(2) - 0.8105784530576404277872603e-3 * c(3) - 0.7348845587775519698437916e-2 * c(4) + 0.1063601949723906997026904e-1 * c(5) - 0.1315967038382618382356495e-1 * c(6) - 0.2117936478838753524581943e-1 * c(7) 0.1911888563316170927411831e-2 * c(4) - 0.4068130355529149936100229e-1 * c(5) + 0.1319674981073749167009902e-1 * c(6) + 0.2557266518123783676349144e-1 * c(7) 0.1559652871136785763960685e-1 * c(5) - 0.6486184157331537899459796e-2 * c(6) - 0.9110344554036319740147054e-2 * c(7) 0.5593983696629863059347067e-3 * c(6) - 0.1384822535100796372263822e-2 * c(5) + 0.8254241654378100663291154e-3 * c(7); -0.1016689339350338144430605e1 * c(1) - 0.2845627370491611369031341e-1 * c(3) - 0.4128029838349298819825156e-1 * c(4) - 0.1392281451620140507549866e0 * c(5) - 0.1195777325611201766551392e0 * c(6) - 0.1194267756529333410855186e0 * c(7) 0.1306332157111667628555907e1 * c(1) + 0.2542001760457345743492403e0 * c(3) + 0.1043897828092562609502636e0 * c(4) + 0.6672328021032112950919876e0 * c(5) + 0.4681819359722749441073885e0 * c(6) + 0.4676415410195836920069412e0 * c(7) -0.9091410269992464604926176e-1 * c(1) + 0.1103611313171476425250639e0 * c(4) - 0.1290397544997518887000350e1 * c(5) - 0.6639605248735044787146222e0 * c(6) - 0.6615974464005206184151509e0 * c(7) -0.2893557395653431666593814e0 * c(1) - 0.2421320004064592721552708e0 * c(3) + 0.1187670255028031027693374e1 * c(5) + 0.3956598149904136332753521e0 * c(6) + 0.3860048921755800000681479e0 * c(7) 0.6712774475803763988977355e-1 * c(1) + 0.9147192682075630179962131e-2 * c(3) - 0.1872196143003808021730728e0 * c(4) - 0.1319358558853174530078498e0 * c(6) - 0.4871575736811911887376923e0 * c(5) - 0.1047516312275448138054418e0 * c(7) 0.2349927974590068869356781e-1 * c(1) + 0.7240905383565181316381731e-2 * c(3) + 0.1858378996391679448655070e-1 * c(4) - 0.9289616133938676174345208e-1 * c(5) + 0.1223513270418807666970488e0 * c(6) + 0.1113520320436295033894092e0 * c(7) -0.4834791406446907590553793e-2 * c(4) + 0.2310683832687820403062715e0 * c(5) - 0.1080774142196007991746827e0 * c(6) - 0.1181561776427343335410349e0 * c(7) -0.8368141434403455353724691e-1 * c(5) + 0.4093499466767054661591066e-1 * c(6) + 0.4274641967636400692133626e-1 * c(7) -0.3576545132696983143406173e-2 * c(6) + 0.7389399124121078682094445e-2 * c(5) - 0.3812853991424095538688273e-2 * c(7); 0.7075642937243715046279337e-1 * c(1) - 0.1845476106024151050283847e0 * c(2) - 0.4364163147111892346990101e-1 * c(4) + 0.2432367907207732460874765e0 * c(5) + 0.1582127073537215443965653e0 * c(6) + 0.1602348578364786307613271e0 * c(7) -0.9091410269992464604926176e-1 * c(1) + 0.1103611313171476425250639e0 * c(4) - 0.1290397544997518887000350e1 * c(5) - 0.6639605248735044787146222e0 * c(6) - 0.6615974464005206184151509e0 * c(7) 0.6327161147136873807796515e-2 * c(1) + 0.1147318200715868527529827e0 * c(2) + 0.1166740554279680007487795e0 * c(4) + 0.2766610808285444037240703e1 * c(5) + 0.1070920689960817104203947e1 * c(6) + 0.1013161391032973057171717e1 * c(7) 0.2013769413884797246646959e-1 * c(1) + 0.1033717994630886401730470e0 * c(2) - 0.2913221621151742724258117e1 * c(5) - 0.8755807343482262259774782e0 * c(6) - 0.6909957183488812426508351e0 * c(7) -0.4671751091575462868310238e-2 * c(1) - 0.2760353365637712827793337e-1 * c(2) - 0.1979290298620869974478871e0 * c(4) + 0.5402985338373433052255418e0 * c(6) + 0.1239177593031911077924537e1 * c(5) + 0.2628038050247358227280031e0 * c(7) -0.1635430866921887819487473e-2 * c(1) - 0.5952475275883259619711594e-2 * c(2) + 0.1964682777744275219350831e-1 * c(4) + 0.3236640012639046600590714e0 * c(5) - 0.4659516693228870973898560e0 * c(6) - 0.2217272720941736859420432e0 * c(7) -0.5111353189352474549563559e-2 * c(4) - 0.5355878163774754346032096e0 * c(5) + 0.3328335104489738933610597e0 * c(6) + 0.2078656591178540157917135e0 * c(7) 0.1824328174134289562208038e0 * c(5) - 0.1059816030196818445908057e0 * c(6) - 0.7645121439374711162999809e-1 * c(7) 0.9209089963443799485648361e-2 * c(6) - 0.1591502818872493167091475e-1 * c(5) + 0.6705938225281132185266388e-2 * c(7); 0.2251991532891353212689574e0 * c(1) - 0.1662748711097054895317080e0 * c(2) + 0.2710530961648671297733465e-1 * c(3) - 0.1916646185968439909125616e0 * c(5) - 0.7684117160199014594442072e-1 * c(6) - 0.8219586949831697575883635e-1 * c(7) -0.2893557395653431666593814e0 * c(1) - 0.2421320004064592721552708e0 * c(3) + 0.1187670255028031027693374e1 * c(5) + 0.3956598149904136332753521e0 * c(6) + 0.3860048921755800000681479e0 * c(7) 0.2013769413884797246646959e-1 * c(1) + 0.1033717994630886401730470e0 * c(2) - 0.2913221621151742724258117e1 * c(5) - 0.8755807343482262259774782e0 * c(6) - 0.6909957183488812426508351e0 * c(7) 0.6409299775987186986730499e-1 * c(1) + 0.9313657638804699948929701e-1 * c(2) + 0.2306367624634749229113646e0 * c(3) + 0.3689440308283716620260816e1 * c(5) + 0.1190550338687608873798462e1 * c(6) + 0.5912479546888856519443605e0 * c(7) -0.1486895819265604128572498e-1 * c(1) - 0.2487040599390160764166412e-1 * c(2) - 0.8712928907711754187084757e-2 * c(3) - 0.1263507837371824205693950e1 * c(6) - 0.3058317397843997326920898e0 * c(7) - 0.1470691926045802954795783e1 * c(5) -0.5205147429855955657625694e-2 * c(1) - 0.5363098747528542488971874e-2 * c(2) - 0.6897142765790609546343709e-2 * c(3) - 0.7857524521667450101721993e0 * c(5) + 0.2291148005423734600066709e0 * c(7) + 0.9977064356292750529201981e0 * c(6) 0.6697297488067662265210608e0 * c(5) - 0.5013247356072127938999311e0 * c(6) - 0.1795161243106645437322408e0 * c(7) -0.2022909060111751565150958e0 * c(5) + 0.1453421858063658498587377e0 * c(6) + 0.5694872020480930665635812e-1 * c(7) -0.1200429618441003833696998e-1 * c(6) - 0.4776915669385923841535432e-2 * c(7) + 0.1678121185379596217850541e-1 * c(5); -0.5224403464202056316702078e-1 * c(1) + 0.4440063948509876221050939e-1 * c(2) - 0.1023976547309387874453988e-2 * c(3) + 0.7403484645316174090533193e-1 * c(4) + 0.1241625568998496895352046e-1 * c(6) + 0.7188652847892601282652228e-1 * c(5) + 0.1379362997104735503447960e-1 * c(7) 0.6712774475803763988977355e-1 * c(1) + 0.9147192682075630179962131e-2 * c(3) - 0.1872196143003808021730728e0 * c(4) - 0.1319358558853174530078498e0 * c(6) - 0.4871575736811911887376923e0 * c(5) - 0.1047516312275448138054418e0 * c(7) -0.4671751091575462868310238e-2 * c(1) - 0.2760353365637712827793337e-1 * c(2) - 0.1979290298620869974478871e0 * c(4) + 0.5402985338373433052255418e0 * c(6) + 0.1239177593031911077924537e1 * c(5) + 0.2628038050247358227280031e0 * c(7) -0.1486895819265604128572498e-1 * c(1) - 0.2487040599390160764166412e-1 * c(2) - 0.8712928907711754187084757e-2 * c(3) - 0.1263507837371824205693950e1 * c(6) - 0.3058317397843997326920898e0 * c(7) - 0.1470691926045802954795783e1 * c(5) 0.3449455095910233625229891e-2 * c(1) + 0.6641183499427826101618457e-2 * c(2) + 0.3291545083271862858501887e-3 * c(3) + 0.3357721707576477199985656e0 * c(4) + 0.2096413329579026439044119e1 * c(6) + 0.2317323204183126854954203e0 * c(7) + 0.6107825764368264576481962e-2 * c(8) + 0.7109125850683376695640722e0 * c(5) 0.1207544072304193806052558e-2 * c(1) + 0.1432116665752147607469646e-2 * c(2) + 0.2605582646183255957264249e-3 * c(3) - 0.3332941113251635390801278e-1 * c(4) - 0.2808241697385532683612407e0 * c(7) - 0.2720908083525083608370563e-1 * c(8) + 0.1045865435682921987447929e0 * c(5) - 0.1348436986667115543203552e1 * c(6) 0.8671038084174692625075159e-2 * c(4) + 0.1736073411355428563685818e0 * c(6) + 0.5331362125287625412555844e-1 * c(8) - 0.2424935262404526301801157e0 * c(5) + 0.1569015257678588270609004e0 * c(7) -0.8631683980217122275970376e-1 * c(6) + 0.2698842360470999243492629e-1 * c(7) + 0.8098194147715651085292754e-1 * c(5) - 0.3276463639080639163926118e-1 * c(8) 0.7462059484530855073291365e-2 * c(6) - 0.8121640361668678949573496e-3 * c(7) + 0.5522702088127090209264064e-3 * c(8) - 0.7202165657176696199260422e-2 * c(5); -0.1828896813877197352675410e-1 * c(1) + 0.9574633163221758060736592e-2 * c(2) - 0.8105784530576404277872603e-3 * c(3) - 0.7348845587775519698437916e-2 * c(4) + 0.1063601949723906997026904e-1 * c(5) - 0.1315967038382618382356495e-1 * c(6) - 0.2117936478838753524581943e-1 * c(7) 0.2349927974590068869356781e-1 * c(1) + 0.7240905383565181316381731e-2 * c(3) + 0.1858378996391679448655070e-1 * c(4) - 0.9289616133938676174345208e-1 * c(5) + 0.1223513270418807666970488e0 * c(6) + 0.1113520320436295033894092e0 * c(7) -0.1635430866921887819487473e-2 * c(1) - 0.5952475275883259619711594e-2 * c(2) + 0.1964682777744275219350831e-1 * c(4) + 0.3236640012639046600590714e0 * c(5) - 0.4659516693228870973898560e0 * c(6) - 0.2217272720941736859420432e0 * c(7) -0.5205147429855955657625694e-2 * c(1) - 0.5363098747528542488971874e-2 * c(2) - 0.6897142765790609546343709e-2 * c(3) - 0.7857524521667450101721993e0 * c(5) + 0.2291148005423734600066709e0 * c(7) + 0.9977064356292750529201981e0 * c(6) 0.1207544072304193806052558e-2 * c(1) + 0.1432116665752147607469646e-2 * c(2) + 0.2605582646183255957264249e-3 * c(3) - 0.3332941113251635390801278e-1 * c(4) - 0.2808241697385532683612407e0 * c(7) - 0.2720908083525083608370563e-1 * c(8) + 0.1045865435682921987447929e0 * c(5) - 0.1348436986667115543203552e1 * c(6) 0.4227226173449345042468960e-3 * c(1) + 0.3088241944378964404772302e-3 * c(2) + 0.2062575706647430620228133e-3 * c(3) + 0.3308343404200968256656458e-2 * c(4) + 0.5828047016405001815804837e0 * c(5) + 0.8054174220366215473556835e0 * c(7) + 0.1338363233410033443348225e0 * c(8) + 0.5555555555555555555555556e-2 * c(9) + 0.1190362071861893051132274e1 * c(6) -0.8607044252686413302647675e-3 * c(4) - 0.1748074708673904989293256e0 * c(5) - 0.3132544850115050165022338e0 * c(8) - 0.2500000000000000000000000e-1 * c(9) - 0.3169166305310429271303167e0 * c(7) - 0.6691607091647929161078591e0 * c(6) 0.3354661791693352108660900e-1 * c(5) - 0.3343620022386971405018586e0 * c(7) + 0.5000000000000000000000000e-1 * c(9) + 0.2169790609807602750804271e0 * c(6) + 0.1838363233410033443348225e0 * c(8) 0.2912518476823004642951502e-1 * c(7) + 0.2279091916474916391629437e-1 * c(8) - 0.3068985997518740530511593e-1 * c(6) - 0.1781799513347360596249022e-2 * c(5) - 0.3055555555555555555555556e-1 * c(9); 0.1911888563316170927411831e-2 * c(4) - 0.4068130355529149936100229e-1 * c(5) + 0.1319674981073749167009902e-1 * c(6) + 0.2557266518123783676349144e-1 * c(7) -0.4834791406446907590553793e-2 * c(4) + 0.2310683832687820403062715e0 * c(5) - 0.1080774142196007991746827e0 * c(6) - 0.1181561776427343335410349e0 * c(7) -0.5111353189352474549563559e-2 * c(4) - 0.5355878163774754346032096e0 * c(5) + 0.3328335104489738933610597e0 * c(6) + 0.2078656591178540157917135e0 * c(7) 0.6697297488067662265210608e0 * c(5) - 0.5013247356072127938999311e0 * c(6) - 0.1795161243106645437322408e0 * c(7) 0.8671038084174692625075159e-2 * c(4) + 0.1736073411355428563685818e0 * c(6) + 0.5331362125287625412555844e-1 * c(8) - 0.2424935262404526301801157e0 * c(5) + 0.1569015257678588270609004e0 * c(7) -0.8607044252686413302647675e-3 * c(4) - 0.1748074708673904989293256e0 * c(5) - 0.3132544850115050165022338e0 * c(8) - 0.2500000000000000000000000e-1 * c(9) - 0.3169166305310429271303167e0 * c(7) - 0.6691607091647929161078591e0 * c(6) 0.2239223735771599178951297e-3 * c(4) + 0.1275437785430956673825710e0 * c(5) + 0.1011699483929608164601067e1 * c(6) + 0.9698817275172575247533506e0 * c(8) + 0.1250000000000000000000000e0 * c(9) + 0.5555555555555555555555556e-2 * c(10) + 0.4823177543031281500117826e0 * c(7) -0.3784113973033012949863031e-1 * c(5) - 0.2997556885134827361576001e0 * c(6) - 0.3000000000000000000000000e0 * c(9) - 0.2500000000000000000000000e-1 * c(10) - 0.3991486867446821178415359e0 * c(7) - 0.4382544850115050165022338e0 * c(8) 0.4698146218022683933926520e-1 * c(6) - 0.2966863787471237458744416e0 * c(8) + 0.5000000000000000000000000e-1 * c(10) + 0.1716355704146006481727960e0 * c(7) + 0.3069346152296258362380356e-2 * c(5) + 0.1750000000000000000000000e0 * c(9); 0.1559652871136785763960685e-1 * c(5) - 0.6486184157331537899459796e-2 * c(6) - 0.9110344554036319740147054e-2 * c(7) -0.8368141434403455353724691e-1 * c(5) + 0.4093499466767054661591066e-1 * c(6) + 0.4274641967636400692133626e-1 * c(7) 0.1824328174134289562208038e0 * c(5) - 0.1059816030196818445908057e0 * c(6) - 0.7645121439374711162999809e-1 * c(7) -0.2022909060111751565150958e0 * c(5) + 0.1453421858063658498587377e0 * c(6) + 0.5694872020480930665635812e-1 * c(7) -0.8631683980217122275970376e-1 * c(6) + 0.2698842360470999243492629e-1 * c(7) + 0.8098194147715651085292754e-1 * c(5) - 0.3276463639080639163926118e-1 * c(8) 0.3354661791693352108660900e-1 * c(5) - 0.3343620022386971405018586e0 * c(7) + 0.5000000000000000000000000e-1 * c(9) + 0.2169790609807602750804271e0 * c(6) + 0.1838363233410033443348225e0 * c(8) -0.3784113973033012949863031e-1 * c(5) - 0.2997556885134827361576001e0 * c(6) - 0.3000000000000000000000000e0 * c(9) - 0.2500000000000000000000000e-1 * c(10) - 0.3991486867446821178415359e0 * c(7) - 0.4382544850115050165022338e0 * c(8) 0.1230328942716804455358698e-1 * c(5) + 0.1183647529645898332481833e0 * c(6) + 0.9410511898227943334189628e0 * c(7) + 0.9500000000000000000000000e0 * c(9) + 0.1250000000000000000000000e0 * c(10) + 0.5555555555555555555555556e-2 * c(11) + 0.5699474344521144554459336e0 * c(8) -0.2308067892671916339568942e-1 * c(6) - 0.2986625053775149497180439e0 * c(7) - 0.3000000000000000000000000e0 * c(10) - 0.2500000000000000000000000e-1 * c(11) - 0.1047734860515050802561078e-2 * c(5) - 0.4272090808352508360837056e0 * c(8) - 0.4250000000000000000000000e0 * c(9); 0.5593983696629863059347067e-3 * c(6) - 0.1384822535100796372263822e-2 * c(5) + 0.8254241654378100663291154e-3 * c(7) -0.3576545132696983143406173e-2 * c(6) + 0.7389399124121078682094445e-2 * c(5) - 0.3812853991424095538688273e-2 * c(7) 0.9209089963443799485648361e-2 * c(6) - 0.1591502818872493167091475e-1 * c(5) + 0.6705938225281132185266388e-2 * c(7) -0.1200429618441003833696998e-1 * c(6) - 0.4776915669385923841535432e-2 * c(7) + 0.1678121185379596217850541e-1 * c(5) 0.7462059484530855073291365e-2 * c(6) - 0.8121640361668678949573496e-3 * c(7) + 0.5522702088127090209264064e-3 * c(8) - 0.7202165657176696199260422e-2 * c(5) 0.2912518476823004642951502e-1 * c(7) + 0.2279091916474916391629437e-1 * c(8) - 0.3068985997518740530511593e-1 * c(6) - 0.1781799513347360596249022e-2 * c(5) - 0.3055555555555555555555556e-1 * c(9) 0.4698146218022683933926520e-1 * c(6) - 0.2966863787471237458744416e0 * c(8) + 0.5000000000000000000000000e-1 * c(10) + 0.1716355704146006481727960e0 * c(7) + 0.3069346152296258362380356e-2 * c(5) + 0.1750000000000000000000000e0 * c(9) -0.2308067892671916339568942e-1 * c(6) - 0.2986625053775149497180439e0 * c(7) - 0.3000000000000000000000000e0 * c(10) - 0.2500000000000000000000000e-1 * c(11) - 0.1047734860515050802561078e-2 * c(5) - 0.4272090808352508360837056e0 * c(8) - 0.4250000000000000000000000e0 * c(9) 0.5139370221149109977041877e-2 * c(6) + 0.1247723215009422001393184e0 * c(7) + 0.9505522702088127090209264e0 * c(8) + 0.9500000000000000000000000e0 * c(10) + 0.1250000000000000000000000e0 * c(11) + 0.5555555555555555555555556e-2 * c(12) + 0.9159362465153641826887659e-4 * c(5) + 0.5611111111111111111111111e0 * c(9);]; + +M(m-8:m,m-8:m)=[0.5555555555555555555555556e-2 * c(m-11) + 0.1250000000000000000000000e0 * c(m-10) + 0.9500000000000000000000000e0 * c(m-9) + 0.9505522702088127090209264e0 * c(m-7) + 0.1247205076844361998744053e0 * c(m-6) + 0.5139370221149109977041877e-2 * c(m-5) + 0.5611111111111111111111111e0 * c(m-8) + 0.1434074411575366831819799e-3 * c(m-4) -0.2500000000000000000000000e-1 * c(m-10) - 0.3000000000000000000000000e0 * c(m-9) - 0.2980649679116425253322056e0 * c(m-6) - 0.2308067892671916339568942e-1 * c(m-5) - 0.4250000000000000000000000e0 * c(m-8) - 0.4272090808352508360837056e0 * c(m-7) - 0.1645272326387475188399322e-2 * c(m-4) 0.5000000000000000000000000e-1 * c(m-9) - 0.2966863787471237458744416e0 * c(m-7) + 0.4698146218022683933926520e-1 * c(m-5) + 0.1750000000000000000000000e0 * c(m-8) + 0.1700291833903489463825077e0 * c(m-6) + 0.4675733176547960152668626e-2 * c(m-4) 0.2279091916474916391629437e-1 * c(m-7) + 0.3097763128598982561225538e-1 * c(m-6) - 0.3055555555555555555555556e-1 * c(m-8) - 0.3068985997518740530511593e-1 * c(m-5) - 0.3634246031107139778989373e-2 * c(m-4) 0.5522702088127090209264064e-3 * c(m-7) - 0.3265435411305071914756373e-2 * c(m-6) + 0.7462059484530855073291365e-2 * c(m-5) - 0.4748894282038492179461399e-2 * c(m-4) 0.6272075574042975468177820e-3 * c(m-6) - 0.1200429618441003833696998e-1 * c(m-5) + 0.1137708862700574079015220e-1 * c(m-4) 0.9209089963443799485648361e-2 * c(m-5) - 0.3129629392354775191148163e-3 * c(m-6) - 0.8896127024208321966533544e-2 * c(m-4) -0.3576545132696983143406173e-2 * c(m-5) + 0.4335019854436220306755673e-3 * c(m-6) + 0.3143043147253361112730605e-2 * c(m-4) 0.5593983696629863059347067e-3 * c(m-5) - 0.1446656414398166805849327e-3 * c(m-6) - 0.4147327282231696253497740e-3 * c(m-4); -0.2500000000000000000000000e-1 * c(m-10) - 0.3000000000000000000000000e0 * c(m-9) - 0.2980649679116425253322056e0 * c(m-6) - 0.2308067892671916339568942e-1 * c(m-5) - 0.4250000000000000000000000e0 * c(m-8) - 0.4272090808352508360837056e0 * c(m-7) - 0.1645272326387475188399322e-2 * c(m-4) 0.5555555555555555555555556e-2 * c(m-10) + 0.1250000000000000000000000e0 * c(m-9) + 0.9500000000000000000000000e0 * c(m-8) + 0.9341601509609901526962449e0 * c(m-6) + 0.1183647529645898332481833e0 * c(m-5) + 0.1919432828897222527630486e-1 * c(m-4) + 0.5699474344521144554459336e0 * c(m-7) -0.2500000000000000000000000e-1 * c(m-9) - 0.3000000000000000000000000e0 * c(m-8) - 0.2997556885134827361576001e0 * c(m-5) - 0.5636663150858098975790317e-1 * c(m-4) - 0.4382544850115050165022338e0 * c(m-7) - 0.3806231949664312575822630e0 * c(m-6) 0.5000000000000000000000000e-1 * c(m-8) - 0.3557251496099816106154206e0 * c(m-6) + 0.5490976528821799120017102e-1 * c(m-4) + 0.1838363233410033443348225e0 * c(m-7) + 0.2169790609807602750804271e0 * c(m-5) 0.5528052133944605740009217e-1 * c(m-6) - 0.8631683980217122275970376e-1 * c(m-5) - 0.3276463639080639163926118e-1 * c(m-7) + 0.5268984374242044588776166e-1 * c(m-4) -0.5373770512016897565958305e-2 * c(m-6) + 0.1453421858063658498587377e0 * c(m-5) - 0.1399684152943489522927794e0 * c(m-4) -0.1059816030196818445908057e0 * c(m-5) + 0.1014880675788250237247178e0 * c(m-4) + 0.4493535440856820866087846e-2 * c(m-6) 0.4093499466767054661591066e-1 * c(m-5) - 0.3471075437892810033585296e-1 * c(m-4) - 0.6224240288742446280057699e-2 * c(m-6) -0.6486184157331537899459796e-2 * c(m-5) + 0.4409068609809831485979484e-2 * c(m-4) + 0.2077115547521706413480312e-2 * c(m-6); 0.5000000000000000000000000e-1 * c(m-9) - 0.2966863787471237458744416e0 * c(m-7) + 0.4698146218022683933926520e-1 * c(m-5) + 0.1750000000000000000000000e0 * c(m-8) + 0.1700291833903489463825077e0 * c(m-6) + 0.4675733176547960152668626e-2 * c(m-4) -0.2500000000000000000000000e-1 * c(m-9) - 0.3000000000000000000000000e0 * c(m-8) - 0.2997556885134827361576001e0 * c(m-5) - 0.5636663150858098975790317e-1 * c(m-4) - 0.4382544850115050165022338e0 * c(m-7) - 0.3806231949664312575822630e0 * c(m-6) 0.5555555555555555555555556e-2 * c(m-9) + 0.1250000000000000000000000e0 * c(m-8) + 0.9698817275172575247533506e0 * c(m-7) + 0.1011699483929608164601067e1 * c(m-5) + 0.1773466968705924819112984e0 * c(m-4) + 0.2239223735771599178951297e-3 * c(m-3) + 0.4325148359756313354830552e0 * c(m-6) -0.2500000000000000000000000e-1 * c(m-8) - 0.3132544850115050165022338e0 * c(m-7) - 0.2322389872063761557916742e0 * c(m-4) - 0.8607044252686413302647675e-3 * c(m-3) - 0.2594851141920572702679681e0 * c(m-6) - 0.6691607091647929161078591e0 * c(m-5) 0.5331362125287625412555844e-1 * c(m-7) + 0.1736073411355428563685818e0 * c(m-5) + 0.8671038084174692625075159e-2 * c(m-3) + 0.8084259844422177692569663e-1 * c(m-6) - 0.1664345989168155800449120e0 * c(m-4) -0.5013247356072127938999311e0 * c(m-5) + 0.5021853752328231128475915e0 * c(m-4) - 0.1197175073672143005877150e-1 * c(m-6) 0.3328335104489738933610597e0 * c(m-5) - 0.3179803804558436283847901e0 * c(m-4) - 0.5111353189352474549563559e-2 * c(m-3) - 0.9741776803777790426705996e-2 * c(m-6) -0.1080774142196007991746827e0 * c(m-5) + 0.9941834083648937298100811e-1 * c(m-4) - 0.4834791406446907590553793e-2 * c(m-3) + 0.1349386478955833378422842e-1 * c(m-6) 0.1319674981073749167009902e-1 * c(m-5) - 0.1060554802883657391328704e-1 * c(m-4) + 0.1911888563316170927411831e-2 * c(m-3) - 0.4503090345217088684223814e-2 * c(m-6); 0.2279091916474916391629437e-1 * c(m-7) + 0.3097763128598982561225538e-1 * c(m-6) - 0.3055555555555555555555556e-1 * c(m-8) - 0.3068985997518740530511593e-1 * c(m-5) - 0.3634246031107139778989373e-2 * c(m-4) 0.5000000000000000000000000e-1 * c(m-8) - 0.3557251496099816106154206e0 * c(m-6) + 0.5490976528821799120017102e-1 * c(m-4) + 0.1838363233410033443348225e0 * c(m-7) + 0.2169790609807602750804271e0 * c(m-5) -0.2500000000000000000000000e-1 * c(m-8) - 0.3132544850115050165022338e0 * c(m-7) - 0.2322389872063761557916742e0 * c(m-4) - 0.8607044252686413302647675e-3 * c(m-3) - 0.2594851141920572702679681e0 * c(m-6) - 0.6691607091647929161078591e0 * c(m-5) 0.5555555555555555555555556e-2 * c(m-8) + 0.1338363233410033443348225e0 * c(m-7) + 0.7391887916719206077121040e0 * c(m-6) + 0.6490333320052011212240632e0 * c(m-4) + 0.3308343404200968256656458e-2 * c(m-3) + 0.2062575706647430620228133e-3 * c(m-2) + 0.3088241944378964404772302e-3 * c(m-1) + 0.4227226173449345042468960e-3 * c(m) + 0.1190362071861893051132274e1 * c(m-5) -0.2720908083525083608370563e-1 * c(m-7) - 0.1931148612480615118957263e0 * c(m-6) - 0.3332941113251635390801278e-1 * c(m-3) + 0.2605582646183255957264249e-3 * c(m-2) + 0.1432116665752147607469646e-2 * c(m-1) + 0.1207544072304193806052558e-2 * c(m) - 0.1348436986667115543203552e1 * c(m-5) + 0.1687723507780044227927853e-1 * c(m-4) 0.3590669644811151307464697e-1 * c(m-6) - 0.5925443480724830632401754e0 * c(m-4) - 0.6897142765790609546343709e-2 * c(m-2) - 0.5363098747528542488971874e-2 * c(m-1) - 0.5205147429855955657625694e-2 * c(m) + 0.9977064356292750529201981e0 * c(m-5) 0.7272438906214475928744770e-1 * c(m-4) + 0.1964682777744275219350831e-1 * c(m-3) - 0.5952475275883259619711594e-2 * c(m-1) - 0.1635430866921887819487473e-2 * c(m) + 0.2921234010758621482958052e-1 * c(m-6) - 0.4659516693228870973898560e0 * c(m-5) 0.5891947149681041048896399e-1 * c(m-4) + 0.1858378996391679448655070e-1 * c(m-3) + 0.7240905383565181316381731e-2 * c(m-2) + 0.2349927974590068869356781e-1 * c(m) - 0.4046360079256766884300687e-1 * c(m-6) + 0.1223513270418807666970488e0 * c(m-5) -0.2404661162020836566908542e-1 * c(m-4) - 0.7348845587775519698437916e-2 * c(m-3) - 0.8105784530576404277872603e-3 * c(m-2) + 0.9574633163221758060736592e-2 * c(m-1) - 0.1828896813877197352675410e-1 * c(m) + 0.1350326632905990039353503e-1 * c(m-6) - 0.1315967038382618382356495e-1 * c(m-5); 0.5522702088127090209264064e-3 * c(m-7) - 0.3265435411305071914756373e-2 * c(m-6) + 0.7462059484530855073291365e-2 * c(m-5) - 0.4748894282038492179461399e-2 * c(m-4) 0.5528052133944605740009217e-1 * c(m-6) - 0.8631683980217122275970376e-1 * c(m-5) - 0.3276463639080639163926118e-1 * c(m-7) + 0.5268984374242044588776166e-1 * c(m-4) 0.5331362125287625412555844e-1 * c(m-7) + 0.1736073411355428563685818e0 * c(m-5) + 0.8671038084174692625075159e-2 * c(m-3) + 0.8084259844422177692569663e-1 * c(m-6) - 0.1664345989168155800449120e0 * c(m-4) -0.2720908083525083608370563e-1 * c(m-7) - 0.1931148612480615118957263e0 * c(m-6) - 0.3332941113251635390801278e-1 * c(m-3) + 0.2605582646183255957264249e-3 * c(m-2) + 0.1432116665752147607469646e-2 * c(m-1) + 0.1207544072304193806052558e-2 * c(m) - 0.1348436986667115543203552e1 * c(m-5) + 0.1687723507780044227927853e-1 * c(m-4) 0.6107825764368264576481962e-2 * c(m-7) + 0.1155752633643216628010304e0 * c(m-6) + 0.2096413329579026439044119e1 * c(m-5) + 0.3357721707576477199985656e0 * c(m-3) + 0.3291545083271862858501887e-3 * c(m-2) + 0.6641183499427826101618457e-2 * c(m-1) + 0.3449455095910233625229891e-2 * c(m) + 0.8270696421223286922584620e0 * c(m-4) -0.4995827370863505253765970e-1 * c(m-6) - 0.1263507837371824205693950e1 * c(m-5) - 0.8712928907711754187084757e-2 * c(m-2) - 0.2487040599390160764166412e-1 * c(m-1) - 0.1486895819265604128572498e-1 * c(m) - 0.1726565392121567634950213e1 * c(m-4) 0.5402985338373433052255418e0 * c(m-5) - 0.1979290298620869974478871e0 * c(m-3) - 0.2760353365637712827793337e-1 * c(m-1) - 0.4671751091575462868310238e-2 * c(m) - 0.6952587985456154591014641e-1 * c(m-6) + 0.1571507277911208446562686e1 * c(m-4) -0.1319358558853174530078498e0 * c(m-5) - 0.1872196143003808021730728e0 * c(m-3) + 0.9147192682075630179962131e-2 * c(m-2) + 0.6712774475803763988977355e-1 * c(m) + 0.9630407686703666967100804e-1 * c(m-6) - 0.6882132817757726722141421e0 * c(m-4) 0.1241625568998496895352046e-1 * c(m-5) + 0.7403484645316174090533193e-1 * c(m-3) - 0.1023976547309387874453988e-2 * c(m-2) + 0.4440063948509876221050939e-1 * c(m-1) - 0.5224403464202056316702078e-1 * c(m) - 0.3213800979246298453953842e-1 * c(m-6) + 0.1178181682424363524005403e0 * c(m-4); 0.6272075574042975468177820e-3 * c(m-6) - 0.1200429618441003833696998e-1 * c(m-5) + 0.1137708862700574079015220e-1 * c(m-4) -0.5373770512016897565958305e-2 * c(m-6) + 0.1453421858063658498587377e0 * c(m-5) - 0.1399684152943489522927794e0 * c(m-4) -0.5013247356072127938999311e0 * c(m-5) + 0.5021853752328231128475915e0 * c(m-4) - 0.1197175073672143005877150e-1 * c(m-6) 0.3590669644811151307464697e-1 * c(m-6) - 0.5925443480724830632401754e0 * c(m-4) - 0.6897142765790609546343709e-2 * c(m-2) - 0.5363098747528542488971874e-2 * c(m-1) - 0.5205147429855955657625694e-2 * c(m) + 0.9977064356292750529201981e0 * c(m-5) -0.4995827370863505253765970e-1 * c(m-6) - 0.1263507837371824205693950e1 * c(m-5) - 0.8712928907711754187084757e-2 * c(m-2) - 0.2487040599390160764166412e-1 * c(m-1) - 0.1486895819265604128572498e-1 * c(m) - 0.1726565392121567634950213e1 * c(m-4) 0.2760393423824887721078848e-1 * c(m-6) + 0.1190550338687608873798462e1 * c(m-5) + 0.4253084328734353394994388e1 * c(m-4) + 0.2306367624634749229113646e0 * c(m-2) + 0.9313657638804699948929701e-1 * c(m-1) + 0.6409299775987186986730499e-1 * c(m) -0.8755807343482262259774782e0 * c(m-5) - 0.3645285178085761821545207e1 * c(m-4) + 0.1033717994630886401730470e0 * c(m-1) + 0.2013769413884797246646959e-1 * c(m) + 0.4106783858513785463625543e-1 * c(m-6) 0.3956598149904136332753521e0 * c(m-5) + 0.1630560443616104907615866e1 * c(m-4) - 0.2421320004064592721552708e0 * c(m-2) - 0.2893557395653431666593814e0 * c(m) - 0.5688529641249387985434413e-1 * c(m-6) -0.7684117160199014594442072e-1 * c(m-5) - 0.2928439026361256842196229e0 * c(m-4) + 0.2710530961648671297733465e-1 * c(m-2) - 0.1662748711097054895317080e0 * c(m-1) + 0.2251991532891353212689574e0 * c(m) + 0.1898341454096471754822498e-1 * c(m-6); 0.9209089963443799485648361e-2 * c(m-5) - 0.3129629392354775191148163e-3 * c(m-6) - 0.8896127024208321966533544e-2 * c(m-4) -0.1059816030196818445908057e0 * c(m-5) + 0.1014880675788250237247178e0 * c(m-4) + 0.4493535440856820866087846e-2 * c(m-6) 0.3328335104489738933610597e0 * c(m-5) - 0.3179803804558436283847901e0 * c(m-4) - 0.5111353189352474549563559e-2 * c(m-3) - 0.9741776803777790426705996e-2 * c(m-6) 0.7272438906214475928744770e-1 * c(m-4) + 0.1964682777744275219350831e-1 * c(m-3) - 0.5952475275883259619711594e-2 * c(m-1) - 0.1635430866921887819487473e-2 * c(m) + 0.2921234010758621482958052e-1 * c(m-6) - 0.4659516693228870973898560e0 * c(m-5) 0.5402985338373433052255418e0 * c(m-5) - 0.1979290298620869974478871e0 * c(m-3) - 0.2760353365637712827793337e-1 * c(m-1) - 0.4671751091575462868310238e-2 * c(m) - 0.6952587985456154591014641e-1 * c(m-6) + 0.1571507277911208446562686e1 * c(m-4) -0.8755807343482262259774782e0 * c(m-5) - 0.3645285178085761821545207e1 * c(m-4) + 0.1033717994630886401730470e0 * c(m-1) + 0.2013769413884797246646959e-1 * c(m) + 0.4106783858513785463625543e-1 * c(m-6) 0.1070920689960817104203947e1 * c(m-5) + 0.3717418466925056542408153e1 * c(m-4) + 0.1166740554279680007487795e0 * c(m-3) + 0.1147318200715868527529827e0 * c(m-1) + 0.6327161147136873807796515e-2 * c(m) + 0.6235373239336055200426697e-1 * c(m-6) -0.6639605248735044787146222e0 * c(m-5) - 0.1865625445986772763641423e1 * c(m-4) + 0.1103611313171476425250639e0 * c(m-3) - 0.9091410269992464604926176e-1 * c(m) - 0.8636954541126674177407762e-1 * c(m-6) 0.1582127073537215443965653e0 * c(m-5) + 0.3746489300753517635549495e0 * c(m-4) - 0.4364163147111892346990101e-1 * c(m-3) - 0.1845476106024151050283847e0 * c(m-1) + 0.7075642937243715046279337e-1 * c(m) + 0.2882271848190011329385407e-1 * c(m-6); -0.3576545132696983143406173e-2 * c(m-5) + 0.4335019854436220306755673e-3 * c(m-6) + 0.3143043147253361112730605e-2 * c(m-4) 0.4093499466767054661591066e-1 * c(m-5) - 0.3471075437892810033585296e-1 * c(m-4) - 0.6224240288742446280057699e-2 * c(m-6) -0.1080774142196007991746827e0 * c(m-5) + 0.9941834083648937298100811e-1 * c(m-4) - 0.4834791406446907590553793e-2 * c(m-3) + 0.1349386478955833378422842e-1 * c(m-6) 0.5891947149681041048896399e-1 * c(m-4) + 0.1858378996391679448655070e-1 * c(m-3) + 0.7240905383565181316381731e-2 * c(m-2) + 0.2349927974590068869356781e-1 * c(m) - 0.4046360079256766884300687e-1 * c(m-6) + 0.1223513270418807666970488e0 * c(m-5) -0.1319358558853174530078498e0 * c(m-5) - 0.1872196143003808021730728e0 * c(m-3) + 0.9147192682075630179962131e-2 * c(m-2) + 0.6712774475803763988977355e-1 * c(m) + 0.9630407686703666967100804e-1 * c(m-6) - 0.6882132817757726722141421e0 * c(m-4) 0.3956598149904136332753521e0 * c(m-5) + 0.1630560443616104907615866e1 * c(m-4) - 0.2421320004064592721552708e0 * c(m-2) - 0.2893557395653431666593814e0 * c(m) - 0.5688529641249387985434413e-1 * c(m-6) -0.6639605248735044787146222e0 * c(m-5) - 0.1865625445986772763641423e1 * c(m-4) + 0.1103611313171476425250639e0 * c(m-3) - 0.9091410269992464604926176e-1 * c(m) - 0.8636954541126674177407762e-1 * c(m-6) 0.4681819359722749441073885e0 * c(m-5) + 0.1015239189167790053447110e1 * c(m-4) + 0.1043897828092562609502636e0 * c(m-3) + 0.2542001760457345743492403e0 * c(m-2) + 0.1306332157111667628555907e1 * c(m) + 0.1196351539550049336518187e0 * c(m-6) -0.1195777325611201766551392e0 * c(m-5) - 0.2187310061229745694542609e0 * c(m-4) - 0.4128029838349298819825156e-1 * c(m-3) - 0.2845627370491611369031341e-1 * c(m-2) - 0.1016689339350338144430605e1 * c(m) - 0.3992391469197282238624438e-1 * c(m-6); 0.5593983696629863059347067e-3 * c(m-5) - 0.1446656414398166805849327e-3 * c(m-6) - 0.4147327282231696253497740e-3 * c(m-4) -0.6486184157331537899459796e-2 * c(m-5) + 0.4409068609809831485979484e-2 * c(m-4) + 0.2077115547521706413480312e-2 * c(m-6) 0.1319674981073749167009902e-1 * c(m-5) - 0.1060554802883657391328704e-1 * c(m-4) + 0.1911888563316170927411831e-2 * c(m-3) - 0.4503090345217088684223814e-2 * c(m-6) -0.2404661162020836566908542e-1 * c(m-4) - 0.7348845587775519698437916e-2 * c(m-3) - 0.8105784530576404277872603e-3 * c(m-2) + 0.9574633163221758060736592e-2 * c(m-1) - 0.1828896813877197352675410e-1 * c(m) + 0.1350326632905990039353503e-1 * c(m-6) - 0.1315967038382618382356495e-1 * c(m-5) 0.1241625568998496895352046e-1 * c(m-5) + 0.7403484645316174090533193e-1 * c(m-3) - 0.1023976547309387874453988e-2 * c(m-2) + 0.4440063948509876221050939e-1 * c(m-1) - 0.5224403464202056316702078e-1 * c(m) - 0.3213800979246298453953842e-1 * c(m-6) + 0.1178181682424363524005403e0 * c(m-4) -0.7684117160199014594442072e-1 * c(m-5) - 0.2928439026361256842196229e0 * c(m-4) + 0.2710530961648671297733465e-1 * c(m-2) - 0.1662748711097054895317080e0 * c(m-1) + 0.2251991532891353212689574e0 * c(m) + 0.1898341454096471754822498e-1 * c(m-6) 0.1582127073537215443965653e0 * c(m-5) + 0.3746489300753517635549495e0 * c(m-4) - 0.4364163147111892346990101e-1 * c(m-3) - 0.1845476106024151050283847e0 * c(m-1) + 0.7075642937243715046279337e-1 * c(m) + 0.2882271848190011329385407e-1 * c(m-6) -0.1195777325611201766551392e0 * c(m-5) - 0.2187310061229745694542609e0 * c(m-4) - 0.4128029838349298819825156e-1 * c(m-3) - 0.2845627370491611369031341e-1 * c(m-2) - 0.1016689339350338144430605e1 * c(m) - 0.3992391469197282238624438e-1 * c(m-6) 0.3167964748016105299646518e-1 * c(m-5) + 0.4976563420877041544013670e-1 * c(m-4) + 0.1632404042590951953384672e-1 * c(m-3) + 0.3185519088796429015220016e-2 * c(m-2) + 0.2968472090638000742888467e0 * c(m-1) + 0.7912667594695582093926295e0 * c(m) + 0.1332316557164627464149716e-1 * c(m-6);]; + +M(5,10)=M(10,5); +M(m-4,m-9)=M(m-9,m-4); + +M=M/h; + +D2=HI*(-M-diag(c)*e_1*S_1+diag(c)*e_m*S_m); + +S2_U=[0.35e2 / 0.12e2 -0.26e2 / 0.3e1 0.19e2 / 0.2e1 -0.14e2 / 0.3e1 0.11e2 / 0.12e2;]/h^2; +S2_1=zeros(1,m); +S2_1(1:5)=S2_U; +S2_m=zeros(1,m); +S2_m(m-4:m)=fliplr(S2_U); + + + + + +% Fourth derivative, 1th order accurate at first 8 boundary points (still +% yield 5th order convergence if stable: for example u_tt=-u_xxxx + +m4=7/240;m3=-2/5;m2=169/60;m1=-122/15;m0=91/8; +M4=m4*(diag(ones(m-4,1),4)+diag(ones(m-4,1),-4))+m3*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3))+m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0); + +%M4=(-1/6*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3) ) + 2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2)) -13/2*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)) + 28/3*diag(ones(m,1),0)); + +M4_U=[0.3504379e7 / 0.907200e6 -0.4613983e7 / 0.453600e6 0.4260437e7 / 0.453600e6 -0.418577e6 / 0.113400e6 0.524579e6 / 0.907200e6 0.535e3 / 0.18144e5; -0.4613983e7 / 0.453600e6 0.5186159e7 / 0.181440e6 -0.81121e5 / 0.2835e4 0.218845e6 / 0.18144e5 -0.159169e6 / 0.90720e5 -0.94669e5 / 0.907200e6; 0.4260437e7 / 0.453600e6 -0.81121e5 / 0.2835e4 0.147695e6 / 0.4536e4 -0.384457e6 / 0.22680e5 0.339653e6 / 0.90720e5 -0.18233e5 / 0.113400e6; -0.418577e6 / 0.113400e6 0.218845e6 / 0.18144e5 -0.384457e6 / 0.22680e5 0.65207e5 / 0.4536e4 -0.22762e5 / 0.2835e4 0.1181753e7 / 0.453600e6; 0.524579e6 / 0.907200e6 -0.159169e6 / 0.90720e5 0.339653e6 / 0.90720e5 -0.22762e5 / 0.2835e4 0.2006171e7 / 0.181440e6 -0.3647647e7 / 0.453600e6; 0.535e3 / 0.18144e5 -0.94669e5 / 0.907200e6 -0.18233e5 / 0.113400e6 0.1181753e7 / 0.453600e6 -0.3647647e7 / 0.453600e6 0.10305271e8 / 0.907200e6;]; + +M4(1:6,1:6)=M4_U; + +M4(m-5:m,m-5:m)=flipud( fliplr( M4_U ) ); +M4=M4/h^3; + +S3_U=[-0.5e1 / 0.2e1 9 -12 7 -0.3e1 / 0.2e1;]/h^3; +S3_1=zeros(1,m); +S3_1(1:5)=S3_U; +S3_m=zeros(1,m); +S3_m(m-4:m)=fliplr(-S3_U); + +D4=HI*(M4-e_1*S3_1+e_m*S3_m + S_1'*S2_1-S_m'*S2_m); + +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/higher_variable8_higher_boundary_order.