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In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
author Martin Almquist <martin.almquist@it.uu.se>
date Sun, 29 Nov 2015 22:23:09 +0100
parents 32b39dc44474
children
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29
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1 function [H, HI, D1, D2, D3, D4, e_1, e_m, M, M4,Q, Q3, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = higher4(m,h)
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2
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3
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4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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5 %%% 4:de ordn. SBP Finita differens %%%
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6 %%% operatorer framtagna av Ken Mattsson %%%
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7 %%% %%%
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8 %%% 6 randpunkter, diagonal norm %%%
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9 %%% %%%
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10 %%% Datum: 2013-11-11 %%%
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11 %%% %%%
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12 %%% %%%
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13 %%% H (Normen) %%%
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14 %%% D1 (approx f?rsta derivatan) %%%
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15 %%% D2 (approx andra derivatan) %%%
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16 %%% D3 (approx tredje derivatan) %%%
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17 %%% D2 (approx fj?rde derivatan) %%%
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18 %%% %%%
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19 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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20
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21 % M?ste ange antal punkter (m) och stegl?ngd (h)
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22 % Notera att dessa opetratorer ?r framtagna f?r att anv?ndas n?r
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23 % vi har 3de och 4de derivator i v?r PDE
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24 % I annat fall anv?nd de "traditionella" som har noggrannare
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25 % randsplutningar f?r D1 och D2
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27 % Vi b?rjar med normen. Notera att alla SBP operatorer delar samma norm,
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28 % vilket ?r n?dv?ndigt f?r stabilitet
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29
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30 H=diag(ones(m,1),0);
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31 H_U=[0.35809e5 / 0.100800e6 0 0 0 0 0; 0 0.13297e5 / 0.11200e5 0 0 0 0; 0 0 0.5701e4 / 0.5600e4 0 0 0; 0 0 0 0.45109e5 / 0.50400e5 0 0; 0 0 0 0 0.35191e5 / 0.33600e5 0; 0 0 0 0 0 0.33503e5 / 0.33600e5;];
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32
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33 H(1:6,1:6)=H_U;
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34 H(m-5:m,m-5:m)=fliplr(flipud(H_U));
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35 H=H*h;
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36 HI=inv(H);
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39 % First derivative SBP operator, 1st order accurate at first 6 boundary points
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40
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41 q2=-1/12;q1=8/12;
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42 Q=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));
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43
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44 %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));
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45
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46 Q_U = [0 0.526249e6 / 0.907200e6 -0.10819e5 / 0.777600e6 -0.50767e5 / 0.907200e6 -0.631e3 / 0.28800e5 0.91e2 / 0.7776e4; -0.526249e6 / 0.907200e6 0 0.1421209e7 / 0.2721600e7 0.16657e5 / 0.201600e6 -0.8467e4 / 0.453600e6 -0.33059e5 / 0.5443200e7; 0.10819e5 / 0.777600e6 -0.1421209e7 / 0.2721600e7 0 0.631187e6 / 0.1360800e7 0.400139e6 / 0.5443200e7 -0.8789e4 / 0.302400e6; 0.50767e5 / 0.907200e6 -0.16657e5 / 0.201600e6 -0.631187e6 / 0.1360800e7 0 0.496403e6 / 0.907200e6 -0.308533e6 / 0.5443200e7; 0.631e3 / 0.28800e5 0.8467e4 / 0.453600e6 -0.400139e6 / 0.5443200e7 -0.496403e6 / 0.907200e6 0 0.1805647e7 / 0.2721600e7; -0.91e2 / 0.7776e4 0.33059e5 / 0.5443200e7 0.8789e4 / 0.302400e6 0.308533e6 / 0.5443200e7 -0.1805647e7 / 0.2721600e7 0;];
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47 Q(1:6,1:6)=Q_U;
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48 Q(m-5:m,m-5:m)=flipud( fliplr( -Q_U ) );