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,103 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% 8:te ordn. SBP Finita differens %%% +%%% operatorer med diagonal norm %%% +%%% %%% +%%% %%% +%%% H (Normen) %%% +%%% D1=H^(-1)Q (approx f?rsta derivatan) %%% +%%% D2 (approx andra derivatan) %%% +%%% D2=HI*(R+C*D*S %%% +%%% %%% +%%% R=-D1'*H*C*D1-RR %%% +%%% %%% +%%% RR ?r dissipation) %%% +%%% Dissipationen uppbyggd av D4: %%% +%%% DI=D4*B*H*D4 %%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%This is 3rd order accurate at the boundary. Not same norm as D1 operator + +%m=10; %problemstorlek +%h=1/(m-1); + +% x=1:h:m*h;x=x'; +x0=x.^0/factorial(1); +x1=x.^1/factorial(1); +x2=x.^2/factorial(2); +x3=x.^3/factorial(3); +x4=x.^4/factorial(4); +x5=x.^5/factorial(5); +x6=x.^6/factorial(6); +x7=x.^7/factorial(7); + +H=diag(ones(m,1),0); +H(1:8,1:8)=[0.7488203e7 / 0.25401600e8 0 0 0 0 0 0 0; 0 0.5539027e7 / 0.3628800e7 0 0 0 0 0 0; 0 0 0.308923e6 / 0.1209600e7 0 0 0 0 0; 0 0 0 0.1307491e7 / 0.725760e6 0 0 0 0; 0 0 0 0 0.59407e5 / 0.145152e6 0 0 0; 0 0 0 0 0 0.1548947e7 / 0.1209600e7 0 0; 0 0 0 0 0 0 0.3347963e7 / 0.3628800e7 0; 0 0 0 0 0 0 0 0.25641187e8 / 0.25401600e8;]; + +H(m-7:m,m-7:m)=fliplr(flipud(H(1:8,1:8))); + +e_1=zeros(m,1);e_1(1)=1; +e_m=zeros(m,1);e_m(m)=1; + +S_U=[-0.49e2 / 0.20e2 6 -0.15e2 / 0.2e1 0.20e2 / 0.3e1 -0.15e2 / 0.4e1 0.6e1 / 0.5e1 -0.1e1 / 0.6e1;]/h; +S_1=zeros(1,m); +S_1(1:7)=S_U; +S_m=zeros(1,m); +S_m(m-6:m)=fliplr(-S_U); + +H=h*H; +HI=inv(H); + + +% M=zeros(m,m); +% +% for i=6:m-5 +% M(i,i-3:i+3)=[c(i-2) / 0.40e2 + c(i-1) / 0.40e2 - 0.11e2 / 0.360e3 * c(i-3) - 0.11e2 / 0.360e3 * c(i) c(i-3) / 0.20e2 - 0.3e1 / 0.10e2 * c(i-1) + c(i+1) / 0.20e2 + 0.7e1 / 0.40e2 * c(i) + 0.7e1 / 0.40e2 * c(i-2) -c(i-3) / 0.40e2 - 0.3e1 / 0.10e2 * c(i-2) - 0.3e1 / 0.10e2 * c(i+1) - c(i+2) / 0.40e2 - 0.17e2 / 0.40e2 * c(i) - 0.17e2 / 0.40e2 * c(i-1) c(i-3) / 0.180e3 + c(i-2) / 0.8e1 + 0.19e2 / 0.20e2 * c(i-1) + 0.19e2 / 0.20e2 * c(i+1) + c(i+2) / 0.8e1 + c(i+3) / 0.180e3 + 0.101e3 / 0.180e3 * c(i) -c(i-2) / 0.40e2 - 0.3e1 / 0.10e2 * c(i-1) - 0.3e1 / 0.10e2 * c(i+2) - c(i+3) / 0.40e2 - 0.17e2 / 0.40e2 * c(i) - 0.17e2 / 0.40e2 * c(i+1) c(i-1) / 0.20e2 - 0.3e1 / 0.10e2 * c(i+1) + c(i+3) / 0.20e2 + 0.7e1 / 0.40e2 * c(i) + 0.7e1 / 0.40e2 * c(i+2) c(i+1) / 0.40e2 + c(i+2) / 0.40e2 - 0.11e2 / 0.360e3 * c(i) - 0.11e2 / 0.360e3 * c(i+3);]; +% end +% +% +% M(1:9,1:9)=[0.7912667594695582093926295e0 * c(1) + 0.2968472090638000742888467e0 * c(2) + 0.3185519088796429015220016e-2 * c(3) + 0.1632404042590951953384672e-1 * c(4) + 0.3160302244094415087693968e-1 * c(5) + 0.3167964748016105299646518e-1 * c(6) + 0.3148577733947253920469418e-1 * c(7) -0.1016689339350338144430605e1 * c(1) - 0.2845627370491611369031341e-1 * c(3) - 0.4128029838349298819825156e-1 * c(4) - 0.1392281451620140507549866e0 * c(5) - 0.1195777325611201766551392e0 * c(6) - 0.1194267756529333410855186e0 * c(7) 0.7075642937243715046279337e-1 * c(1) - 0.1845476106024151050283847e0 * c(2) - 0.4364163147111892346990101e-1 * c(4) + 0.2432367907207732460874765e0 * c(5) + 0.1582127073537215443965653e0 * c(6) + 0.1602348578364786307613271e0 * c(7) 0.2251991532891353212689574e0 * c(1) - 0.1662748711097054895317080e0 * c(2) + 0.2710530961648671297733465e-1 * c(3) - 0.1916646185968439909125616e0 * c(5) - 0.7684117160199014594442072e-1 * c(6) - 0.8219586949831697575883635e-1 * c(7) -0.5224403464202056316702078e-1 * c(1) + 0.4440063948509876221050939e-1 * c(2) - 0.1023976547309387874453988e-2 * c(3) + 0.7403484645316174090533193e-1 * c(4) + 0.1241625568998496895352046e-1 * c(6) + 0.7188652847892601282652228e-1 * c(5) + 0.1379362997104735503447960e-1 * c(7) -0.1828896813877197352675410e-1 * c(1) + 0.9574633163221758060736592e-2 * c(2) - 0.8105784530576404277872603e-3 * c(3) - 0.7348845587775519698437916e-2 * c(4) + 0.1063601949723906997026904e-1 * c(5) - 0.1315967038382618382356495e-1 * c(6) - 0.2117936478838753524581943e-1 * c(7) 0.1911888563316170927411831e-2 * c(4) - 0.4068130355529149936100229e-1 * c(5) + 0.1319674981073749167009902e-1 * c(6) + 0.2557266518123783676349144e-1 * c(7) 0.1559652871136785763960685e-1 * c(5) - 0.6486184157331537899459796e-2 * c(6) - 0.9110344554036319740147054e-2 * c(7) 0.5593983696629863059347067e-3 * c(6) - 0.1384822535100796372263822e-2 * c(5) + 0.8254241654378100663291154e-3 * c(7); -0.1016689339350338144430605e1 * c(1) - 0.2845627370491611369031341e-1 * c(3) - 0.4128029838349298819825156e-1 * c(4) - 0.1392281451620140507549866e0 * c(5) - 0.1195777325611201766551392e0 * c(6) - 0.1194267756529333410855186e0 * c(7) 0.1306332157111667628555907e1 * c(1) + 0.2542001760457345743492403e0 * c(3) + 0.1043897828092562609502636e0 * c(4) + 0.6672328021032112950919876e0 * c(5) + 0.4681819359722749441073885e0 * c(6) + 0.4676415410195836920069412e0 * c(7) -0.9091410269992464604926176e-1 * c(1) + 0.1103611313171476425250639e0 * c(4) - 0.1290397544997518887000350e1 * c(5) - 0.6639605248735044787146222e0 * c(6) - 0.6615974464005206184151509e0 * c(7) -0.2893557395653431666593814e0 * c(1) - 0.2421320004064592721552708e0 * c(3) + 0.1187670255028031027693374e1 * c(5) + 0.3956598149904136332753521e0 * c(6) + 0.3860048921755800000681479e0 * c(7) 0.6712774475803763988977355e-1 * c(1) + 0.9147192682075630179962131e-2 * c(3) - 0.1872196143003808021730728e0 * c(4) - 0.1319358558853174530078498e0 * c(6) - 0.4871575736811911887376923e0 * c(5) - 0.1047516312275448138054418e0 * c(7) 0.2349927974590068869356781e-1 * c(1) + 0.7240905383565181316381731e-2 * c(3) + 0.1858378996391679448655070e-1 * c(4) - 0.9289616133938676174345208e-1 * c(5) + 0.1223513270418807666970488e0 * c(6) + 0.1113520320436295033894092e0 * c(7) -0.4834791406446907590553793e-2 * c(4) + 0.2310683832687820403062715e0 * c(5) - 0.1080774142196007991746827e0 * c(6) - 0.1181561776427343335410349e0 * c(7) -0.8368141434403455353724691e-1 * c(5) + 0.4093499466767054661591066e-1 * c(6) + 0.4274641967636400692133626e-1 * c(7) -0.3576545132696983143406173e-2 * c(6) + 0.7389399124121078682094445e-2 * c(5) - 0.3812853991424095538688273e-2 * c(7); 0.7075642937243715046279337e-1 * c(1) - 0.1845476106024151050283847e0 * c(2) - 0.4364163147111892346990101e-1 * c(4) + 0.2432367907207732460874765e0 * c(5) + 0.1582127073537215443965653e0 * c(6) + 0.1602348578364786307613271e0 * c(7) -0.9091410269992464604926176e-1 * c(1) + 0.1103611313171476425250639e0 * c(4) - 0.1290397544997518887000350e1 * c(5) - 0.6639605248735044787146222e0 * c(6) - 0.6615974464005206184151509e0 * c(7) 0.6327161147136873807796515e-2 * c(1) + 0.1147318200715868527529827e0 * c(2) + 0.1166740554279680007487795e0 * c(4) + 0.2766610808285444037240703e1 * c(5) + 0.1070920689960817104203947e1 * c(6) + 0.1013161391032973057171717e1 * c(7) 0.2013769413884797246646959e-1 * c(1) + 0.1033717994630886401730470e0 * c(2) - 0.2913221621151742724258117e1 * c(5) - 0.8755807343482262259774782e0 * c(6) - 0.6909957183488812426508351e0 * c(7) -0.4671751091575462868310238e-2 * c(1) - 0.2760353365637712827793337e-1 * c(2) - 0.1979290298620869974478871e0 * c(4) + 0.5402985338373433052255418e0 * c(6) + 0.1239177593031911077924537e1 * c(5) + 0.2628038050247358227280031e0 * c(7) -0.1635430866921887819487473e-2 * c(1) - 0.5952475275883259619711594e-2 * c(2) + 0.1964682777744275219350831e-1 * c(4) + 0.3236640012639046600590714e0 * c(5) - 0.4659516693228870973898560e0 * c(6) - 0.2217272720941736859420432e0 * c(7) -0.5111353189352474549563559e-2 * c(4) - 0.5355878163774754346032096e0 * c(5) + 0.3328335104489738933610597e0 * c(6) + 0.2078656591178540157917135e0 * c(7) 0.1824328174134289562208038e0 * c(5) - 0.1059816030196818445908057e0 * c(6) - 0.7645121439374711162999809e-1 * c(7) 0.9209089963443799485648361e-2 * c(6) - 0.1591502818872493167091475e-1 * c(5) + 0.6705938225281132185266388e-2 * c(7); 0.2251991532891353212689574e0 * c(1) - 0.1662748711097054895317080e0 * c(2) + 0.2710530961648671297733465e-1 * c(3) - 0.1916646185968439909125616e0 * c(5) - 0.7684117160199014594442072e-1 * c(6) - 0.8219586949831697575883635e-1 * c(7) -0.2893557395653431666593814e0 * c(1) - 0.2421320004064592721552708e0 * c(3) + 0.1187670255028031027693374e1 * c(5) + 0.3956598149904136332753521e0 * c(6) + 0.3860048921755800000681479e0 * c(7) 0.2013769413884797246646959e-1 * c(1) + 0.1033717994630886401730470e0 * c(2) - 0.2913221621151742724258117e1 * c(5) - 0.8755807343482262259774782e0 * c(6) - 0.6909957183488812426508351e0 * c(7) 0.6409299775987186986730499e-1 * c(1) + 0.9313657638804699948929701e-1 * c(2) + 0.2306367624634749229113646e0 * c(3) + 0.3689440308283716620260816e1 * c(5) + 0.1190550338687608873798462e1 * c(6) + 0.5912479546888856519443605e0 * c(7) -0.1486895819265604128572498e-1 * c(1) - 0.2487040599390160764166412e-1 * c(2) - 0.8712928907711754187084757e-2 * c(3) - 0.1263507837371824205693950e1 * c(6) - 0.3058317397843997326920898e0 * c(7) - 0.1470691926045802954795783e1 * c(5) -0.5205147429855955657625694e-2 * c(1) - 0.5363098747528542488971874e-2 * c(2) - 0.6897142765790609546343709e-2 * c(3) - 0.7857524521667450101721993e0 * c(5) + 0.2291148005423734600066709e0 * c(7) + 0.9977064356292750529201981e0 * c(6) 0.6697297488067662265210608e0 * c(5) - 0.5013247356072127938999311e0 * c(6) - 0.1795161243106645437322408e0 * c(7) -0.2022909060111751565150958e0 * c(5) + 0.1453421858063658498587377e0 * c(6) + 0.5694872020480930665635812e-1 * c(7) -0.1200429618441003833696998e-1 * c(6) - 0.4776915669385923841535432e-2 * c(7) + 0.1678121185379596217850541e-1 * c(5); -0.5224403464202056316702078e-1 * c(1) + 0.4440063948509876221050939e-1 * c(2) - 0.1023976547309387874453988e-2 * c(3) + 0.7403484645316174090533193e-1 * c(4) + 0.1241625568998496895352046e-1 * c(6) + 0.7188652847892601282652228e-1 * c(5) + 0.1379362997104735503447960e-1 * c(7) 0.6712774475803763988977355e-1 * c(1) + 0.9147192682075630179962131e-2 * c(3) - 0.1872196143003808021730728e0 * c(4) - 0.1319358558853174530078498e0 * c(6) - 0.4871575736811911887376923e0 * c(5) - 0.1047516312275448138054418e0 * c(7) -0.4671751091575462868310238e-2 * c(1) - 0.2760353365637712827793337e-1 * c(2) - 0.1979290298620869974478871e0 * c(4) + 0.5402985338373433052255418e0 * c(6) + 0.1239177593031911077924537e1 * c(5) + 0.2628038050247358227280031e0 * c(7) -0.1486895819265604128572498e-1 * c(1) - 0.2487040599390160764166412e-1 * c(2) - 0.8712928907711754187084757e-2 * c(3) - 0.1263507837371824205693950e1 * c(6) - 0.3058317397843997326920898e0 * c(7) - 0.1470691926045802954795783e1 * c(5) 0.3449455095910233625229891e-2 * c(1) + 0.6641183499427826101618457e-2 * c(2) + 0.3291545083271862858501887e-3 * c(3) + 0.3357721707576477199985656e0 * c(4) + 0.2096413329579026439044119e1 * c(6) + 0.2317323204183126854954203e0 * c(7) + 0.6107825764368264576481962e-2 * c(8) + 0.7109125850683376695640722e0 * c(5) 0.1207544072304193806052558e-2 * c(1) + 0.1432116665752147607469646e-2 * c(2) + 0.2605582646183255957264249e-3 * c(3) - 0.3332941113251635390801278e-1 * c(4) - 0.2808241697385532683612407e0 * c(7) - 0.2720908083525083608370563e-1 * c(8) + 0.1045865435682921987447929e0 * c(5) - 0.1348436986667115543203552e1 * c(6) 0.8671038084174692625075159e-2 * c(4) + 0.1736073411355428563685818e0 * c(6) + 0.5331362125287625412555844e-1 * c(8) - 0.2424935262404526301801157e0 * c(5) + 0.1569015257678588270609004e0 * c(7) -0.8631683980217122275970376e-1 * c(6) + 0.2698842360470999243492629e-1 * c(7) + 0.8098194147715651085292754e-1 * c(5) - 0.3276463639080639163926118e-1 * c(8) 0.7462059484530855073291365e-2 * c(6) - 0.8121640361668678949573496e-3 * c(7) + 0.5522702088127090209264064e-3 * c(8) - 0.7202165657176696199260422e-2 * c(5); -0.1828896813877197352675410e-1 * c(1) + 0.9574633163221758060736592e-2 * c(2) - 0.8105784530576404277872603e-3 * c(3) - 0.7348845587775519698437916e-2 * c(4) + 0.1063601949723906997026904e-1 * c(5) - 0.1315967038382618382356495e-1 * c(6) - 0.2117936478838753524581943e-1 * c(7) 0.2349927974590068869356781e-1 * c(1) + 0.7240905383565181316381731e-2 * c(3) + 0.1858378996391679448655070e-1 * c(4) - 0.9289616133938676174345208e-1 * c(5) + 0.1223513270418807666970488e0 * c(6) + 0.1113520320436295033894092e0 * c(7) -0.1635430866921887819487473e-2 * c(1) - 0.5952475275883259619711594e-2 * c(2) + 0.1964682777744275219350831e-1 * c(4) + 0.3236640012639046600590714e0 * c(5) - 0.4659516693228870973898560e0 * c(6) - 0.2217272720941736859420432e0 * c(7) -0.5205147429855955657625694e-2 * c(1) - 0.5363098747528542488971874e-2 * c(2) - 0.6897142765790609546343709e-2 * c(3) - 0.7857524521667450101721993e0 * c(5) + 0.2291148005423734600066709e0 * c(7) + 0.9977064356292750529201981e0 * c(6) 0.1207544072304193806052558e-2 * c(1) + 0.1432116665752147607469646e-2 * c(2) + 0.2605582646183255957264249e-3 * c(3) - 0.3332941113251635390801278e-1 * c(4) - 0.2808241697385532683612407e0 * c(7) - 0.2720908083525083608370563e-1 * c(8) + 0.1045865435682921987447929e0 * c(5) - 0.1348436986667115543203552e1 * c(6) 0.4227226173449345042468960e-3 * c(1) + 0.3088241944378964404772302e-3 * c(2) + 0.2062575706647430620228133e-3 * c(3) + 0.3308343404200968256656458e-2 * c(4) + 0.5828047016405001815804837e0 * c(5) + 0.8054174220366215473556835e0 * c(7) + 0.1338363233410033443348225e0 * c(8) + 0.5555555555555555555555556e-2 * c(9) + 0.1190362071861893051132274e1 * c(6) -0.8607044252686413302647675e-3 * c(4) - 0.1748074708673904989293256e0 * c(5) - 0.3132544850115050165022338e0 * c(8) - 0.2500000000000000000000000e-1 * c(9) - 0.3169166305310429271303167e0 * c(7) - 0.6691607091647929161078591e0 * c(6) 0.3354661791693352108660900e-1 * c(5) - 0.3343620022386971405018586e0 * c(7) + 0.5000000000000000000000000e-1 * c(9) + 0.2169790609807602750804271e0 * c(6) + 0.1838363233410033443348225e0 * c(8) 0.2912518476823004642951502e-1 * c(7) + 0.2279091916474916391629437e-1 * c(8) - 0.3068985997518740530511593e-1 * c(6) - 0.1781799513347360596249022e-2 * c(5) - 0.3055555555555555555555556e-1 * c(9); 0.1911888563316170927411831e-2 * c(4) - 0.4068130355529149936100229e-1 * c(5) + 0.1319674981073749167009902e-1 * c(6) + 0.2557266518123783676349144e-1 * c(7) -0.4834791406446907590553793e-2 * c(4) + 0.2310683832687820403062715e0 * c(5) - 0.1080774142196007991746827e0 * c(6) - 0.1181561776427343335410349e0 * c(7) -0.5111353189352474549563559e-2 * c(4) - 0.5355878163774754346032096e0 * c(5) + 0.3328335104489738933610597e0 * c(6) + 0.2078656591178540157917135e0 * c(7) 0.6697297488067662265210608e0 * c(5) - 0.5013247356072127938999311e0 * c(6) - 0.1795161243106645437322408e0 * c(7) 0.8671038084174692625075159e-2 * c(4) + 0.1736073411355428563685818e0 * c(6) + 0.5331362125287625412555844e-1 * c(8) - 0.2424935262404526301801157e0 * c(5) + 0.1569015257678588270609004e0 * c(7) -0.8607044252686413302647675e-3 * c(4) - 0.1748074708673904989293256e0 * c(5) - 0.3132544850115050165022338e0 * c(8) - 0.2500000000000000000000000e-1 * c(9) - 0.3169166305310429271303167e0 * c(7) - 0.6691607091647929161078591e0 * c(6) 0.2239223735771599178951297e-3 * c(4) + 0.1275437785430956673825710e0 * c(5) + 0.1011699483929608164601067e1 * c(6) + 0.9698817275172575247533506e0 * c(8) + 0.1250000000000000000000000e0 * c(9) + 0.5555555555555555555555556e-2 * c(10) + 0.4823177543031281500117826e0 * c(7) -0.3784113973033012949863031e-1 * c(5) - 0.2997556885134827361576001e0 * c(6) - 0.3000000000000000000000000e0 * c(9) - 0.2500000000000000000000000e-1 * c(10) - 0.3991486867446821178415359e0 * c(7) - 0.4382544850115050165022338e0 * c(8) 0.4698146218022683933926520e-1 * c(6) - 0.2966863787471237458744416e0 * c(8) + 0.5000000000000000000000000e-1 * c(10) + 0.1716355704146006481727960e0 * c(7) + 0.3069346152296258362380356e-2 * c(5) + 0.1750000000000000000000000e0 * c(9); 0.1559652871136785763960685e-1 * c(5) - 0.6486184157331537899459796e-2 * c(6) - 0.9110344554036319740147054e-2 * c(7) -0.8368141434403455353724691e-1 * c(5) + 0.4093499466767054661591066e-1 * c(6) + 0.4274641967636400692133626e-1 * c(7) 0.1824328174134289562208038e0 * c(5) - 0.1059816030196818445908057e0 * c(6) - 0.7645121439374711162999809e-1 * c(7) -0.2022909060111751565150958e0 * c(5) + 0.1453421858063658498587377e0 * c(6) + 0.5694872020480930665635812e-1 * c(7) -0.8631683980217122275970376e-1 * c(6) + 0.2698842360470999243492629e-1 * c(7) + 0.8098194147715651085292754e-1 * c(5) - 0.3276463639080639163926118e-1 * c(8) 0.3354661791693352108660900e-1 * c(5) - 0.3343620022386971405018586e0 * c(7) + 0.5000000000000000000000000e-1 * c(9) + 0.2169790609807602750804271e0 * c(6) + 0.1838363233410033443348225e0 * c(8) -0.3784113973033012949863031e-1 * c(5) - 0.2997556885134827361576001e0 * c(6) - 0.3000000000000000000000000e0 * c(9) - 0.2500000000000000000000000e-1 * c(10) - 0.3991486867446821178415359e0 * c(7) - 0.4382544850115050165022338e0 * c(8) 0.1230328942716804455358698e-1 * c(5) + 0.1183647529645898332481833e0 * c(6) + 0.9410511898227943334189628e0 * c(7) + 0.9500000000000000000000000e0 * c(9) + 0.1250000000000000000000000e0 * c(10) + 0.5555555555555555555555556e-2 * c(11) + 0.5699474344521144554459336e0 * c(8) -0.2308067892671916339568942e-1 * c(6) - 0.2986625053775149497180439e0 * c(7) - 0.3000000000000000000000000e0 * c(10) - 0.2500000000000000000000000e-1 * c(11) - 0.1047734860515050802561078e-2 * c(5) - 0.4272090808352508360837056e0 * c(8) - 0.4250000000000000000000000e0 * c(9); 0.5593983696629863059347067e-3 * c(6) - 0.1384822535100796372263822e-2 * c(5) + 0.8254241654378100663291154e-3 * c(7) -0.3576545132696983143406173e-2 * c(6) + 0.7389399124121078682094445e-2 * c(5) - 0.3812853991424095538688273e-2 * c(7) 0.9209089963443799485648361e-2 * c(6) - 0.1591502818872493167091475e-1 * c(5) + 0.6705938225281132185266388e-2 * c(7) -0.1200429618441003833696998e-1 * c(6) - 0.4776915669385923841535432e-2 * c(7) + 0.1678121185379596217850541e-1 * c(5) 0.7462059484530855073291365e-2 * c(6) - 0.8121640361668678949573496e-3 * c(7) + 0.5522702088127090209264064e-3 * c(8) - 0.7202165657176696199260422e-2 * c(5) 0.2912518476823004642951502e-1 * c(7) + 0.2279091916474916391629437e-1 * c(8) - 0.3068985997518740530511593e-1 * c(6) - 0.1781799513347360596249022e-2 * c(5) - 0.3055555555555555555555556e-1 * c(9) 0.4698146218022683933926520e-1 * c(6) - 0.2966863787471237458744416e0 * c(8) + 0.5000000000000000000000000e-1 * c(10) + 0.1716355704146006481727960e0 * c(7) + 0.3069346152296258362380356e-2 * c(5) + 0.1750000000000000000000000e0 * c(9) -0.2308067892671916339568942e-1 * c(6) - 0.2986625053775149497180439e0 * c(7) - 0.3000000000000000000000000e0 * c(10) - 0.2500000000000000000000000e-1 * c(11) - 0.1047734860515050802561078e-2 * c(5) - 0.4272090808352508360837056e0 * c(8) - 0.4250000000000000000000000e0 * c(9) 0.5139370221149109977041877e-2 * c(6) + 0.1247723215009422001393184e0 * c(7) + 0.9505522702088127090209264e0 * c(8) + 0.9500000000000000000000000e0 * c(10) + 0.1250000000000000000000000e0 * c(11) + 0.5555555555555555555555556e-2 * c(12) + 0.9159362465153641826887659e-4 * c(5) + 0.5611111111111111111111111e0 * c(9);]; +% +% M(m-8:m,m-8:m)=[0.5555555555555555555555556e-2 * c(m-11) + 0.1250000000000000000000000e0 * c(m-10) + 0.9500000000000000000000000e0 * c(m-9) + 0.9505522702088127090209264e0 * c(m-7) + 0.1247205076844361998744053e0 * c(m-6) + 0.5139370221149109977041877e-2 * c(m-5) + 0.5611111111111111111111111e0 * c(m-8) + 0.1434074411575366831819799e-3 * c(m-4) -0.2500000000000000000000000e-1 * c(m-10) - 0.3000000000000000000000000e0 * c(m-9) - 0.2980649679116425253322056e0 * c(m-6) - 0.2308067892671916339568942e-1 * c(m-5) - 0.4250000000000000000000000e0 * c(m-8) - 0.4272090808352508360837056e0 * c(m-7) - 0.1645272326387475188399322e-2 * c(m-4) 0.5000000000000000000000000e-1 * c(m-9) - 0.2966863787471237458744416e0 * c(m-7) + 0.4698146218022683933926520e-1 * c(m-5) + 0.1750000000000000000000000e0 * c(m-8) + 0.1700291833903489463825077e0 * c(m-6) + 0.4675733176547960152668626e-2 * c(m-4) 0.2279091916474916391629437e-1 * c(m-7) + 0.3097763128598982561225538e-1 * c(m-6) - 0.3055555555555555555555556e-1 * c(m-8) - 0.3068985997518740530511593e-1 * c(m-5) - 0.3634246031107139778989373e-2 * c(m-4) 0.5522702088127090209264064e-3 * c(m-7) - 0.3265435411305071914756373e-2 * c(m-6) + 0.7462059484530855073291365e-2 * c(m-5) - 0.4748894282038492179461399e-2 * c(m-4) 0.6272075574042975468177820e-3 * c(m-6) - 0.1200429618441003833696998e-1 * c(m-5) + 0.1137708862700574079015220e-1 * c(m-4) 0.9209089963443799485648361e-2 * c(m-5) - 0.3129629392354775191148163e-3 * c(m-6) - 0.8896127024208321966533544e-2 * c(m-4) -0.3576545132696983143406173e-2 * c(m-5) + 0.4335019854436220306755673e-3 * c(m-6) + 0.3143043147253361112730605e-2 * c(m-4) 0.5593983696629863059347067e-3 * c(m-5) - 0.1446656414398166805849327e-3 * c(m-6) - 0.4147327282231696253497740e-3 * c(m-4); -0.2500000000000000000000000e-1 * c(m-10) - 0.3000000000000000000000000e0 * c(m-9) - 0.2980649679116425253322056e0 * c(m-6) - 0.2308067892671916339568942e-1 * c(m-5) - 0.4250000000000000000000000e0 * c(m-8) - 0.4272090808352508360837056e0 * c(m-7) - 0.1645272326387475188399322e-2 * c(m-4) 0.5555555555555555555555556e-2 * c(m-10) + 0.1250000000000000000000000e0 * c(m-9) + 0.9500000000000000000000000e0 * c(m-8) + 0.9341601509609901526962449e0 * c(m-6) + 0.1183647529645898332481833e0 * c(m-5) + 0.1919432828897222527630486e-1 * c(m-4) + 0.5699474344521144554459336e0 * c(m-7) -0.2500000000000000000000000e-1 * c(m-9) - 0.3000000000000000000000000e0 * c(m-8) - 0.2997556885134827361576001e0 * c(m-5) - 0.5636663150858098975790317e-1 * c(m-4) - 0.4382544850115050165022338e0 * c(m-7) - 0.3806231949664312575822630e0 * c(m-6) 0.5000000000000000000000000e-1 * c(m-8) - 0.3557251496099816106154206e0 * c(m-6) + 0.5490976528821799120017102e-1 * c(m-4) + 0.1838363233410033443348225e0 * c(m-7) + 0.2169790609807602750804271e0 * c(m-5) 0.5528052133944605740009217e-1 * c(m-6) - 0.8631683980217122275970376e-1 * c(m-5) - 0.3276463639080639163926118e-1 * c(m-7) + 0.5268984374242044588776166e-1 * c(m-4) -0.5373770512016897565958305e-2 * c(m-6) + 0.1453421858063658498587377e0 * c(m-5) - 0.1399684152943489522927794e0 * c(m-4) -0.1059816030196818445908057e0 * c(m-5) + 0.1014880675788250237247178e0 * c(m-4) + 0.4493535440856820866087846e-2 * c(m-6) 0.4093499466767054661591066e-1 * c(m-5) - 0.3471075437892810033585296e-1 * c(m-4) - 0.6224240288742446280057699e-2 * c(m-6) -0.6486184157331537899459796e-2 * c(m-5) + 0.4409068609809831485979484e-2 * c(m-4) + 0.2077115547521706413480312e-2 * c(m-6); 0.5000000000000000000000000e-1 * c(m-9) - 0.2966863787471237458744416e0 * c(m-7) + 0.4698146218022683933926520e-1 * c(m-5) + 0.1750000000000000000000000e0 * c(m-8) + 0.1700291833903489463825077e0 * c(m-6) + 0.4675733176547960152668626e-2 * c(m-4) -0.2500000000000000000000000e-1 * c(m-9) - 0.3000000000000000000000000e0 * c(m-8) - 0.2997556885134827361576001e0 * c(m-5) - 0.5636663150858098975790317e-1 * c(m-4) - 0.4382544850115050165022338e0 * c(m-7) - 0.3806231949664312575822630e0 * c(m-6) 0.5555555555555555555555556e-2 * c(m-9) + 0.1250000000000000000000000e0 * c(m-8) + 0.9698817275172575247533506e0 * c(m-7) + 0.1011699483929608164601067e1 * c(m-5) + 0.1773466968705924819112984e0 * c(m-4) + 0.2239223735771599178951297e-3 * c(m-3) + 0.4325148359756313354830552e0 * c(m-6) -0.2500000000000000000000000e-1 * c(m-8) - 0.3132544850115050165022338e0 * c(m-7) - 0.2322389872063761557916742e0 * c(m-4) - 0.8607044252686413302647675e-3 * c(m-3) - 0.2594851141920572702679681e0 * c(m-6) - 0.6691607091647929161078591e0 * c(m-5) 0.5331362125287625412555844e-1 * c(m-7) + 0.1736073411355428563685818e0 * c(m-5) + 0.8671038084174692625075159e-2 * c(m-3) + 0.8084259844422177692569663e-1 * c(m-6) - 0.1664345989168155800449120e0 * c(m-4) -0.5013247356072127938999311e0 * c(m-5) + 0.5021853752328231128475915e0 * c(m-4) - 0.1197175073672143005877150e-1 * c(m-6) 0.3328335104489738933610597e0 * c(m-5) - 0.3179803804558436283847901e0 * c(m-4) - 0.5111353189352474549563559e-2 * c(m-3) - 0.9741776803777790426705996e-2 * c(m-6) -0.1080774142196007991746827e0 * c(m-5) + 0.9941834083648937298100811e-1 * c(m-4) - 0.4834791406446907590553793e-2 * c(m-3) + 0.1349386478955833378422842e-1 * c(m-6) 0.1319674981073749167009902e-1 * c(m-5) - 0.1060554802883657391328704e-1 * c(m-4) + 0.1911888563316170927411831e-2 * c(m-3) - 0.4503090345217088684223814e-2 * c(m-6); 0.2279091916474916391629437e-1 * c(m-7) + 0.3097763128598982561225538e-1 * c(m-6) - 0.3055555555555555555555556e-1 * c(m-8) - 0.3068985997518740530511593e-1 * c(m-5) - 0.3634246031107139778989373e-2 * c(m-4) 0.5000000000000000000000000e-1 * c(m-8) - 0.3557251496099816106154206e0 * c(m-6) + 0.5490976528821799120017102e-1 * c(m-4) + 0.1838363233410033443348225e0 * c(m-7) + 0.2169790609807602750804271e0 * c(m-5) -0.2500000000000000000000000e-1 * c(m-8) - 0.3132544850115050165022338e0 * c(m-7) - 0.2322389872063761557916742e0 * c(m-4) - 0.8607044252686413302647675e-3 * c(m-3) - 0.2594851141920572702679681e0 * c(m-6) - 0.6691607091647929161078591e0 * c(m-5) 0.5555555555555555555555556e-2 * c(m-8) + 0.1338363233410033443348225e0 * c(m-7) + 0.7391887916719206077121040e0 * c(m-6) + 0.6490333320052011212240632e0 * c(m-4) + 0.3308343404200968256656458e-2 * c(m-3) + 0.2062575706647430620228133e-3 * c(m-2) + 0.3088241944378964404772302e-3 * c(m-1) + 0.4227226173449345042468960e-3 * c(m) + 0.1190362071861893051132274e1 * c(m-5) -0.2720908083525083608370563e-1 * c(m-7) - 0.1931148612480615118957263e0 * c(m-6) - 0.3332941113251635390801278e-1 * c(m-3) + 0.2605582646183255957264249e-3 * c(m-2) + 0.1432116665752147607469646e-2 * c(m-1) + 0.1207544072304193806052558e-2 * c(m) - 0.1348436986667115543203552e1 * c(m-5) + 0.1687723507780044227927853e-1 * c(m-4) 0.3590669644811151307464697e-1 * c(m-6) - 0.5925443480724830632401754e0 * c(m-4) - 0.6897142765790609546343709e-2 * c(m-2) - 0.5363098747528542488971874e-2 * c(m-1) - 0.5205147429855955657625694e-2 * c(m) + 0.9977064356292750529201981e0 * c(m-5) 0.7272438906214475928744770e-1 * c(m-4) + 0.1964682777744275219350831e-1 * c(m-3) - 0.5952475275883259619711594e-2 * c(m-1) - 0.1635430866921887819487473e-2 * c(m) + 0.2921234010758621482958052e-1 * c(m-6) - 0.4659516693228870973898560e0 * c(m-5) 0.5891947149681041048896399e-1 * c(m-4) + 0.1858378996391679448655070e-1 * c(m-3) + 0.7240905383565181316381731e-2 * c(m-2) + 0.2349927974590068869356781e-1 * c(m) - 0.4046360079256766884300687e-1 * c(m-6) + 0.1223513270418807666970488e0 * c(m-5) -0.2404661162020836566908542e-1 * c(m-4) - 0.7348845587775519698437916e-2 * c(m-3) - 0.8105784530576404277872603e-3 * c(m-2) + 0.9574633163221758060736592e-2 * c(m-1) - 0.1828896813877197352675410e-1 * c(m) + 0.1350326632905990039353503e-1 * c(m-6) - 0.1315967038382618382356495e-1 * c(m-5); 0.5522702088127090209264064e-3 * c(m-7) - 0.3265435411305071914756373e-2 * c(m-6) + 0.7462059484530855073291365e-2 * c(m-5) - 0.4748894282038492179461399e-2 * c(m-4) 0.5528052133944605740009217e-1 * c(m-6) - 0.8631683980217122275970376e-1 * c(m-5) - 0.3276463639080639163926118e-1 * c(m-7) + 0.5268984374242044588776166e-1 * c(m-4) 0.5331362125287625412555844e-1 * c(m-7) + 0.1736073411355428563685818e0 * c(m-5) + 0.8671038084174692625075159e-2 * c(m-3) + 0.8084259844422177692569663e-1 * c(m-6) - 0.1664345989168155800449120e0 * c(m-4) -0.2720908083525083608370563e-1 * c(m-7) - 0.1931148612480615118957263e0 * c(m-6) - 0.3332941113251635390801278e-1 * c(m-3) + 0.2605582646183255957264249e-3 * c(m-2) + 0.1432116665752147607469646e-2 * c(m-1) + 0.1207544072304193806052558e-2 * c(m) - 0.1348436986667115543203552e1 * c(m-5) + 0.1687723507780044227927853e-1 * c(m-4) 0.6107825764368264576481962e-2 * c(m-7) + 0.1155752633643216628010304e0 * c(m-6) + 0.2096413329579026439044119e1 * c(m-5) + 0.3357721707576477199985656e0 * c(m-3) + 0.3291545083271862858501887e-3 * c(m-2) + 0.6641183499427826101618457e-2 * c(m-1) + 0.3449455095910233625229891e-2 * c(m) + 0.8270696421223286922584620e0 * c(m-4) -0.4995827370863505253765970e-1 * c(m-6) - 0.1263507837371824205693950e1 * c(m-5) - 0.8712928907711754187084757e-2 * c(m-2) - 0.2487040599390160764166412e-1 * c(m-1) - 0.1486895819265604128572498e-1 * c(m) - 0.1726565392121567634950213e1 * c(m-4) 0.5402985338373433052255418e0 * c(m-5) - 0.1979290298620869974478871e0 * c(m-3) - 0.2760353365637712827793337e-1 * c(m-1) - 0.4671751091575462868310238e-2 * c(m) - 0.6952587985456154591014641e-1 * c(m-6) + 0.1571507277911208446562686e1 * c(m-4) -0.1319358558853174530078498e0 * c(m-5) - 0.1872196143003808021730728e0 * c(m-3) + 0.9147192682075630179962131e-2 * c(m-2) + 0.6712774475803763988977355e-1 * c(m) + 0.9630407686703666967100804e-1 * c(m-6) - 0.6882132817757726722141421e0 * c(m-4) 0.1241625568998496895352046e-1 * c(m-5) + 0.7403484645316174090533193e-1 * c(m-3) - 0.1023976547309387874453988e-2 * c(m-2) + 0.4440063948509876221050939e-1 * c(m-1) - 0.5224403464202056316702078e-1 * c(m) - 0.3213800979246298453953842e-1 * c(m-6) + 0.1178181682424363524005403e0 * c(m-4); 0.6272075574042975468177820e-3 * c(m-6) - 0.1200429618441003833696998e-1 * c(m-5) + 0.1137708862700574079015220e-1 * c(m-4) -0.5373770512016897565958305e-2 * c(m-6) + 0.1453421858063658498587377e0 * c(m-5) - 0.1399684152943489522927794e0 * c(m-4) -0.5013247356072127938999311e0 * c(m-5) + 0.5021853752328231128475915e0 * c(m-4) - 0.1197175073672143005877150e-1 * c(m-6) 0.3590669644811151307464697e-1 * c(m-6) - 0.5925443480724830632401754e0 * c(m-4) - 0.6897142765790609546343709e-2 * c(m-2) - 0.5363098747528542488971874e-2 * c(m-1) - 0.5205147429855955657625694e-2 * c(m) + 0.9977064356292750529201981e0 * c(m-5) -0.4995827370863505253765970e-1 * c(m-6) - 0.1263507837371824205693950e1 * c(m-5) - 0.8712928907711754187084757e-2 * c(m-2) - 0.2487040599390160764166412e-1 * c(m-1) - 0.1486895819265604128572498e-1 * c(m) - 0.1726565392121567634950213e1 * c(m-4) 0.2760393423824887721078848e-1 * c(m-6) + 0.1190550338687608873798462e1 * c(m-5) + 0.4253084328734353394994388e1 * c(m-4) + 0.2306367624634749229113646e0 * c(m-2) + 0.9313657638804699948929701e-1 * c(m-1) + 0.6409299775987186986730499e-1 * c(m) -0.8755807343482262259774782e0 * c(m-5) - 0.3645285178085761821545207e1 * c(m-4) + 0.1033717994630886401730470e0 * c(m-1) + 0.2013769413884797246646959e-1 * c(m) + 0.4106783858513785463625543e-1 * c(m-6) 0.3956598149904136332753521e0 * c(m-5) + 0.1630560443616104907615866e1 * c(m-4) - 0.2421320004064592721552708e0 * c(m-2) - 0.2893557395653431666593814e0 * c(m) - 0.5688529641249387985434413e-1 * c(m-6) -0.7684117160199014594442072e-1 * c(m-5) - 0.2928439026361256842196229e0 * c(m-4) + 0.2710530961648671297733465e-1 * c(m-2) - 0.1662748711097054895317080e0 * c(m-1) + 0.2251991532891353212689574e0 * c(m) + 0.1898341454096471754822498e-1 * c(m-6); 0.9209089963443799485648361e-2 * c(m-5) - 0.3129629392354775191148163e-3 * c(m-6) - 0.8896127024208321966533544e-2 * c(m-4) -0.1059816030196818445908057e0 * c(m-5) + 0.1014880675788250237247178e0 * c(m-4) + 0.4493535440856820866087846e-2 * c(m-6) 0.3328335104489738933610597e0 * c(m-5) - 0.3179803804558436283847901e0 * c(m-4) - 0.5111353189352474549563559e-2 * c(m-3) - 0.9741776803777790426705996e-2 * c(m-6) 0.7272438906214475928744770e-1 * c(m-4) + 0.1964682777744275219350831e-1 * c(m-3) - 0.5952475275883259619711594e-2 * c(m-1) - 0.1635430866921887819487473e-2 * c(m) + 0.2921234010758621482958052e-1 * c(m-6) - 0.4659516693228870973898560e0 * c(m-5) 0.5402985338373433052255418e0 * c(m-5) - 0.1979290298620869974478871e0 * c(m-3) - 0.2760353365637712827793337e-1 * c(m-1) - 0.4671751091575462868310238e-2 * c(m) - 0.6952587985456154591014641e-1 * c(m-6) + 0.1571507277911208446562686e1 * c(m-4) -0.8755807343482262259774782e0 * c(m-5) - 0.3645285178085761821545207e1 * c(m-4) + 0.1033717994630886401730470e0 * c(m-1) + 0.2013769413884797246646959e-1 * c(m) + 0.4106783858513785463625543e-1 * c(m-6) 0.1070920689960817104203947e1 * c(m-5) + 0.3717418466925056542408153e1 * c(m-4) + 0.1166740554279680007487795e0 * c(m-3) + 0.1147318200715868527529827e0 * c(m-1) + 0.6327161147136873807796515e-2 * c(m) + 0.6235373239336055200426697e-1 * c(m-6) -0.6639605248735044787146222e0 * c(m-5) - 0.1865625445986772763641423e1 * c(m-4) + 0.1103611313171476425250639e0 * c(m-3) - 0.9091410269992464604926176e-1 * c(m) - 0.8636954541126674177407762e-1 * c(m-6) 0.1582127073537215443965653e0 * c(m-5) + 0.3746489300753517635549495e0 * c(m-4) - 0.4364163147111892346990101e-1 * c(m-3) - 0.1845476106024151050283847e0 * c(m-1) + 0.7075642937243715046279337e-1 * c(m) + 0.2882271848190011329385407e-1 * c(m-6); -0.3576545132696983143406173e-2 * c(m-5) + 0.4335019854436220306755673e-3 * c(m-6) + 0.3143043147253361112730605e-2 * c(m-4) 0.4093499466767054661591066e-1 * c(m-5) - 0.3471075437892810033585296e-1 * c(m-4) - 0.6224240288742446280057699e-2 * c(m-6) -0.1080774142196007991746827e0 * c(m-5) + 0.9941834083648937298100811e-1 * c(m-4) - 0.4834791406446907590553793e-2 * c(m-3) + 0.1349386478955833378422842e-1 * c(m-6) 0.5891947149681041048896399e-1 * c(m-4) + 0.1858378996391679448655070e-1 * c(m-3) + 0.7240905383565181316381731e-2 * c(m-2) + 0.2349927974590068869356781e-1 * c(m) - 0.4046360079256766884300687e-1 * c(m-6) + 0.1223513270418807666970488e0 * c(m-5) -0.1319358558853174530078498e0 * c(m-5) - 0.1872196143003808021730728e0 * c(m-3) + 0.9147192682075630179962131e-2 * c(m-2) + 0.6712774475803763988977355e-1 * c(m) + 0.9630407686703666967100804e-1 * c(m-6) - 0.6882132817757726722141421e0 * c(m-4) 0.3956598149904136332753521e0 * c(m-5) + 0.1630560443616104907615866e1 * c(m-4) - 0.2421320004064592721552708e0 * c(m-2) - 0.2893557395653431666593814e0 * c(m) - 0.5688529641249387985434413e-1 * c(m-6) -0.6639605248735044787146222e0 * c(m-5) - 0.1865625445986772763641423e1 * c(m-4) + 0.1103611313171476425250639e0 * c(m-3) - 0.9091410269992464604926176e-1 * c(m) - 0.8636954541126674177407762e-1 * c(m-6) 0.4681819359722749441073885e0 * c(m-5) + 0.1015239189167790053447110e1 * c(m-4) + 0.1043897828092562609502636e0 * c(m-3) + 0.2542001760457345743492403e0 * c(m-2) + 0.1306332157111667628555907e1 * c(m) + 0.1196351539550049336518187e0 * c(m-6) -0.1195777325611201766551392e0 * c(m-5) - 0.2187310061229745694542609e0 * c(m-4) - 0.4128029838349298819825156e-1 * c(m-3) - 0.2845627370491611369031341e-1 * c(m-2) - 0.1016689339350338144430605e1 * c(m) - 0.3992391469197282238624438e-1 * c(m-6); 0.5593983696629863059347067e-3 * c(m-5) - 0.1446656414398166805849327e-3 * c(m-6) - 0.4147327282231696253497740e-3 * c(m-4) -0.6486184157331537899459796e-2 * c(m-5) + 0.4409068609809831485979484e-2 * c(m-4) + 0.2077115547521706413480312e-2 * c(m-6) 0.1319674981073749167009902e-1 * c(m-5) - 0.1060554802883657391328704e-1 * c(m-4) + 0.1911888563316170927411831e-2 * c(m-3) - 0.4503090345217088684223814e-2 * c(m-6) -0.2404661162020836566908542e-1 * c(m-4) - 0.7348845587775519698437916e-2 * c(m-3) - 0.8105784530576404277872603e-3 * c(m-2) + 0.9574633163221758060736592e-2 * c(m-1) - 0.1828896813877197352675410e-1 * c(m) + 0.1350326632905990039353503e-1 * c(m-6) - 0.1315967038382618382356495e-1 * c(m-5) 0.1241625568998496895352046e-1 * c(m-5) + 0.7403484645316174090533193e-1 * c(m-3) - 0.1023976547309387874453988e-2 * c(m-2) + 0.4440063948509876221050939e-1 * c(m-1) - 0.5224403464202056316702078e-1 * c(m) - 0.3213800979246298453953842e-1 * c(m-6) + 0.1178181682424363524005403e0 * c(m-4) -0.7684117160199014594442072e-1 * c(m-5) - 0.2928439026361256842196229e0 * c(m-4) + 0.2710530961648671297733465e-1 * c(m-2) - 0.1662748711097054895317080e0 * c(m-1) + 0.2251991532891353212689574e0 * c(m) + 0.1898341454096471754822498e-1 * c(m-6) 0.1582127073537215443965653e0 * c(m-5) + 0.3746489300753517635549495e0 * c(m-4) - 0.4364163147111892346990101e-1 * c(m-3) - 0.1845476106024151050283847e0 * c(m-1) + 0.7075642937243715046279337e-1 * c(m) + 0.2882271848190011329385407e-1 * c(m-6) -0.1195777325611201766551392e0 * c(m-5) - 0.2187310061229745694542609e0 * c(m-4) - 0.4128029838349298819825156e-1 * c(m-3) - 0.2845627370491611369031341e-1 * c(m-2) - 0.1016689339350338144430605e1 * c(m) - 0.3992391469197282238624438e-1 * c(m-6) 0.3167964748016105299646518e-1 * c(m-5) + 0.4976563420877041544013670e-1 * c(m-4) + 0.1632404042590951953384672e-1 * c(m-3) + 0.3185519088796429015220016e-2 * c(m-2) + 0.2968472090638000742888467e0 * c(m-1) + 0.7912667594695582093926295e0 * c(m) + 0.1332316557164627464149716e-1 * c(m-6);]; +% +% M(5,10)=M(10,5); +% M(m-4,m-9)=M(m-9,m-4); +% +% M=M/h; +% +% D2=HI*(-M-diag(c)*e_1*S_1+diag(c)*e_m*S_m); + +S2_U=[0.203e3 / 0.45e2 -0.87e2 / 0.5e1 0.117e3 / 0.4e1 -0.254e3 / 0.9e1 0.33e2 / 0.2e1 -0.27e2 / 0.5e1 0.137e3 / 0.180e3;]/h^2; +S2_1=zeros(1,m); +S2_1(1:7)=S2_U; +S2_m=zeros(1,m); +S2_m(m-6:m)=fliplr(S2_U); + + + + + +% Fourth derivative, 1th order accurate at first 8 boundary points (still +% yield 5th order convergence if stable: for example u_tt=-u_xxxx + +m5=-0.41e2 / 0.7560e4;m4= 0.1261e4 / 0.15120e5;m3= -0.541e3 / 0.840e3; m2= 0.4369e4 / 0.1260e4; m1= -0.1669e4 / 0.180e3; m0= 0.1529e4 / 0.120e3; + + +M4=m5*(diag(ones(m-5,1),5)+diag(ones(m-5,1),-5))+m4*(diag(ones(m-4,1),4)+diag(ones(m-4,1),-4))+m3*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3))+m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0); + +%M4=(-1/6*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3) ) + 2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2)) -13/2*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)) + 28/3*diag(ones(m,1),0)); + +M4_U=[0.1031569831e10 / 0.155675520e9 -0.32874237931e11 / 0.1452971520e10 0.3069551773e10 / 0.90810720e8 -0.658395212131e12 / 0.21794572800e11 0.31068454007e11 / 0.1816214400e10 -0.39244130657e11 / 0.7264857600e10 0.1857767503e10 / 0.2724321600e10 0.1009939e7 / 0.49420800e8; -0.32874237931e11 / 0.1452971520e10 0.12799022387e11 / 0.155675520e9 -0.134456503627e12 / 0.1037836800e10 0.15366749479e11 / 0.129729600e9 -0.207640325549e12 / 0.3113510400e10 0.5396424073e10 / 0.259459200e9 -0.858079351e9 / 0.345945600e9 -0.19806607e8 / 0.170270100e9; 0.3069551773e10 / 0.90810720e8 -0.134456503627e12 / 0.1037836800e10 0.6202056779e10 / 0.28828800e8 -0.210970327081e12 / 0.1037836800e10 0.2127730129e10 / 0.18532800e8 -0.4048692749e10 / 0.115315200e9 0.1025943959e10 / 0.259459200e9 0.71054663e8 / 0.290594304e9; -0.658395212131e12 / 0.21794572800e11 0.15366749479e11 / 0.129729600e9 -0.210970327081e12 / 0.1037836800e10 0.31025293213e11 / 0.155675520e9 -0.1147729001e10 / 0.9884160e7 0.1178067773e10 / 0.32432400e8 -0.13487255581e11 / 0.3113510400e10 -0.231082547e9 / 0.1816214400e10; 0.31068454007e11 / 0.1816214400e10 -0.207640325549e12 / 0.3113510400e10 0.2127730129e10 / 0.18532800e8 -0.1147729001e10 / 0.9884160e7 0.11524865123e11 / 0.155675520e9 -0.29754506009e11 / 0.1037836800e10 0.14231221e8 / 0.2316600e7 -0.15030629699e11 / 0.21794572800e11; -0.39244130657e11 / 0.7264857600e10 0.5396424073e10 / 0.259459200e9 -0.4048692749e10 / 0.115315200e9 0.1178067773e10 / 0.32432400e8 -0.29754506009e11 / 0.1037836800e10 0.572247737e9 / 0.28828800e8 -0.11322059051e11 / 0.1037836800e10 0.3345834083e10 / 0.908107200e9; 0.1857767503e10 / 0.2724321600e10 -0.858079351e9 / 0.345945600e9 0.1025943959e10 / 0.259459200e9 -0.13487255581e11 / 0.3113510400e10 0.14231221e8 / 0.2316600e7 -0.11322059051e11 / 0.1037836800e10 0.10478882597e11 / 0.778377600e9 -0.68446325191e11 / 0.7264857600e10; 0.1009939e7 / 0.49420800e8 -0.19806607e8 / 0.170270100e9 0.71054663e8 / 0.290594304e9 -0.231082547e9 / 0.1816214400e10 -0.15030629699e11 / 0.21794572800e11 0.3345834083e10 / 0.908107200e9 -0.68446325191e11 / 0.7264857600e10 0.9944747557e10 / 0.778377600e9;]; + +M4(1:8,1:8)=M4_U; + +M4(m-7:m,m-7:m)=flipud( fliplr( M4_U ) ); +M4=M4/h^3; + +S3_U=[-0.49e2 / 0.8e1 29 -0.461e3 / 0.8e1 62 -0.307e3 / 0.8e1 13 -0.15e2 / 0.8e1;]/h^3; +S3_1=zeros(1,m); +S3_1(1:7)=S3_U; +S3_m=zeros(1,m); +S3_m(m-6:m)=fliplr(-S3_U); + +D4=HI*(M4-e_1*S3_1+e_m*S3_m + S_1'*S2_1-S_m'*S2_m); + +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/higher_variable8_min_boundary_points.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,102 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% 8:te ordn. SBP Finita differens %%% +%%% operatorer med diagonal norm %%% +%%% %%% +%%% %%% +%%% H (Normen) %%% +%%% D1=H^(-1)Q (approx f?rsta derivatan) %%% +%%% D2 (approx andra derivatan) %%% +%%% D2=HI*(R+C*D*S %%% +%%% %%% +%%% R=-D1'*H*C*D1-RR %%% +%%% %%% +%%% RR ?r dissipation) %%% +%%% Dissipationen uppbyggd av D4: %%% +%%% DI=D4*B*H*D4 %%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +%m=10; %problemstorlek +%h=1/(m-1); + +% x=1:h:m*h;x=x'; +% x0=x.^0/factorial(1); +% x1=x.^1/factorial(1); +% x2=x.^2/factorial(2); +% x3=x.^3/factorial(3); +% x4=x.^4/factorial(4); +% x5=x.^5/factorial(5); +% x6=x.^6/factorial(6); + +H=diag(ones(m,1),0); +H(1:8,1:8)=diag([1498139/5080320, 1107307/725760, 20761/80640, 1304999/725760, 299527/725760, 103097/80640, 670091/725760, 5127739/5080320]); +H(m-7:m,m-7:m)=fliplr(flipud(diag([1498139/5080320, 1107307/725760, 20761/80640, 1304999/725760, 299527/725760, 103097/80640, 670091/725760, 5127739/5080320]))); + + +e_1=zeros(m,1);e_1(1)=1; +e_m=zeros(m,1);e_m(m)=1; + +S_U=[-0.137e3 / 0.60e2 5 -5 0.10e2 / 0.3e1 -0.5e1 / 0.4e1 0.1e1 / 0.5e1;]/h; +S_1=zeros(1,m); +S_1(1:6)=S_U; +S_m=zeros(1,m); +S_m(m-5:m)=fliplr(-S_U); + +H=h*H; +HI=inv(H); + + +% M=zeros(m,m); +% +% for i=6:m-5 +% M(i,i-3:i+3)=[c(i-2) / 0.40e2 + c(i-1) / 0.40e2 - 0.11e2 / 0.360e3 * c(i-3) - 0.11e2 / 0.360e3 * c(i) c(i-3) / 0.20e2 - 0.3e1 / 0.10e2 * c(i-1) + c(i+1) / 0.20e2 + 0.7e1 / 0.40e2 * c(i) + 0.7e1 / 0.40e2 * c(i-2) -c(i-3) / 0.40e2 - 0.3e1 / 0.10e2 * c(i-2) - 0.3e1 / 0.10e2 * c(i+1) - c(i+2) / 0.40e2 - 0.17e2 / 0.40e2 * c(i) - 0.17e2 / 0.40e2 * c(i-1) c(i-3) / 0.180e3 + c(i-2) / 0.8e1 + 0.19e2 / 0.20e2 * c(i-1) + 0.19e2 / 0.20e2 * c(i+1) + c(i+2) / 0.8e1 + c(i+3) / 0.180e3 + 0.101e3 / 0.180e3 * c(i) -c(i-2) / 0.40e2 - 0.3e1 / 0.10e2 * c(i-1) - 0.3e1 / 0.10e2 * c(i+2) - c(i+3) / 0.40e2 - 0.17e2 / 0.40e2 * c(i) - 0.17e2 / 0.40e2 * c(i+1) c(i-1) / 0.20e2 - 0.3e1 / 0.10e2 * c(i+1) + c(i+3) / 0.20e2 + 0.7e1 / 0.40e2 * c(i) + 0.7e1 / 0.40e2 * c(i+2) c(i+1) / 0.40e2 + c(i+2) / 0.40e2 - 0.11e2 / 0.360e3 * c(i) - 0.11e2 / 0.360e3 * c(i+3);]; +% end +% +% +% M(1:9,1:9)=[0.7912667594695582093926295e0 * c(1) + 0.2968472090638000742888467e0 * c(2) + 0.3185519088796429015220016e-2 * c(3) + 0.1632404042590951953384672e-1 * c(4) + 0.3160302244094415087693968e-1 * c(5) + 0.3167964748016105299646518e-1 * c(6) + 0.3148577733947253920469418e-1 * c(7) -0.1016689339350338144430605e1 * c(1) - 0.2845627370491611369031341e-1 * c(3) - 0.4128029838349298819825156e-1 * c(4) - 0.1392281451620140507549866e0 * c(5) - 0.1195777325611201766551392e0 * c(6) - 0.1194267756529333410855186e0 * c(7) 0.7075642937243715046279337e-1 * c(1) - 0.1845476106024151050283847e0 * c(2) - 0.4364163147111892346990101e-1 * c(4) + 0.2432367907207732460874765e0 * c(5) + 0.1582127073537215443965653e0 * c(6) + 0.1602348578364786307613271e0 * c(7) 0.2251991532891353212689574e0 * c(1) - 0.1662748711097054895317080e0 * c(2) + 0.2710530961648671297733465e-1 * c(3) - 0.1916646185968439909125616e0 * c(5) - 0.7684117160199014594442072e-1 * c(6) - 0.8219586949831697575883635e-1 * c(7) -0.5224403464202056316702078e-1 * c(1) + 0.4440063948509876221050939e-1 * c(2) - 0.1023976547309387874453988e-2 * c(3) + 0.7403484645316174090533193e-1 * c(4) + 0.1241625568998496895352046e-1 * c(6) + 0.7188652847892601282652228e-1 * c(5) + 0.1379362997104735503447960e-1 * c(7) -0.1828896813877197352675410e-1 * c(1) + 0.9574633163221758060736592e-2 * c(2) - 0.8105784530576404277872603e-3 * c(3) - 0.7348845587775519698437916e-2 * c(4) + 0.1063601949723906997026904e-1 * c(5) - 0.1315967038382618382356495e-1 * c(6) - 0.2117936478838753524581943e-1 * c(7) 0.1911888563316170927411831e-2 * c(4) - 0.4068130355529149936100229e-1 * c(5) + 0.1319674981073749167009902e-1 * c(6) + 0.2557266518123783676349144e-1 * c(7) 0.1559652871136785763960685e-1 * c(5) - 0.6486184157331537899459796e-2 * c(6) - 0.9110344554036319740147054e-2 * c(7) 0.5593983696629863059347067e-3 * c(6) - 0.1384822535100796372263822e-2 * c(5) + 0.8254241654378100663291154e-3 * c(7); -0.1016689339350338144430605e1 * c(1) - 0.2845627370491611369031341e-1 * c(3) - 0.4128029838349298819825156e-1 * c(4) - 0.1392281451620140507549866e0 * c(5) - 0.1195777325611201766551392e0 * c(6) - 0.1194267756529333410855186e0 * c(7) 0.1306332157111667628555907e1 * c(1) + 0.2542001760457345743492403e0 * c(3) + 0.1043897828092562609502636e0 * c(4) + 0.6672328021032112950919876e0 * c(5) + 0.4681819359722749441073885e0 * c(6) + 0.4676415410195836920069412e0 * c(7) -0.9091410269992464604926176e-1 * c(1) + 0.1103611313171476425250639e0 * c(4) - 0.1290397544997518887000350e1 * c(5) - 0.6639605248735044787146222e0 * c(6) - 0.6615974464005206184151509e0 * c(7) -0.2893557395653431666593814e0 * c(1) - 0.2421320004064592721552708e0 * c(3) + 0.1187670255028031027693374e1 * c(5) + 0.3956598149904136332753521e0 * c(6) + 0.3860048921755800000681479e0 * c(7) 0.6712774475803763988977355e-1 * c(1) + 0.9147192682075630179962131e-2 * c(3) - 0.1872196143003808021730728e0 * c(4) - 0.1319358558853174530078498e0 * c(6) - 0.4871575736811911887376923e0 * c(5) - 0.1047516312275448138054418e0 * c(7) 0.2349927974590068869356781e-1 * c(1) + 0.7240905383565181316381731e-2 * c(3) + 0.1858378996391679448655070e-1 * c(4) - 0.9289616133938676174345208e-1 * c(5) + 0.1223513270418807666970488e0 * c(6) + 0.1113520320436295033894092e0 * c(7) -0.4834791406446907590553793e-2 * c(4) + 0.2310683832687820403062715e0 * c(5) - 0.1080774142196007991746827e0 * c(6) - 0.1181561776427343335410349e0 * c(7) -0.8368141434403455353724691e-1 * c(5) + 0.4093499466767054661591066e-1 * c(6) + 0.4274641967636400692133626e-1 * c(7) -0.3576545132696983143406173e-2 * c(6) + 0.7389399124121078682094445e-2 * c(5) - 0.3812853991424095538688273e-2 * c(7); 0.7075642937243715046279337e-1 * c(1) - 0.1845476106024151050283847e0 * c(2) - 0.4364163147111892346990101e-1 * c(4) + 0.2432367907207732460874765e0 * c(5) + 0.1582127073537215443965653e0 * c(6) + 0.1602348578364786307613271e0 * c(7) -0.9091410269992464604926176e-1 * c(1) + 0.1103611313171476425250639e0 * c(4) - 0.1290397544997518887000350e1 * c(5) - 0.6639605248735044787146222e0 * c(6) - 0.6615974464005206184151509e0 * c(7) 0.6327161147136873807796515e-2 * c(1) + 0.1147318200715868527529827e0 * c(2) + 0.1166740554279680007487795e0 * c(4) + 0.2766610808285444037240703e1 * c(5) + 0.1070920689960817104203947e1 * c(6) + 0.1013161391032973057171717e1 * c(7) 0.2013769413884797246646959e-1 * c(1) + 0.1033717994630886401730470e0 * c(2) - 0.2913221621151742724258117e1 * c(5) - 0.8755807343482262259774782e0 * c(6) - 0.6909957183488812426508351e0 * c(7) -0.4671751091575462868310238e-2 * c(1) - 0.2760353365637712827793337e-1 * c(2) - 0.1979290298620869974478871e0 * c(4) + 0.5402985338373433052255418e0 * c(6) + 0.1239177593031911077924537e1 * c(5) + 0.2628038050247358227280031e0 * c(7) -0.1635430866921887819487473e-2 * c(1) - 0.5952475275883259619711594e-2 * c(2) + 0.1964682777744275219350831e-1 * c(4) + 0.3236640012639046600590714e0 * c(5) - 0.4659516693228870973898560e0 * c(6) - 0.2217272720941736859420432e0 * c(7) -0.5111353189352474549563559e-2 * c(4) - 0.5355878163774754346032096e0 * c(5) + 0.3328335104489738933610597e0 * c(6) + 0.2078656591178540157917135e0 * c(7) 0.1824328174134289562208038e0 * c(5) - 0.1059816030196818445908057e0 * c(6) - 0.7645121439374711162999809e-1 * c(7) 0.9209089963443799485648361e-2 * c(6) - 0.1591502818872493167091475e-1 * c(5) + 0.6705938225281132185266388e-2 * c(7); 0.2251991532891353212689574e0 * c(1) - 0.1662748711097054895317080e0 * c(2) + 0.2710530961648671297733465e-1 * c(3) - 0.1916646185968439909125616e0 * c(5) - 0.7684117160199014594442072e-1 * c(6) - 0.8219586949831697575883635e-1 * c(7) -0.2893557395653431666593814e0 * c(1) - 0.2421320004064592721552708e0 * c(3) + 0.1187670255028031027693374e1 * c(5) + 0.3956598149904136332753521e0 * c(6) + 0.3860048921755800000681479e0 * c(7) 0.2013769413884797246646959e-1 * c(1) + 0.1033717994630886401730470e0 * c(2) - 0.2913221621151742724258117e1 * c(5) - 0.8755807343482262259774782e0 * c(6) - 0.6909957183488812426508351e0 * c(7) 0.6409299775987186986730499e-1 * c(1) + 0.9313657638804699948929701e-1 * c(2) + 0.2306367624634749229113646e0 * c(3) + 0.3689440308283716620260816e1 * c(5) + 0.1190550338687608873798462e1 * c(6) + 0.5912479546888856519443605e0 * c(7) -0.1486895819265604128572498e-1 * c(1) - 0.2487040599390160764166412e-1 * c(2) - 0.8712928907711754187084757e-2 * c(3) - 0.1263507837371824205693950e1 * c(6) - 0.3058317397843997326920898e0 * c(7) - 0.1470691926045802954795783e1 * c(5) -0.5205147429855955657625694e-2 * c(1) - 0.5363098747528542488971874e-2 * c(2) - 0.6897142765790609546343709e-2 * c(3) - 0.7857524521667450101721993e0 * c(5) + 0.2291148005423734600066709e0 * c(7) + 0.9977064356292750529201981e0 * c(6) 0.6697297488067662265210608e0 * c(5) - 0.5013247356072127938999311e0 * c(6) - 0.1795161243106645437322408e0 * c(7) -0.2022909060111751565150958e0 * c(5) + 0.1453421858063658498587377e0 * c(6) + 0.5694872020480930665635812e-1 * c(7) -0.1200429618441003833696998e-1 * c(6) - 0.4776915669385923841535432e-2 * c(7) + 0.1678121185379596217850541e-1 * c(5); -0.5224403464202056316702078e-1 * c(1) + 0.4440063948509876221050939e-1 * c(2) - 0.1023976547309387874453988e-2 * c(3) + 0.7403484645316174090533193e-1 * c(4) + 0.1241625568998496895352046e-1 * c(6) + 0.7188652847892601282652228e-1 * c(5) + 0.1379362997104735503447960e-1 * c(7) 0.6712774475803763988977355e-1 * c(1) + 0.9147192682075630179962131e-2 * c(3) - 0.1872196143003808021730728e0 * c(4) - 0.1319358558853174530078498e0 * c(6) - 0.4871575736811911887376923e0 * c(5) - 0.1047516312275448138054418e0 * c(7) -0.4671751091575462868310238e-2 * c(1) - 0.2760353365637712827793337e-1 * c(2) - 0.1979290298620869974478871e0 * c(4) + 0.5402985338373433052255418e0 * c(6) + 0.1239177593031911077924537e1 * c(5) + 0.2628038050247358227280031e0 * c(7) -0.1486895819265604128572498e-1 * c(1) - 0.2487040599390160764166412e-1 * c(2) - 0.8712928907711754187084757e-2 * c(3) - 0.1263507837371824205693950e1 * c(6) - 0.3058317397843997326920898e0 * c(7) - 0.1470691926045802954795783e1 * c(5) 0.3449455095910233625229891e-2 * c(1) + 0.6641183499427826101618457e-2 * c(2) + 0.3291545083271862858501887e-3 * c(3) + 0.3357721707576477199985656e0 * c(4) + 0.2096413329579026439044119e1 * c(6) + 0.2317323204183126854954203e0 * c(7) + 0.6107825764368264576481962e-2 * c(8) + 0.7109125850683376695640722e0 * c(5) 0.1207544072304193806052558e-2 * c(1) + 0.1432116665752147607469646e-2 * c(2) + 0.2605582646183255957264249e-3 * c(3) - 0.3332941113251635390801278e-1 * c(4) - 0.2808241697385532683612407e0 * c(7) - 0.2720908083525083608370563e-1 * c(8) + 0.1045865435682921987447929e0 * c(5) - 0.1348436986667115543203552e1 * c(6) 0.8671038084174692625075159e-2 * c(4) + 0.1736073411355428563685818e0 * c(6) + 0.5331362125287625412555844e-1 * c(8) - 0.2424935262404526301801157e0 * c(5) + 0.1569015257678588270609004e0 * c(7) -0.8631683980217122275970376e-1 * c(6) + 0.2698842360470999243492629e-1 * c(7) + 0.8098194147715651085292754e-1 * c(5) - 0.3276463639080639163926118e-1 * c(8) 0.7462059484530855073291365e-2 * c(6) - 0.8121640361668678949573496e-3 * c(7) + 0.5522702088127090209264064e-3 * c(8) - 0.7202165657176696199260422e-2 * c(5); -0.1828896813877197352675410e-1 * c(1) + 0.9574633163221758060736592e-2 * c(2) - 0.8105784530576404277872603e-3 * c(3) - 0.7348845587775519698437916e-2 * c(4) + 0.1063601949723906997026904e-1 * c(5) - 