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49
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50 e_1=zeros(m,1);e_1(1)=1;
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51 e_m=zeros(m,1);e_m(m)=1;
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52
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53
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54 D1=HI*(Q-1/2*e_1*e_1'+1/2*e_m*e_m') ;
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55
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56 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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57
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58
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59
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60 % Second derivative, 1st order accurate at first 6 boundary points
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61 m2=1/12;m1=-16/12;m0=30/12;
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62 M=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);
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63 %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));
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64 M_U=[0.2386127e7 / 0.2177280e7 -0.515449e6 / 0.453600e6 -0.10781e5 / 0.777600e6 0.61567e5 / 0.1360800e7 0.6817e4 / 0.403200e6 -0.1069e4 / 0.136080e6; -0.515449e6 / 0.453600e6 0.4756039e7 / 0.2177280e7 -0.1270009e7 / 0.1360800e7 -0.3751e4 / 0.28800e5 0.3067e4 / 0.680400e6 0.119459e6 / 0.10886400e8; -0.10781e5 / 0.777600e6 -0.1270009e7 / 0.1360800e7 0.111623e6 / 0.60480e5 -0.555587e6 / 0.680400e6 -0.551339e6 / 0.5443200e7 0.8789e4 / 0.453600e6; 0.61567e5 / 0.1360800e7 -0.3751e4 / 0.28800e5 -0.555587e6 / 0.680400e6 0.1025327e7 / 0.544320e6 -0.464003e6 / 0.453600e6 0.222133e6 / 0.5443200e7; 0.6817e4 / 0.403200e6 0.3067e4 / 0.680400e6 -0.551339e6 / 0.5443200e7 -0.464003e6 / 0.453600e6 0.5074159e7 / 0.2177280e7 -0.1784047e7 / 0.1360800e7; -0.1069e4 / 0.136080e6 0.119459e6 / 0.10886400e8 0.8789e4 / 0.453600e6 0.222133e6 / 0.5443200e7 -0.1784047e7 / 0.1360800e7 0.1812749e7 / 0.725760e6;];
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65
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66 M(1:6,1:6)=M_U;
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67
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68 M(m-5:m,m-5:m)=flipud( fliplr( M_U ) );
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69 M=M/h;
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70
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71 S_U=[-0.11e2 / 0.6e1 3 -0.3e1 / 0.2e1 0.1e1 / 0.3e1;]/h;
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72 S_1=zeros(1,m);
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73 S_1(1:4)=S_U;
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74 S_m=zeros(1,m);
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75
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76 S_m(m-3:m)=fliplr(-S_U);
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77
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78 D2=HI*(-M-e_1*S_1+e_m*S_m);
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81 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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82
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84
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85 % Third derivative, 1st order accurate at first 6 boundary points
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86
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87 q3=-1/8;q2=1;q1=-13/8;
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88 Q3=q3*(diag(ones(m-3,1),3)-diag(ones(m-3,1),-3))+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));
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89
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90 %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));
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93 Q3_U = [0 -0.88471e5 / 0.67200e5 0.58139e5 / 0.33600e5 -0.1167e4 / 0.2800e4 -0.89e2 / 0.11200e5 0.7e1 / 0.640e3; 0.88471e5 / 0.67200e5 0 -0.43723e5 / 0.16800e5 0.46783e5 / 0.33600e5 -0.191e3 / 0.3200e4 -0.1567e4 / 0.33600e5; -0.58139e5 / 0.33600e5 0.43723e5 / 0.16800e5 0 -0.4049e4 / 0.2400e4 0.29083e5 / 0.33600e5 -0.71e2 / 0.1400e4; 0.1167e4 / 0.2800e4 -0.46783e5 / 0.33600e5 0.4049e4 / 0.2400e4 0 -0.8591e4 / 0.5600e4 0.10613e5 / 0.11200e5; 0.89e2 / 0.11200e5 0.191e3 / 0.3200e4 -0.29083e5 / 0.33600e5 0.8591e4 / 0.5600e4 0 -0.108271e6 / 0.67200e5; -0.7e1 / 0.640e3 0.1567e4 / 0.33600e5 0.71e2 / 0.1400e4 -0.10613e5 / 0.11200e5 0.108271e6 / 0.67200e5 0;];
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94
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95 Q3(1:6,1:6)=Q3_U;
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96 Q3(m-5:m,m-5:m)=flipud( fliplr( -Q3_U ) );
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97 Q3=Q3/h^2;
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98