0.1315967038382618382356495e-1 * c(6) - 0.2117936478838753524581943e-1 * c(7) 0.2349927974590068869356781e-1 * c(1) + 0.7240905383565181316381731e-2 * c(3) + 0.1858378996391679448655070e-1 * c(4) - 0.9289616133938676174345208e-1 * c(5) + 0.1223513270418807666970488e0 * c(6) + 0.1113520320436295033894092e0 * c(7) -0.1635430866921887819487473e-2 * c(1) - 0.5952475275883259619711594e-2 * c(2) + 0.1964682777744275219350831e-1 * c(4) + 0.3236640012639046600590714e0 * c(5) - 0.4659516693228870973898560e0 * c(6) - 0.2217272720941736859420432e0 * c(7) -0.5205147429855955657625694e-2 * c(1) - 0.5363098747528542488971874e-2 * c(2) - 0.6897142765790609546343709e-2 * c(3) - 0.7857524521667450101721993e0 * c(5) + 0.2291148005423734600066709e0 * c(7) + 0.9977064356292750529201981e0 * c(6) 0.1207544072304193806052558e-2 * c(1) + 0.1432116665752147607469646e-2 * c(2) + 0.2605582646183255957264249e-3 * c(3) - 0.3332941113251635390801278e-1 * c(4) - 0.2808241697385532683612407e0 * c(7) - 0.2720908083525083608370563e-1 * c(8) + 0.1045865435682921987447929e0 * c(5) - 0.1348436986667115543203552e1 * c(6) 0.4227226173449345042468960e-3 * c(1) + 0.3088241944378964404772302e-3 * c(2) + 0.2062575706647430620228133e-3 * c(3) + 0.3308343404200968256656458e-2 * c(4) + 0.5828047016405001815804837e0 * c(5) + 0.8054174220366215473556835e0 * c(7) + 0.1338363233410033443348225e0 * c(8) + 0.5555555555555555555555556e-2 * c(9) + 0.1190362071861893051132274e1 * c(6) -0.8607044252686413302647675e-3 * c(4) - 0.1748074708673904989293256e0 * c(5) - 0.3132544850115050165022338e0 * c(8) - 0.2500000000000000000000000e-1 * c(9) - 0.3169166305310429271303167e0 * c(7) - 0.6691607091647929161078591e0 * c(6) 0.3354661791693352108660900e-1 * c(5) - 0.3343620022386971405018586e0 * c(7) + 0.5000000000000000000000000e-1 * c(9) + 0.2169790609807602750804271e0 * c(6) + 0.1838363233410033443348225e0 * c(8) 0.2912518476823004642951502e-1 * c(7) + 0.2279091916474916391629437e-1 * c(8) - 0.3068985997518740530511593e-1 * c(6) - 0.1781799513347360596249022e-2 * c(5) - 0.3055555555555555555555556e-1 * c(9); 0.1911888563316170927411831e-2 * c(4) - 0.4068130355529149936100229e-1 * c(5) + 0.1319674981073749167009902e-1 * c(6) + 0.2557266518123783676349144e-1 * c(7) -0.4834791406446907590553793e-2 * c(4) + 0.2310683832687820403062715e0 * c(5) - 0.1080774142196007991746827e0 * c(6) - 0.1181561776427343335410349e0 * c(7) -0.5111353189352474549563559e-2 * c(4) - 0.5355878163774754346032096e0 * c(5) + 0.3328335104489738933610597e0 * c(6) + 0.2078656591178540157917135e0 * c(7) 0.6697297488067662265210608e0 * c(5) - 0.5013247356072127938999311e0 * c(6) - 0.1795161243106645437322408e0 * c(7) 0.8671038084174692625075159e-2 * c(4) + 0.1736073411355428563685818e0 * c(6) + 0.5331362125287625412555844e-1 * c(8) - 0.2424935262404526301801157e0 * c(5) + 0.1569015257678588270609004e0 * c(7) -0.8607044252686413302647675e-3 * c(4) - 0.1748074708673904989293256e0 * c(5) - 0.3132544850115050165022338e0 * c(8) - 0.2500000000000000000000000e-1 * c(9) - 0.3169166305310429271303167e0 * c(7) - 0.6691607091647929161078591e0 * c(6) 0.2239223735771599178951297e-3 * c(4) + 0.1275437785430956673825710e0 * c(5) + 0.1011699483929608164601067e1 * c(6) + 0.9698817275172575247533506e0 * c(8) + 0.1250000000000000000000000e0 * c(9) + 0.5555555555555555555555556e-2 * c(10) + 0.4823177543031281500117826e0 * c(7) -0.3784113973033012949863031e-1 * c(5) - 0.2997556885134827361576001e0 * c(6) - 0.3000000000000000000000000e0 * c(9) - 0.2500000000000000000000000e-1 * c(10) - 0.3991486867446821178415359e0 * c(7) - 0.4382544850115050165022338e0 * c(8) 0.4698146218022683933926520e-1 * c(6) - 0.2966863787471237458744416e0 * c(8) + 0.5000000000000000000000000e-1 * c(10) + 0.1716355704146006481727960e0 * c(7) + 0.3069346152296258362380356e-2 * c(5) + 0.1750000000000000000000000e0 * c(9); 0.1559652871136785763960685e-1 * c(5) - 0.6486184157331537899459796e-2 * c(6) - 0.9110344554036319740147054e-2 * c(7) -0.8368141434403455353724691e-1 * c(5) + 0.4093499466767054661591066e-1 * c(6) + 0.4274641967636400692133626e-1 * c(7) 0.1824328174134289562208038e0 * c(5) - 0.1059816030196818445908057e0 * c(6) - 0.7645121439374711162999809e-1 * c(7) -0.2022909060111751565150958e0 * c(5) + 0.1453421858063658498587377e0 * c(6) + 0.5694872020480930665635812e-1 * c(7) -0.8631683980217122275970376e-1 * c(6) + 0.2698842360470999243492629e-1 * c(7) + 0.8098194147715651085292754e-1 * c(5) - 0.3276463639080639163926118e-1 * c(8) 0.3354661791693352108660900e-1 * c(5) - 0.3343620022386971405018586e0 * c(7) + 0.5000000000000000000000000e-1 * c(9) + 0.2169790609807602750804271e0 * c(6) + 0.1838363233410033443348225e0 * c(8) -0.3784113973033012949863031e-1 * c(5) - 0.2997556885134827361576001e0 * c(6) - 0.3000000000000000000000000e0 * c(9) - 0.2500000000000000000000000e-1 * c(10) - 0.3991486867446821178415359e0 * c(7) - 0.4382544850115050165022338e0 * c(8) 0.1230328942716804455358698e-1 * c(5) + 0.1183647529645898332481833e0 * c(6) + 0.9410511898227943334189628e0 * c(7) + 0.9500000000000000000000000e0 * c(9) + 0.1250000000000000000000000e0 * c(10) + 0.5555555555555555555555556e-2 * c(11) + 0.5699474344521144554459336e0 * c(8) -0.2308067892671916339568942e-1 * c(6) - 0.2986625053775149497180439e0 * c(7) - 0.3000000000000000000000000e0 * c(10) - 0.2500000000000000000000000e-1 * c(11) - 0.1047734860515050802561078e-2 * c(5) - 0.4272090808352508360837056e0 * c(8) - 0.4250000000000000000000000e0 * c(9); 0.5593983696629863059347067e-3 * c(6) - 0.1384822535100796372263822e-2 * c(5) + 0.8254241654378100663291154e-3 * c(7) -0.3576545132696983143406173e-2 * c(6) + 0.7389399124121078682094445e-2 * c(5) - 0.3812853991424095538688273e-2 * c(7) 0.9209089963443799485648361e-2 * c(6) - 0.1591502818872493167091475e-1 * c(5) + 0.6705938225281132185266388e-2 * c(7) -0.1200429618441003833696998e-1 * c(6) - 0.4776915669385923841535432e-2 * c(7) + 0.1678121185379596217850541e-1 * c(5) 0.7462059484530855073291365e-2 * c(6) - 0.8121640361668678949573496e-3 * c(7) + 0.5522702088127090209264064e-3 * c(8) - 0.7202165657176696199260422e-2 * c(5) 0.2912518476823004642951502e-1 * c(7) + 0.2279091916474916391629437e-1 * c(8) - 0.3068985997518740530511593e-1 * c(6) - 0.1781799513347360596249022e-2 * c(5) - 0.3055555555555555555555556e-1 * c(9) 0.4698146218022683933926520e-1 * c(6) - 0.2966863787471237458744416e0 * c(8) + 0.5000000000000000000000000e-1 * c(10) + 0.1716355704146006481727960e0 * c(7) + 0.3069346152296258362380356e-2 * c(5) + 0.1750000000000000000000000e0 * c(9) -0.2308067892671916339568942e-1 * c(6) - 0.2986625053775149497180439e0 * c(7) - 0.3000000000000000000000000e0 * c(10) - 0.2500000000000000000000000e-1 * c(11) - 0.1047734860515050802561078e-2 * c(5) - 0.4272090808352508360837056e0 * c(8) - 0.4250000000000000000000000e0 * c(9) 0.5139370221149109977041877e-2 * c(6) + 0.1247723215009422001393184e0 * c(7) + 0.9505522702088127090209264e0 * c(8) + 0.9500000000000000000000000e0 * c(10) + 0.1250000000000000000000000e0 * c(11) + 0.5555555555555555555555556e-2 * c(12) + 0.9159362465153641826887659e-4 * c(5) + 0.5611111111111111111111111e0 * c(9);]; +% +% M(m-8:m,m-8:m)=[0.5555555555555555555555556e-2 * c(m-11) + 0.1250000000000000000000000e0 * c(m-10) + 0.9500000000000000000000000e0 * c(m-9) + 0.9505522702088127090209264e0 * c(m-7) + 0.1247205076844361998744053e0 * c(m-6) + 0.5139370221149109977041877e-2 * c(m-5) + 0.5611111111111111111111111e0 * c(m-8) + 0.1434074411575366831819799e-3 * c(m-4) -0.2500000000000000000000000e-1 * c(m-10) - 0.3000000000000000000000000e0 * c(m-9) - 0.2980649679116425253322056e0 * c(m-6) - 0.2308067892671916339568942e-1 * c(m-5) - 0.4250000000000000000000000e0 * c(m-8) - 0.4272090808352508360837056e0 * c(m-7) - 0.1645272326387475188399322e-2 * c(m-4) 0.5000000000000000000000000e-1 * c(m-9) - 0.2966863787471237458744416e0 * c(m-7) + 0.4698146218022683933926520e-1 * c(m-5) + 0.1750000000000000000000000e0 * c(m-8) + 0.1700291833903489463825077e0 * c(m-6) + 0.4675733176547960152668626e-2 * c(m-4) 0.2279091916474916391629437e-1 * c(m-7) + 0.3097763128598982561225538e-1 * c(m-6) - 0.3055555555555555555555556e-1 * c(m-8) - 0.3068985997518740530511593e-1 * c(m-5) - 0.3634246031107139778989373e-2 * c(m-4) 0.5522702088127090209264064e-3 * c(m-7) - 0.3265435411305071914756373e-2 * c(m-6) + 0.7462059484530855073291365e-2 * c(m-5) - 0.4748894282038492179461399e-2 * c(m-4) 0.6272075574042975468177820e-3 * c(m-6) - 0.1200429618441003833696998e-1 * c(m-5) + 0.1137708862700574079015220e-1 * c(m-4) 0.9209089963443799485648361e-2 * c(m-5) - 0.3129629392354775191148163e-3 * c(m-6) - 0.8896127024208321966533544e-2 * c(m-4) -0.3576545132696983143406173e-2 * c(m-5) + 0.4335019854436220306755673e-3 * c(m-6) + 0.3143043147253361112730605e-2 * c(m-4) 0.5593983696629863059347067e-3 * c(m-5) - 0.1446656414398166805849327e-3 * c(m-6) - 0.4147327282231696253497740e-3 * c(m-4); -0.2500000000000000000000000e-1 * c(m-10) - 0.3000000000000000000000000e0 * c(m-9) - 0.2980649679116425253322056e0 * c(m-6) - 0.2308067892671916339568942e-1 * c(m-5) - 0.4250000000000000000000000e0 * c(m-8) - 0.4272090808352508360837056e0 * c(m-7) - 0.1645272326387475188399322e-2 * c(m-4) 0.5555555555555555555555556e-2 * c(m-10) + 0.1250000000000000000000000e0 * c(m-9) + 0.9500000000000000000000000e0 * c(m-8) + 0.9341601509609901526962449e0 * c(m-6) + 0.1183647529645898332481833e0 * c(m-5) + 0.1919432828897222527630486e-1 * c(m-4) + 0.5699474344521144554459336e0 * c(m-7) -0.2500000000000000000000000e-1 * c(m-9) - 0.3000000000000000000000000e0 * c(m-8) - 0.2997556885134827361576001e0 * c(m-5) - 0.5636663150858098975790317e-1 * c(m-4) - 0.4382544850115050165022338e0 * c(m-7) - 0.3806231949664312575822630e0 * c(m-6) 0.5000000000000000000000000e-1 * c(m-8) - 0.3557251496099816106154206e0 * c(m-6) + 0.5490976528821799120017102e-1 * c(m-4) + 0.1838363233410033443348225e0 * c(m-7) + 0.2169790609807602750804271e0 * c(m-5) 0.5528052133944605740009217e-1 * c(m-6) - 0.8631683980217122275970376e-1 * c(m-5) - 0.3276463639080639163926118e-1 * c(m-7) + 0.5268984374242044588776166e-1 * c(m-4) -0.5373770512016897565958305e-2 * c(m-6) + 0.1453421858063658498587377e0 * c(m-5) - 0.1399684152943489522927794e0 * c(m-4) -0.1059816030196818445908057e0 * c(m-5) + 0.1014880675788250237247178e0 * c(m-4) + 0.4493535440856820866087846e-2 * c(m-6) 0.4093499466767054661591066e-1 * c(m-5) - 0.3471075437892810033585296e-1 * c(m-4) - 0.6224240288742446280057699e-2 * c(m-6) -0.6486184157331537899459796e-2 * c(m-5) + 0.4409068609809831485979484e-2 * c(m-4) + 0.2077115547521706413480312e-2 * c(m-6); 0.5000000000000000000000000e-1 * c(m-9) - 0.2966863787471237458744416e0 * c(m-7) + 0.4698146218022683933926520e-1 * c(m-5) + 0.1750000000000000000000000e0 * c(m-8) + 0.1700291833903489463825077e0 * c(m-6) + 0.4675733176547960152668626e-2 * c(m-4) -0.2500000000000000000000000e-1 * c(m-9) - 0.3000000000000000000000000e0 * c(m-8) - 0.2997556885134827361576001e0 * c(m-5) - 0.5636663150858098975790317e-1 * c(m-4) - 0.4382544850115050165022338e0 * c(m-7) - 0.3806231949664312575822630e0 * c(m-6) 0.5555555555555555555555556e-2 * c(m-9) + 0.1250000000000000000000000e0 * c(m-8) + 0.9698817275172575247533506e0 * c(m-7) + 0.1011699483929608164601067e1 * c(m-5) + 0.1773466968705924819112984e0 * c(m-4) + 0.2239223735771599178951297e-3 * c(m-3) + 0.4325148359756313354830552e0 * c(m-6) -0.2500000000000000000000000e-1 * c(m-8) - 0.3132544850115050165022338e0 * c(m-7) - 0.2322389872063761557916742e0 * c(m-4) - 0.8607044252686413302647675e-3 * c(m-3) - 0.2594851141920572702679681e0 * c(m-6) - 0.6691607091647929161078591e0 * c(m-5) 0.5331362125287625412555844e-1 * c(m-7) + 0.1736073411355428563685818e0 * c(m-5) + 0.8671038084174692625075159e-2 * c(m-3) + 0.8084259844422177692569663e-1 * c(m-6) - 0.1664345989168155800449120e0 * c(m-4) -0.5013247356072127938999311e0 * c(m-5) + 0.5021853752328231128475915e0 * c(m-4) - 0.1197175073672143005877150e-1 * c(m-6) 0.3328335104489738933610597e0 * c(m-5) - 0.3179803804558436283847901e0 * c(m-4) - 0.5111353189352474549563559e-2 * c(m-3) - 0.9741776803777790426705996e-2 * c(m-6) -0.1080774142196007991746827e0 * c(m-5) + 0.9941834083648937298100811e-1 * c(m-4) - 0.4834791406446907590553793e-2 * c(m-3) + 0.1349386478955833378422842e-1 * c(m-6) 0.1319674981073749167009902e-1 * c(m-5) - 0.1060554802883657391328704e-1 * c(m-4) + 0.1911888563316170927411831e-2 * c(m-3) - 0.4503090345217088684223814e-2 * c(m-6); 0.2279091916474916391629437e-1 * c(m-7) + 0.3097763128598982561225538e-1 * c(m-6) - 0.3055555555555555555555556e-1 * c(m-8) - 0.3068985997518740530511593e-1 * c(m-5) - 0.3634246031107139778989373e-2 * c(m-4) 0.5000000000000000000000000e-1 * c(m-8) - 0.3557251496099816106154206e0 * c(m-6) + 0.5490976528821799120017102e-1 * c(m-4) + 0.1838363233410033443348225e0 * c(m-7) + 0.2169790609807602750804271e0 * c(m-5) -0.2500000000000000000000000e-1 * c(m-8) - 0.3132544850115050165022338e0 * c(m-7) - 0.2322389872063761557916742e0 * c(m-4) - 0.8607044252686413302647675e-3 * c(m-3) - 0.2594851141920572702679681e0 * c(m-6) - 0.6691607091647929161078591e0 * c(m-5) 0.5555555555555555555555556e-2 * c(m-8) + 0.1338363233410033443348225e0 * c(m-7) + 0.7391887916719206077121040e0 * c(m-6) + 0.6490333320052011212240632e0 * c(m-4) + 0.3308343404200968256656458e-2 * c(m-3) + 0.2062575706647430620228133e-3 * c(m-2) + 0.3088241944378964404772302e-3 * c(m-1) + 0.4227226173449345042468960e-3 * c(m) + 0.1190362071861893051132274e1 * c(m-5) -0.2720908083525083608370563e-1 * c(m-7) - 0.1931148612480615118957263e0 * c(m-6) - 0.3332941113251635390801278e-1 * c(m-3) + 0.2605582646183255957264249e-3 * c(m-2) + 0.1432116665752147607469646e-2 * c(m-1) + 0.1207544072304193806052558e-2 * c(m) - 0.1348436986667115543203552e1 * c(m-5) + 0.1687723507780044227927853e-1 * c(m-4) 0.3590669644811151307464697e-1 * c(m-6) - 0.5925443480724830632401754e0 * c(m-4) - 0.6897142765790609546343709e-2 * c(m-2) - 0.5363098747528542488971874e-2 * c(m-1) - 0.5205147429855955657625694e-2 * c(m) + 0.9977064356292750529201981e0 * c(m-5) 0.7272438906214475928744770e-1 * c(m-4) + 0.1964682777744275219350831e-1 * c(m-3) - 0.5952475275883259619711594e-2 * c(m-1) - 0.1635430866921887819487473e-2 * c(m) + 0.2921234010758621482958052e-1 * c(m-6) - 0.4659516693228870973898560e0 * c(m-5) 0.5891947149681041048896399e-1 * c(m-4) + 0.1858378996391679448655070e-1 * c(m-3) + 0.7240905383565181316381731e-2 * c(m-2) + 0.2349927974590068869356781e-1 * c(m) - 0.4046360079256766884300687e-1 * c(m-6) + 0.1223513270418807666970488e0 * c(m-5) -0.2404661162020836566908542e-1 * c(m-4) - 0.7348845587775519698437916e-2 * c(m-3) - 0.8105784530576404277872603e-3 * c(m-2) + 0.9574633163221758060736592e-2 * c(m-1) - 0.1828896813877197352675410e-1 * c(m) + 0.1350326632905990039353503e-1 * c(m-6) - 0.1315967038382618382356495e-1 * c(m-5); 0.5522702088127090209264064e-3 * c(m-7) - 0.3265435411305071914756373e-2 * c(m-6) + 0.7462059484530855073291365e-2 * c(m-5) - 0.4748894282038492179461399e-2 * c(m-4) 0.5528052133944605740009217e-1 * c(m-6) - 0.8631683980217122275970376e-1 * c(m-5) - 0.3276463639080639163926118e-1 * c(m-7) + 0.5268984374242044588776166e-1 * c(m-4) 0.5331362125287625412555844e-1 * c(m-7) + 0.1736073411355428563685818e0 * c(m-5) + 0.8671038084174692625075159e-2 * c(m-3) + 0.8084259844422177692569663e-1 * c(m-6) - 0.1664345989168155800449120e0 * c(m-4) -0.2720908083525083608370563e-1 * c(m-7) - 0.1931148612480615118957263e0 * c(m-6) - 0.3332941113251635390801278e-1 * c(m-3) + 0.2605582646183255957264249e-3 * c(m-2) + 0.1432116665752147607469646e-2 * c(m-1) + 0.1207544072304193806052558e-2 * c(m) - 0.1348436986667115543203552e1 * c(m-5) + 0.1687723507780044227927853e-1 * c(m-4) 0.6107825764368264576481962e-2 * c(m-7) + 0.1155752633643216628010304e0 * c(m-6) + 0.2096413329579026439044119e1 * c(m-5) + 0.3357721707576477199985656e0 * c(m-3) + 0.3291545083271862858501887e-3 * c(m-2) + 0.6641183499427826101618457e-2 * c(m-1) + 0.3449455095910233625229891e-2 * c(m) + 0.8270696421223286922584620e0 * c(m-4) -0.4995827370863505253765970e-1 * c(m-6) - 0.1263507837371824205693950e1 * c(m-5) - 0.8712928907711754187084757e-2 * c(m-2) - 0.2487040599390160764166412e-1 * c(m-1) - 0.1486895819265604128572498e-1 * c(m) - 0.1726565392121567634950213e1 * c(m-4) 0.5402985338373433052255418e0 * c(m-5) - 0.1979290298620869974478871e0 * c(m-3) - 0.2760353365637712827793337e-1 * c(m-1) - 0.4671751091575462868310238e-2 * c(m) - 0.6952587985456154591014641e-1 * c(m-6) + 0.1571507277911208446562686e1 * c(m-4) -0.1319358558853174530078498e0 * c(m-5) - 0.1872196143003808021730728e0 * c(m-3) + 0.9147192682075630179962131e-2 * c(m-2) + 0.6712774475803763988977355e-1 * c(m) + 0.9630407686703666967100804e-1 * c(m-6) - 0.6882132817757726722141421e0 * c(m-4) 0.1241625568998496895352046e-1 * c(m-5) + 0.7403484645316174090533193e-1 * c(m-3) - 0.1023976547309387874453988e-2 * c(m-2) + 0.4440063948509876221050939e-1 * c(m-1) - 0.5224403464202056316702078e-1 * c(m) - 0.3213800979246298453953842e-1 * c(m-6) + 0.1178181682424363524005403e0 * c(m-4); 0.6272075574042975468177820e-3 * c(m-6) - 0.1200429618441003833696998e-1 * c(m-5) + 0.1137708862700574079015220e-1 * c(m-4) -0.5373770512016897565958305e-2 * c(m-6) + 0.1453421858063658498587377e0 * c(m-5) - 0.1399684152943489522927794e0 * c(m-4) -0.5013247356072127938999311e0 * c(m-5) + 0.5021853752328231128475915e0 * c(m-4) - 0.1197175073672143005877150e-1 * c(m-6) 0.3590669644811151307464697e-1 * c(m-6) - 0.5925443480724830632401754e0 * c(m-4) - 0.6897142765790609546343709e-2 * c(m-2) - 0.5363098747528542488971874e-2 * c(m-1) - 0.5205147429855955657625694e-2 * c(m) + 0.9977064356292750529201981e0 * c(m-5) -0.4995827370863505253765970e-1 * c(m-6) - 0.1263507837371824205693950e1 * c(m-5) - 0.8712928907711754187084757e-2 * c(m-2) - 0.2487040599390160764166412e-1 * c(m-1) - 0.1486895819265604128572498e-1 * c(m) - 0.1726565392121567634950213e1 * c(m-4) 0.2760393423824887721078848e-1 * c(m-6) + 0.1190550338687608873798462e1 * c(m-5) + 0.4253084328734353394994388e1 * c(m-4) + 0.2306367624634749229113646e0 * c(m-2) + 0.9313657638804699948929701e-1 * c(m-1) + 0.6409299775987186986730499e-1 * c(m) -0.8755807343482262259774782e0 * c(m-5) - 0.3645285178085761821545207e1 * c(m-4) + 0.1033717994630886401730470e0 * c(m-1) + 0.2013769413884797246646959e-1 * c(m) + 0.4106783858513785463625543e-1 * c(m-6) 0.3956598149904136332753521e0 * c(m-5) + 0.1630560443616104907615866e1 * c(m-4) - 0.2421320004064592721552708e0 * c(m-2) - 0.2893557395653431666593814e0 * c(m) - 0.5688529641249387985434413e-1 * c(m-6) -0.7684117160199014594442072e-1 * c(m-5) - 0.2928439026361256842196229e0 * c(m-4) + 0.2710530961648671297733465e-1 * c(m-2) - 0.1662748711097054895317080e0 * c(m-1) + 0.2251991532891353212689574e0 * c(m) + 0.1898341454096471754822498e-1 * c(m-6); 0.9209089963443799485648361e-2 * c(m-5) - 0.3129629392354775191148163e-3 * c(m-6) - 0.8896127024208321966533544e-2 * c(m-4) -0.1059816030196818445908057e0 * c(m-5) + 0.1014880675788250237247178e0 * c(m-4) + 0.4493535440856820866087846e-2 * c(m-6) 0.3328335104489738933610597e0 * c(m-5) - 0.3179803804558436283847901e0 * c(m-4) - 0.5111353189352474549563559e-2 * c(m-3) - 0.9741776803777790426705996e-2 * c(m-6) 0.7272438906214475928744770e-1 * c(m-4) + 0.1964682777744275219350831e-1 * c(m-3) - 0.5952475275883259619711594e-2 * c(m-1) - 0.1635430866921887819487473e-2 * c(m) + 0.2921234010758621482958052e-1 * c(m-6) - 0.4659516693228870973898560e0 * c(m-5) 0.5402985338373433052255418e0 * c(m-5) - 0.1979290298620869974478871e0 * c(m-3) - 0.2760353365637712827793337e-1 * c(m-1) - 0.4671751091575462868310238e-2 * c(m) - 0.6952587985456154591014641e-1 * c(m-6) + 0.1571507277911208446562686e1 * c(m-4) -0.8755807343482262259774782e0 * c(m-5) - 0.3645285178085761821545207e1 * c(m-4) + 0.1033717994630886401730470e0 * c(m-1) + 0.2013769413884797246646959e-1 * c(m) + 0.4106783858513785463625543e-1 * c(m-6) 0.1070920689960817104203947e1 * c(m-5) + 0.3717418466925056542408153e1 * c(m-4) + 0.1166740554279680007487795e0 * c(m-3) + 0.1147318200715868527529827e0 * c(m-1) + 0.6327161147136873807796515e-2 * c(m) + 0.6235373239336055200426697e-1 * c(m-6) -0.6639605248735044787146222e0 * c(m-5) - 0.1865625445986772763641423e1 * c(m-4) + 0.1103611313171476425250639e0 * c(m-3) - 0.9091410269992464604926176e-1 * c(m) - 0.8636954541126674177407762e-1 * c(m-6) 0.1582127073537215443965653e0 * c(m-5) + 0.3746489300753517635549495e0 * c(m-4) - 0.4364163147111892346990101e-1 * c(m-3) - 0.1845476106024151050283847e0 * c(m-1) + 0.7075642937243715046279337e-1 * c(m) + 0.2882271848190011329385407e-1 * c(m-6); -0.3576545132696983143406173e-2 * c(m-5) + 0.4335019854436220306755673e-3 * c(m-6) + 0.3143043147253361112730605e-2 * c(m-4) 0.4093499466767054661591066e-1 * c(m-5) - 0.3471075437892810033585296e-1 * c(m-4) - 0.6224240288742446280057699e-2 * c(m-6) -0.1080774142196007991746827e0 * c(m-5) + 0.9941834083648937298100811e-1 * c(m-4) - 0.4834791406446907590553793e-2 * c(m-3) + 0.1349386478955833378422842e-1 * c(m-6) 0.5891947149681041048896399e-1 * c(m-4) + 0.1858378996391679448655070e-1 * c(m-3) + 0.7240905383565181316381731e-2 * c(m-2) + 0.2349927974590068869356781e-1 * c(m) - 0.4046360079256766884300687e-1 * c(m-6) + 0.1223513270418807666970488e0 * c(m-5) -0.1319358558853174530078498e0 * c(m-5) - 0.1872196143003808021730728e0 * c(m-3) + 0.9147192682075630179962131e-2 * c(m-2) + 0.6712774475803763988977355e-1 * c(m) + 0.9630407686703666967100804e-1 * c(m-6) - 0.6882132817757726722141421e0 * c(m-4) 0.3956598149904136332753521e0 * c(m-5) + 0.1630560443616104907615866e1 * c(m-4) - 0.2421320004064592721552708e0 * c(m-2) - 0.2893557395653431666593814e0 * c(m) - 0.5688529641249387985434413e-1 * c(m-6) -0.6639605248735044787146222e0 * c(m-5) - 0.1865625445986772763641423e1 * c(m-4) + 0.1103611313171476425250639e0 * c(m-3) - 0.9091410269992464604926176e-1 * c(m) - 0.8636954541126674177407762e-1 * c(m-6) 0.4681819359722749441073885e0 * c(m-5) + 0.1015239189167790053447110e1 * c(m-4) + 0.1043897828092562609502636e0 * c(m-3) + 0.2542001760457345743492403e0 * c(m-2) + 0.1306332157111667628555907e1 * c(m) + 0.1196351539550049336518187e0 * c(m-6) -0.1195777325611201766551392e0 * c(m-5) - 0.2187310061229745694542609e0 * c(m-4) - 0.4128029838349298819825156e-1 * c(m-3) - 0.2845627370491611369031341e-1 * c(m-2) - 0.1016689339350338144430605e1 * c(m) - 0.3992391469197282238624438e-1 * c(m-6); 0.5593983696629863059347067e-3 * c(m-5) - 0.1446656414398166805849327e-3 * c(m-6) - 0.4147327282231696253497740e-3 * c(m-4) -0.6486184157331537899459796e-2 * c(m-5) + 0.4409068609809831485979484e-2 * c(m-4) + 0.2077115547521706413480312e-2 * c(m-6) 0.1319674981073749167009902e-1 * c(m-5) - 0.1060554802883657391328704e-1 * c(m-4) + 0.1911888563316170927411831e-2 * c(m-3) - 0.4503090345217088684223814e-2 * c(m-6) -0.2404661162020836566908542e-1 * c(m-4) - 0.7348845587775519698437916e-2 * c(m-3) - 0.8105784530576404277872603e-3 * c(m-2) + 0.9574633163221758060736592e-2 * c(m-1) - 0.1828896813877197352675410e-1 * c(m) + 0.1350326632905990039353503e-1 * c(m-6) - 0.1315967038382618382356495e-1 * c(m-5) 0.1241625568998496895352046e-1 * c(m-5) + 0.7403484645316174090533193e-1 * c(m-3) - 0.1023976547309387874453988e-2 * c(m-2) + 0.4440063948509876221050939e-1 * c(m-1) - 0.5224403464202056316702078e-1 * c(m) - 0.3213800979246298453953842e-1 * c(m-6) + 0.1178181682424363524005403e0 * c(m-4) -0.7684117160199014594442072e-1 * c(m-5) - 0.2928439026361256842196229e0 * c(m-4) + 0.2710530961648671297733465e-1 * c(m-2) - 0.1662748711097054895317080e0 * c(m-1) + 0.2251991532891353212689574e0 * c(m) + 0.1898341454096471754822498e-1 * c(m-6) 0.1582127073537215443965653e0 * c(m-5) + 0.3746489300753517635549495e0 * c(m-4) - 0.4364163147111892346990101e-1 * c(m-3) - 0.1845476106024151050283847e0 * c(m-1) + 0.7075642937243715046279337e-1 * c(m) + 0.2882271848190011329385407e-1 * c(m-6) -0.1195777325611201766551392e0 * c(m-5) - 0.2187310061229745694542609e0 * c(m-4) - 0.4128029838349298819825156e-1 * c(m-3) - 0.2845627370491611369031341e-1 * c(m-2) - 0.1016689339350338144430605e1 * c(m) - 0.3992391469197282238624438e-1 * c(m-6) 0.3167964748016105299646518e-1 * c(m-5) + 0.4976563420877041544013670e-1 * c(m-4) + 0.1632404042590951953384672e-1 * c(m-3) + 0.3185519088796429015220016e-2 * c(m-2) + 0.2968472090638000742888467e0 * c(m-1) + 0.7912667594695582093926295e0 * c(m) + 0.1332316557164627464149716e-1 * c(m-6);]; +% +% M(5,10)=M(10,5); +% M(m-4,m-9)=M(m-9,m-4); +% +% M=M/h; +% +% D2=HI*(-M-diag(c)*e_1*S_1+diag(c)*e_m*S_m); + +S2_U=[0.15e2 / 0.4e1 -0.77e2 / 0.6e1 0.107e3 / 0.6e1 -13 0.61e2 / 0.12e2 -0.5e1 / 0.6e1;]/h^2; +S2_1=zeros(1,m); +S2_1(1:6)=S2_U; +S2_m=zeros(1,m); +S2_m(m-5:m)=fliplr(S2_U); + + + + + +% Fourth derivative, 1th order accurate at first 8 boundary points (still +% yield 5th order convergence if stable: for example u_tt=-u_xxxx + +m5=-0.41e2 / 0.7560e4;m4= 0.1261e4 / 0.15120e5;m3= -0.541e3 / 0.840e3; m2= 0.4369e4 / 0.1260e4; m1= -0.1669e4 / 0.180e3; m0= 0.1529e4 / 0.120e3; + + +M4=m5*(diag(ones(m-5,1),5)+diag(ones(m-5,1),-5))+m4*(diag(ones(m-4,1),4)+diag(ones(m-4,1),-4))+m3*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3))+m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0); + +%M4=(-1/6*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3) ) + 2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2)) -13/2*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)) + 28/3*diag(ones(m,1),0)); + +M4_U=[0.151705142321e12 / 0.29189160000e11 -0.25643455801727e14 / 0.1634592960000e13 0.286417898677e12 / 0.15135120000e11 -0.4038072020317e13 / 0.326918592000e12 0.96455968907e11 / 0.20432412000e11 -0.151076916769e12 / 0.181621440000e12 0.14511526363e11 / 0.408648240000e12 -0.196663079e9 / 0.33359040000e11; -0.25643455801727e14 / 0.1634592960000e13 0.735383382473e12 / 0.14594580000e11 -0.5035391734409e13 / 0.77837760000e11 0.20392440917e11 / 0.467026560e9 -0.109540902413e12 / 0.6671808000e10 0.2488686539e10 / 0.884520000e9 -0.2798067539e10 / 0.33359040000e11 0.6433463591e10 / 0.408648240000e12; 0.286417898677e12 / 0.15135120000e11 -0.5035391734409e13 / 0.77837760000e11 0.145019791981e12 / 0.1621620000e10 -0.333577111061e12 / 0.5189184000e10 0.18928722391e11 / 0.778377600e9 -0.93081704557e11 / 0.25945920000e11 -0.372660319e9 / 0.3243240000e10 0.2861399869e10 / 0.544864320000e12; -0.4038072020317e13 / 0.326918592000e12 0.20392440917e11 / 0.467026560e9 -0.333577111061e12 / 0.5189184000e10 0.59368471277e11 / 0.1167566400e10 -0.201168708569e12 / 0.9340531200e10 0.1492314487e10 / 0.432432000e9 0.1911896257e10 / 0.9340531200e10 0.24383341e8 / 0.2554051500e10; 0.96455968907e11 / 0.20432412000e11 -0.109540902413e12 / 0.6671808000e10 0.18928722391e11 / 0.778377600e9 -0.201168708569e12 / 0.9340531200e10 0.1451230301e10 / 0.106142400e9 -0.103548247007e12 / 0.15567552000e11 0.27808437809e11 / 0.11675664000e11 -0.36870830713e11 / 0.65383718400e11; -0.151076916769e12 / 0.181621440000e12 0.2488686539e10 / 0.884520000e9 -0.93081704557e11 / 0.25945920000e11 0.1492314487e10 / 0.432432000e9 -0.103548247007e12 / 0.15567552000e11 0.1229498243e10 / 0.115830000e9 -0.32222519717e11 / 0.3706560000e10 0.470092704233e12 / 0.136216080000e12; 0.14511526363e11 / 0.408648240000e12 -0.2798067539e10 / 0.33359040000e11 -0.372660319e9 / 0.3243240000e10 0.1911896257e10 / 0.9340531200e10 0.27808437809e11 / 0.11675664000e11 -0.32222519717e11 / 0.3706560000e10 0.11547819313e11 / 0.912161250e9 -0.15187033999199e14 / 0.1634592960000e13; -0.196663079e9 / 0.33359040000e11 0.6433463591e10 / 0.408648240000e12 0.2861399869e10 / 0.544864320000e12 0.24383341e8 / 0.2554051500e10 -0.36870830713e11 / 0.65383718400e11 0.470092704233e12 / 0.136216080000e12 -0.15187033999199e14 / 0.1634592960000e13 0.33832994693e11 / 0.2653560000e10;]; + +M4(1:8,1:8)=M4_U; + +M4(m-7:m,m-7:m)=flipud( fliplr( M4_U ) ); +M4=M4/h^3; + +S3_U=[-0.17e2 / 0.4e1 0.71e2 / 0.4e1 -0.59e2 / 0.2e1 0.49e2 / 0.2e1 -0.41e2 / 0.4e1 0.7e1 / 0.4e1;]/h^3; +S3_1=zeros(1,m); +S3_1(1:6)=S3_U; +S3_m=zeros(1,m); +S3_m(m-5:m)=fliplr(-S3_U); + +D4=HI*(M4-e_1*S3_1+e_m*S3_m + S_1'*S2_1-S_m'*S2_m); + +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+scheme/Beam.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,212 @@ +classdef Beam < scheme.Scheme + properties + order % Order accuracy for the approximation + grid + + D % non-stabalized scheme operator + alpha + + h + H % Discrete norm + Hi + + e_l, e_r + d1_l, d1_r + d2_l, d2_r + d3_l, d3_r + gamm + delt + alphaII + alphaIII + + opt + end + + methods + function obj = Beam(grid, order, alpha, opsGen, opt) + default_arg('alpha', -1); + + + opt_default.interface_l.tuning = 1.1; + opt_default.interface_l.tau = []; + opt_default.interface_l.sig = []; + opt_default.interface_r.tuning = 1.1; + opt_default.interface_r.tau = []; + opt_default.interface_r.sig = []; + default_struct('opt', opt_default); + + + % default_arg('opsGen', @sbp.Higher); + default_arg('opsGen', @sbp.HigherVariable); % Supposed to be better + + if ~isa(grid, 'grid.Cartesian') || grid.D() ~= 1 + error('Grid must be 1d cartesian'); + end + + obj.grid = grid; + obj.order = order; + obj.alpha = alpha; + + m = grid.m; + h = grid.scaling(); + + ops = opsGen(m, h, order); + + I = speye(m); + + D4 = sparse(ops.derivatives.D4); + obj.H = sparse(ops.norms.H); + obj.Hi = sparse(ops.norms.HI); + obj.e_l = sparse(ops.boundary.e_1); + obj.e_r = sparse(ops.boundary.e_m); + obj.d1_l = sparse(ops.boundary.S_1); + obj.d1_r = sparse(ops.boundary.S_m); + obj.d2_l = sparse(ops.boundary.S2_1); + obj.d2_r = sparse(ops.boundary.S2_m); + obj.d3_l = sparse(ops.boundary.S3_1); + obj.d3_r = sparse(ops.boundary.S3_m); + + obj.D = alpha*D4; + + alphaII = ops.borrowing.N.S2/2; + alphaIII = ops.borrowing.N.S3/2; + + obj.gamm = h*alphaII; + obj.delt = h^3*alphaIII; + obj.alphaII = alphaII; + obj.alphaIII = alphaIII; + obj.h = h; + obj.opt = opt; + end + + + % Closure functions return the opertors applied to the own doamin to close the boundary + % Penalty functions return the opertors to force the solution. In the case of an interface it returns the operator applied to the other doamin. + % boundary is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'. + % type is a string specifying the type of boundary condition if there are several. + % neighbour_scheme is an instance of Scheme that should be interfaced to. + % neighbour_boundary is a string specifying which boundary to interface to. + function [closure, penalty] = boundary_condition(obj,boundary,type) + default_arg('type','dn'); + + [e, d1, d2, d3, s] = obj.get_boundary_ops(boundary); + gamm = obj.gamm; + delt = obj.delt; + + switch type + case {'dn'} % Dirichlet-neumann boundary condition + alpha = obj.alpha; + + % tau1 < -alpha^2/gamma + % tuning = 2; + tuning = 1.1; + + tau1 = tuning * alpha/delt; + tau4 = s*alpha; + + sig2 = tuning * alpha/gamm; + sig3 = -s*alpha; + + tau = tau1*e+tau4*d3; + sig = sig2*d1+sig3*d2; + + closure = obj.Hi*(tau*e' + sig*d1'); + + penalty{1} = -obj.Hi*tau; + penalty{2} = -obj.Hi*sig; + otherwise % Unknown, boundary condition + error('No such boundary condition: type = %s',type); + end + end + + function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary) + % u denotes the solution in the own domain + % v denotes the solution in the neighbour domain + [e_u,d1_u,d2_u,d3_u,s_u] = obj.get_boundary_ops(boundary); + [e_v,d1_v,d2_v,d3_v,s_v] = neighbour_scheme.get_boundary_ops(neighbour_boundary); + + + alpha_u = obj.alpha; + alpha_v = neighbour_scheme.alpha; + + + switch boundary + case 'l' + interface_opt = obj.opt.interface_l; + case 'r' + interface_opt = obj.opt.interface_r; + end + + + if isempty(interface_opt.tau) && isempty(interface_opt.sig) + gamm_u = obj.gamm; + delt_u = obj.delt; + + gamm_v = neighbour_scheme.gamm; + delt_v = neighbour_scheme.delt; + + tuning = interface_opt.tuning; + + tau1 = ((alpha_u/2)/delt_u + (alpha_v/2)/delt_v)/2*tuning; + sig2 = ((alpha_u/2)/gamm_u + (alpha_v/2)/gamm_v)/2*tuning; + else + h_u = obj.h; + h_v = neighbour_scheme.h; + + switch neighbour_boundary + case 'l' + neighbour_interface_opt = neighbour_scheme.opt.interface_l; + case 'r' + neighbour_interface_opt = neighbour_scheme.opt.interface_r; + end + + tau_u = interface_opt.tau; + sig_u = interface_opt.sig; + tau_v = neighbour_interface_opt.tau; + sig_v = neighbour_interface_opt.sig; + + tau1 = tau_u/h_u^3 + tau_v/h_v^3; + sig2 = sig_u/h_u + sig_v/h_v; + end + + tau4 = s_u*alpha_u/2; + sig3 = -s_u*alpha_u/2; + phi2 = s_u*1/2; + psi1 = -s_u*1/2; + + tau = tau1*e_u + tau4*d3_u; + sig = sig2*d1_u + sig3*d2_u ; + phi = phi2*d1_u ; + psi = psi1*e_u ; + + closure = obj.Hi*(tau*e_u' + sig*d1_u' + phi*alpha_u*d2_u' + psi*alpha_u*d3_u'); + penalty = -obj.Hi*(tau*e_v' + sig*d1_v' + phi*alpha_v*d2_v' + psi*alpha_v*d3_v'); + end + + % Returns the boundary ops and sign for the boundary specified by the string boundary. + % The right boundary is considered the positive boundary + function [e, d1, d2, d3, s] = get_boundary_ops(obj,boundary) + switch boundary + case 'l' + e = obj.e_l; + d1 = obj.d1_l; + d2 = obj.d2_l; + d3 = obj.d3_l; + s = -1; + case 'r' + e = obj.e_r; + d1 = obj.d1_r; + d2 = obj.d2_r; + d3 = obj.d3_r; + s = 1; + otherwise + error('No such boundary: boundary = %s',boundary); + end + end + + function N = size(obj) + N = obj.grid.N; + end + + end +end
--- a/+scheme/Beam2d.m Tue Sep 06 10:36:33 2016 +0200 +++ b/+scheme/Beam2d.m Thu Sep 08 08:47:16 2016 +0200 @@ -1,10 +1,6 @@ classdef Beam2d < scheme.Scheme properties - m % Number of points in each direction, possibly a vector - N % Number of points total - h % Grid spacing - u,v % Grid - x,y % Values of x and y for each grid point + grid order % Order accuracy for the approximation D % non-stabalized scheme operator @@ -27,21 +23,23 @@ methods function obj = Beam2d(m,lim,order,alpha,opsGen) + default_arg('alpha',1); default_arg('opsGen',@sbp.Higher); - default_arg('a',1); - if length(m) == 1 - m = [m m]; + if ~isa(grid, 'grid.Cartesian') || grid.D() ~= 2 + error('Grid must be 2d cartesian'); end - m_x = m(1); - m_y = m(2); + obj.grid = grid; + obj.alpha = alpha; + obj.order = order; - xlim = lim{1}; - ylim = lim{2}; + m_x = grid.m(1); + m_y = grid.m(2); - [x, h_x] = util.get_grid(xlim{:},m_x); - [y, h_y] = util.get_grid(ylim{:},m_y); + h = grid.scaling(); + h_x = h(1); + h_y = h(2); ops_x = opsGen(m_x,h_x,order); ops_y = opsGen(m_y,h_y,order); @@ -49,9 +47,6 @@ I_x = speye(m_x); I_y = speye(m_y); - - - D4_x = sparse(ops_x.derivatives.D4); H_x = sparse(ops_x.norms.H); Hi_x = sparse(ops_x.norms.HI); @@ -105,16 +100,7 @@ obj.d3_s = kr(I_x,d3_l_y); obj.d3_n = kr(I_x,d3_r_y); - obj.m = m; - obj.h = [h_x h_y]; - obj.order = order; - - obj.alpha = alpha; obj.D = alpha*D4; - obj.u = x; - obj.v = y; - obj.x = kr(x,ones(m_y,1)); - obj.y = kr(ones(m_x,1),y); obj.gamm_x = h_x*ops_x.borrowing.N.S2/2; obj.delt_x = h_x^3*ops_x.borrowing.N.S3/2;