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99
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100
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101 S2_U=[2 -5 4 -1;]/h^2;
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102 S2_1=zeros(1,m);
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103 S2_1(1:4)=S2_U;
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104 S2_m=zeros(1,m);
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105 S2_m(m-3:m)=fliplr(S2_U);
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106
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107
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108
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109 D3=HI*(Q3 - e_1*S2_1 + e_m*S2_m +1/2*S_1'*S_1 -1/2*S_m'*S_m ) ;
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110
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111 % Fourth derivative, 0th order accurate at first 6 boundary points (still
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112 % yield 4th order convergence if stable: for example u_tt=-u_xxxx
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113
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114 m3=-1/6;m2=2;m1=-13/2;m0=28/3;
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115 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);
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116
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117 %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));
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118
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119 M4_U=[0.4596181e7 / 0.1814400e7 -0.10307743e8 / 0.1814400e7 0.160961e6 / 0.43200e5 -0.535019e6 / 0.907200e6 0.109057e6 / 0.1814400e7 -0.29273e5 / 0.604800e6; -0.10307743e8 / 0.1814400e7 0.8368543e7 / 0.604800e6 -0.9558943e7 / 0.907200e6 0.2177057e7 / 0.907200e6 -0.11351e5 / 0.86400e5 0.204257e6 / 0.1814400e7; 0.160961e6 / 0.43200e5 -0.9558943e7 / 0.907200e6 0.4938581e7 / 0.453600e6 -0.786473e6 / 0.151200e6 0.1141057e7 / 0.907200e6 -0.120619e6 / 0.907200e6; -0.535019e6 / 0.907200e6 0.2177057e7 / 0.907200e6 -0.786473e6 / 0.151200e6 0.3146581e7 / 0.453600e6 -0.4614143e7 / 0.907200e6 0.24587e5 / 0.14400e5; 0.109057e6 / 0.1814400e7 -0.11351e5 / 0.86400e5 0.1141057e7 / 0.907200e6 -0.4614143e7 / 0.907200e6 0.185709e6 / 0.22400e5 -0.11293343e8 / 0.1814400e7; -0.29273e5 / 0.604800e6 0.204257e6 / 0.1814400e7 -0.120619e6 / 0.907200e6 0.24587e5 / 0.14400e5 -0.11293343e8 / 0.1814400e7 0.16787381e8 / 0.1814400e7;];
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120
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121 M4(1:6,1:6)=M4_U;
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122
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123 M4(m-5:m,m-5:m)=flipud( fliplr( M4_U ) );
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124 M4=M4/h^3;
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125
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126 S3_U=[-1 3 -3 1;]/h^3;
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127 S3_1=zeros(1,m);
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128 S3_1(1:4)=S3_U;
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129 S3_m=zeros(1,m);
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130 S3_m(m-3:m)=fliplr(-S3_U);
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131
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132 D4=HI*(M4-e_1*S3_1+e_m*S3_m + S_1'*S2_1-S_m'*S2_m);
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133
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134
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135 % L=h*(m-1);
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136 %
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137 % x1=linspace(0,L,m)';
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138 % x2=x1.^2/fac(2);
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139 % x3=x1.^3/fac(3);
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140 % x4=x1.^4/fac(4);
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141 % x5=x1.^5/fac(5);
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142 %
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143 % x0=x1.^0/fac(1);
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144
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145 S_1 = S_1';
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146 S2_1 = S2_1';
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147 S3_1 = S3_1';
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148 S_m = S_m';
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149 S2_m = S2_m';
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150 S3_m = S3_m';
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151
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152
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153
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154 end