--- a/+scheme/Scheme.m Tue Sep 06 10:36:33 2016 +0200 +++ b/+scheme/Scheme.m Thu Sep 08 08:47:16 2016 +0200 @@ -1,42 +1,44 @@ -% Start with all matrix returns. When that works see how we should generalize to non-matrix stuff/nonlinear +% Start with all matrix returns. When that works see how we should generalize +% to non-matrix stuff/nonlinear classdef Scheme < handle properties (Abstract) - m % Number of points in each direction, possibly a vector order % Order accuracy for the approximation - % vectors u,v,w depending on dim that gives were gridpoints are in each dimension - % vectors x,y,z containing the x,y,z values corresponding to each grid point - % matrices X,Y,Z with point coordinates as multi dimensional vectors + grid D % non-stabalized scheme operator H % Discrete norm - - % Should also containg: - % the grid points used - % the grid spacing end methods (Abstract) - % Closure functions return the opertors applied to the own doamin to close the boundary - % Penalty functions return the opertors to force the solution. In the case of an interface it returns the operator applied to the other doamin. - % boundary is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'. - % type is a string specifying the type of boundary condition if there are several. - % data is a function returning the data that should be applied at the boundary. - % neighbour_scheme is an instance of Scheme that should be interfaced to. - % neighbour_boundary is a string specifying which boundary to interface to. - m = boundary_condition(obj,boundary,type,data) - m = interface(obj,boundary,neighbour_scheme,neighbour_boundary) - N = size(obj) % Returns the number of degrees of freedom. + % Closure functions return the opertors applied to the own doamin to + % close the boundary Penalty functions return the opertors to force + % the solution. In the case of an interface it returns the operator + % applied to the other doamin. In some cases the penalty return value + % can be ommited and the closure function take care of both parts. + % boundary is a string specifying the boundary e.g. + % 'l','r' or 'e','w','n','s'. + % type is a string specifying the type of + % boundary condition if there are several. + % neighbour_scheme is an instance of Scheme that should be + % interfaced to. + % neighbour_boundary is a string specifying which boundary to + % interface to. + % penalty may be a cell array if there are several penalties with different weights + [closure, penalty] = boundary_condition(obj,boundary,type) + [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary) + % Returns the number of degrees of freedom. + N = size(obj) end methods(Static) - % Calculates the matrcis need for the inteface coupling between boundary bound_u of scheme schm_u - % and bound_v of scheme schm_v. - % [uu, uv, vv, vu] = inteface_couplong(A,'r',B,'l') + % Calculates the matrcis need for the inteface coupling between + % boundary bound_u of scheme schm_u and bound_v of scheme schm_v. + % [uu, uv, vv, vu] = inteface_coupling(A,'r',B,'l') function [uu, uv, vv, vu] = interface_coupling(schm_u,bound_u,schm_v,bound_v) [uu,uv] = schm_u.interface(bound_u,schm_v,bound_v); [vv,vu] = schm_v.interface(bound_v,schm_u,bound_u); end end -end \ No newline at end of file +end
--- a/+time/Cdiff.m Tue Sep 06 10:36:33 2016 +0200 +++ b/+time/Cdiff.m Thu Sep 08 08:47:16 2016 +0200 @@ -12,22 +12,20 @@ methods + % Solves u_tt = Du + Eu_t + S + % D, E, S can either all be constants or all be function handles, + % They can also be omitted by setting them equal to the empty matrix. + % Cdiff(D, E, S, k, t0, n0, v, v_prev) function obj = Cdiff(D, E, S, k, t0, n0, v, v_prev) - % Cdiff(D, E, S, k, t0, n0, v, v_prev) - m = size(D,1); + m = length(v); default_arg('E',sparse(m,m)); default_arg('S',sparse(m,1)); - if ~(issparse(D) && issparse(E) && issparse(S)) - warning('One of the matrices D, E, S is not sparse!') - print_issparse(D) - print_issparse(E) - print_issparse(S) - end - obj.D = D; obj.E = E; obj.S = S; + + obj.k = k; obj.t = t0; obj.n = n0;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+time/CdiffTimeDep.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,51 @@ +classdef CdiffTimeDep < time.Timestepper + properties + D + E + S + k + t + v + v_prev + n + end + + + methods + % Solves u_tt = Du + E(t)u_t + S(t) + % D, E, S can either all be constants or all be function handles, + % They can also be omitted by setting them equal to the empty matrix. + % CdiffTimeDep(D, E, S, k, t0, n0, v, v_prev) + function obj = CdiffTimeDep(D, E, S, k, t0, n0, v, v_prev) + m = length(v); + default_arg('E', @(t)sparse(m,m)); + default_arg('S', @(t)sparse(m,1)); + + obj.D = D; + obj.E = E; + obj.S = S; + + obj.k = k; + obj.t = t0; + obj.n = n0; + obj.v = v; + obj.v_prev = v_prev; + end + + function [v,t] = getV(obj) + v = obj.v; + t = obj.t; + end + + function [vt,t] = getVt(obj) + vt = (obj.v-obj.v_prev)/obj.k; % Could be improved using u_tt = f(u)) + t = obj.t; + end + + function obj = step(obj) + [obj.v, obj.v_prev] = time.cdiff.cdiff(obj.v, obj.v_prev, obj.k, obj.D, obj.E(obj.t), obj.S(obj.t)); + obj.t = obj.t + obj.k; + obj.n = obj.n + 1; + end + end +end \ No newline at end of file
--- a/+time/Rungekutta4SecondOrder.m Tue Sep 06 10:36:33 2016 +0200 +++ b/+time/Rungekutta4SecondOrder.m Thu Sep 08 08:47:16 2016 +0200 @@ -15,6 +15,9 @@ methods + % Solves u_tt = Du + Eu_t + S + % D, E, S can either all be constants or all be function handles, + % They can also be omitted by setting them equal to the empty matrix. function obj = Rungekutta4SecondOrder(D, E, S, k, t0, v0, v0t) obj.D = D; obj.E = E; @@ -22,21 +25,63 @@ obj.m = length(v0); obj.n = 0; - I = speye(obj.m); - O = sparse(obj.m,obj.m); - obj.M = [O, I; D, E*I]; % Multiply with I to allow 0 as input. + + if isa(D, 'function_handle') || isa(E, 'function_handle') || isa(S, 'function_handle') + default_arg('D', @(t)sparse(obj.m, obj.m)); + default_arg('E', @(t)sparse(obj.m, obj.m)); + default_arg('S', @(t)sparse(obj.m, 1) ); + + if ~isa(D, 'function_handle') + D = @(t)D; + end + if ~isa(D, 'function_handle') + E = @(t)E; + end + if ~isa(D, 'function_handle') + S = @(t)S; + end - if S == 0 - obj.C = zeros(2*obj.m,1); + I = speye(obj.m); + O = sparse(obj.m,obj.m); + + obj.M = @(t)[ + O, I; + D(t), E(t); + ]; + obj.C = @(t)[ + zeros(obj.m,1); + S(t); + ]; + + obj.k = k; + obj.t = t0; + obj.w = [v0; v0t]; + + obj.F = @(w,t)(obj.M(t)*w + obj.C(t)); else - obj.C = [zeros(obj.m,1), S]; - end + + default_arg('D', sparse(obj.m, obj.m)); + default_arg('E', sparse(obj.m, obj.m)); + default_arg('S', sparse(obj.m, 1) ); + + I = speye(obj.m); + O = sparse(obj.m,obj.m); - obj.k = k; - obj.t = t0; - obj.w = [v0; v0t]; + obj.M = [ + O, I; + D, E; + ]; + obj.C = [ + zeros(obj.m,1); + S; + ]; - obj.F = @(w,t)(obj.M*w + obj.C); + obj.k = k; + obj.t = t0; + obj.w = [v0; v0t]; + + obj.F = @(w,t)(obj.M*w + obj.C); + end end function [v,t] = getV(obj)
--- a/+util/ReplaceableString.m Tue Sep 06 10:36:33 2016 +0200 +++ b/+util/ReplaceableString.m Thu Sep 08 08:47:16 2016 +0200 @@ -34,10 +34,20 @@ obj.display(); end + function show(obj, varargin) + obj.param = varargin; + obj.display(); + end + + function remove(obj) + obj.update(''); + end + function display(obj) reverseStr = repmat(sprintf('\b'), 1, obj.n); - newStr = padStr(sprintf(obj.fmt, obj.param{:}),obj.n); - fprintf([reverseStr, newStr]); + cleareStr = repmat(sprintf(' '), 1, obj.n); + newStr = sprintf(obj.fmt, obj.param{:}); + fprintf([reverseStr, cleareStr, reverseStr, newStr]); obj.n = length(newStr); end
--- a/cell2sparse.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,54 +0,0 @@ -function A = cell2sparse(C) - - if isempty(C) - A = sparse([]); - return - end - - n = row_height(C); - m = col_width(C); - - N = sum(n); - M = sum(m); - - A = sparse(N,M); - - n_ind = [0 cumsum(n)]; - m_ind = [0 cumsum(m)]; - - for i = 1:size(C,1) - for j = 1:size(C,2) - if ~has_matrix(C{i,j}) - continue - end - A(n_ind(i)+1:n_ind(i+1),m_ind(j)+1:m_ind(j+1)) = C{i,j}; - end - end - -end - -function m = col_width(C) - for j = 1:size(C,2) - for i = 1:size(C,1) - if ~has_matrix(C{i,j}) - continue - end - m(j) = size(C{i,j},2); - end - end -end - -function n = row_height(C) - for i = 1:size(C,1) - for j = 1:size(C,2) - if ~has_matrix(C{i,j}) - continue - end - n(i) = size(C{i,j},1); - end - end -end - -function b = has_matrix(c) - b = ~(isempty(c) || (numel(c)==1 && c == 0)); -end \ No newline at end of file
--- a/cell2vector.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,14 +0,0 @@ -% cell2vector accepts a column cell array of column vectors and returns a columnvector -% with the input concatenated. It also returns the number of elements in each vector. -% cv -- column cell array with column vectors -% v -- vector of the concatenated vectors -% n -- number of elements in each vector before concatenation. Can be used with vector2cell(). -function [v, n] = cell2vector(cv) - v = []; - n = zeros(length(cv),1); - - for i = 1:length(cv) - n(i) = length(cv{i}); - v = [v; cv{i}]; - end -end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/copyWithDefault.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,30 @@ +% Copy the struct src to dest with default values from default +% dest = copyWithDefault(src, default) +function dest = copyWithDefault(src, default) + % src does not have a value => use default + if isempty(src) + dest = default; + return + end + + % src has a value and is not a struct => use src + if ~isstruct(src) + dest = src; + return + end + + + % src has a value and is a struct => add all default fields + dest = src; + + fn = fieldnames(default); + for i = 1:length(fn) + if isfield(src, fn{i}) + srcField = src.(fn{i}); + else + srcField = []; + end + + dest.(fn{i}) = copyWithDefault(srcField, default.(fn{i})); + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/default_struct.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,10 @@ +function default_struct(s, val) + if evalin('caller',sprintf('~exist(''%s'',''var'')',s)) + given = []; + else + given = evalin('caller', s); + end + + final = copyWithDefault(given, val); + assignin('caller', s, final); +end
--- a/minors.m Tue Sep 06 10:36:33 2016 +0200 +++ b/minors.m Thu Sep 08 08:47:16 2016 +0200 @@ -1,4 +1,5 @@ -function [minor, sub] = minors(A) +function [minor, sub] = minors(A, verbose) + default_arg('verbose', true); [n, m] = size(A); if n ~= m @@ -18,8 +19,10 @@ end for i = 1:length(sub) - fprintf('%d:\n', ks{i}); - disp(sub{i}) + if verbose + fprintf('%d:\n', ks{i}); + disp(sub{i}) + end minor(i) = det(sub{i}); end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/reshapeRowMaj.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,12 @@ +% Reshapes a matrix as if it was stored in row major order. +function B = reshapeRowMaj(A, m) + D = length(m); + + if D == 1 + m = [m 1]; + D = 2; + end + + % Reshape and reverse order of indecies + B = permute(reshape(permute(A, ndims(A):-1:1), rot90(m,2)), D:-1:1); +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/reshapeRowMajTest.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,113 @@ +function tests = reshapeRowMajTest() + tests = functiontests(localfunctions); +end + +function test1D(testCase) + in = { + {5,[1; 2; 3; 4; 5]}, + {5,[1 2 3 4 5]}, + }; + out = { + [1; 2; 3; 4; 5], + [1; 2; 3; 4; 5], + }; + for i = 1:length(in) + testCase.verifyEqual(reshapeRowMaj(in{i}{2}, in{i}{1}),out{i}); + end +end + + +function testIdentity(testCase) + in = { + {[2,2], magic(2)}, + {[3,3], magic(3)}, + {[2,3], [1 2 3; 4 5 6]}, + }; + + for i = 1:length(in) + testCase.verifyEqual(reshapeRowMaj(in{i}{2}, in{i}{1}),in{i}{2}); + end +end + +function test2D(testCase) + in = { + {[2,2],[11; 12; 21; 22]}, + {[3,2],[1 2 3; 4 5 6]}, + {[6 1],[1 2 3; 4 5 6]}, + {[1 6],[1 2 3; 4 5 6]}, + }; + + out{1}(1,1) = 11; + out{1}(1,2) = 12; + out{1}(2,1) = 21; + out{1}(2,2) = 22; + + out{2} = [1 2; 3 4; 5 6]; + out{3} = [1; 2; 3; 4; 5; 6]; + out{4} = [1 2 3 4 5 6]; + + for i = 1:length(in) + testCase.verifyEqual(reshapeRowMaj(in{i}{2}, in{i}{1}),out{i}); + end +end + +function test3D(testCase) + in = { + {[2, 2, 2], [111; 112; 121; 122; 211; 212; 221; 222]}, + {[8 1], cat(3,[1 2; 3 4],[5 6; 7 8])}, + {[1 8], cat(3,[1 2; 3 4],[5 6; 7 8])}, + {[2 4], cat(3,[1 2; 3 4],[5 6; 7 8])}, + {[4 2], cat(3,[1 2; 3 4],[5 6; 7 8])}, + }; + + out{1}(1,1,1) = 111; + out{1}(1,1,2) = 112; + out{1}(1,2,1) = 121; + out{1}(1,2,2) = 122; + out{1}(2,1,1) = 211; + out{1}(2,1,2) = 212; + out{1}(2,2,1) = 221; + out{1}(2,2,2) = 222; + + out{2} = [1; 5; 2; 6; 3; 7; 4; 8]; + out{3} = [1 5 2 6 3 7 4 8]; + out{4} = [1 5 2 6; 3 7 4 8]; + out{5} = [1 5; 2 6; 3 7; 4 8]; + + for i = 1:length(in) + testCase.verifyEqual(reshapeRowMaj(in{i}{2}, in{i}{1}),out{i}); + end +end + +function testNonSquare(testCase) + in = { + {[2, 3, 4],[111; 112; 113; 114; 121; 122; 123; 124; 131; 132; 133; 134; 211; 212; 213; 214; 221; 222; 223; 224; 231; 232; 233; 234]}, + }; + out{1}(1,1,1) = 111; + out{1}(1,1,2) = 112; + out{1}(1,1,3) = 113; + out{1}(1,1,4) = 114; + out{1}(1,2,1) = 121; + out{1}(1,2,2) = 122; + out{1}(1,2,3) = 123; + out{1}(1,2,4) = 124; + out{1}(1,3,1) = 131; + out{1}(1,3,2) = 132; + out{1}(1,3,3) = 133; + out{1}(1,3,4) = 134; + out{1}(2,1,1) = 211; + out{1}(2,1,2) = 212; + out{1}(2,1,3) = 213; + out{1}(2,1,4) = 214; + out{1}(2,2,1) = 221; + out{1}(2,2,2) = 222; + out{1}(2,2,3) = 223; + out{1}(2,2,4) = 224; + out{1}(2,3,1) = 231; + out{1}(2,3,2) = 232; + out{1}(2,3,3) = 233; + out{1}(2,3,4) = 234; + for i = 1:length(in) + testCase.verifyEqual(reshapeRowMaj(in{i}{2}, in{i}{1}),out{i}); + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/reshapeToPlotMatrix.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,17 @@ +% Takes a grid function and reshapes it into a matrix of shape m for plotting. +function F = reshapeToPlotMatrix(gf, m) + D = length(m); + + switch D + case 1 + F = gf; + case 2 + F = reshape(gf, rot90(m,2)); + case 3 + % After the reshape the indecies will be M(z,y,x). Plot need them to be M(y,x,z) + p = [2 3 1]; % Permuation + F = permute(reshape(gf,rot90(m,2)), p); + otherwise + error('reshapeToPlotMatrix:NotImplemented','Grid function to matrix is not implemented for dimension = %d', length(m)); + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/reshapeToPlotMatrixTest.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,48 @@ +function tests = reshapeToPlotMatrixTest() + tests = functiontests(localfunctions); +end + +function test1D(testCase) + inGf = [1 2 3 4 5]'; + inM = 5; + out = [1 2 3 4 5]'; + testCase.verifyEqual(reshapeToPlotMatrix(inGf, inM),out); +end + +function test2D(testCase) + x = 1:2; + y = 1:3; + + f = @(x,y) x + y*10; + + xx = [1; 1; 1; 2; 2; 2]; + yy = [1; 2; 3; 1; 2; 3]; + inGf = f(xx,yy); + + [X,Y] = meshgrid(x,y); + out = f(X,Y); + + inM = [2, 3]; + + testCase.verifyEqual(reshapeToPlotMatrix(inGf, inM),out); +end + +function test3D(testCase) + x = 1:2; + y = 1:3; + z = 1:4; + + f = @(x,y,z) x + y*10 + z*100; + + xx = [repmat(1, [12, 1]); repmat(2, [12, 1])]; + yy = repmat([1; 1; 1; 1; 2; 2; 2; 2; 3; 3; 3; 3], [2, 1]); + zz = repmat([1; 2; 3; 4], [6, 1]); + inGf = f(xx,yy,zz); + + [X,Y,Z] = meshgrid(x,y,z); + out = f(X,Y,Z); + + inM = [2, 3, 4]; + + testCase.verifyEqual(reshapeToPlotMatrix(inGf, inM),out); +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/runtestsPackage.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,4 @@ +function res = runtestsPackage(pkgName) + ts = matlab.unittest.TestSuite.fromPackage(pkgName); + res = ts.run(); +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/saveFigureSize.m Thu Sep 08 08:47:16 2016 +0200 @@ -0,0 +1,6 @@ +function saveFigurePosition() + defaultPosition = get(0,'defaultfigureposition'); + f = gcf; + defaultPosition(3:4) = f.Position(3:4); + set(0,'defaultfigureposition',defaultPosition); +end \ No newline at end of file
--- a/semiDefIneq.m Tue Sep 06 10:36:33 2016 +0200 +++ b/semiDefIneq.m Thu Sep 08 08:47:16 2016 +0200 @@ -1,5 +1,6 @@ -function ineq = semiDefIneq(A) - [m, sub] = minors(A); +function ineq = semiDefIneq(A, verbose) + default_arg('verbose', true); + [m, sub] = minors(A, verbose); ineqsys = true; for i = 1:length(m)
--- a/vector2cell.m Tue Sep 06 10:36:33 2016 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,14 +0,0 @@ -% Splits column vector v into segments of length n and returns the result as a column cell array. -% v -- column vector to be split -% n -- number of elements in each part -% -% cv -- cell array of vectors with lenght n(i) -function cv = vector2cell(v,n) - cv = cell(length(n),1); - - ind = [0; cumsum(n)]; - for i = 1:length(n) - ind_i = (ind(i)+1):ind(i+1); - cv{i} = v(ind_i); - end -end \ No newline at end of file