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codisinmyvines

Daha

Mar 10th, 2022 (edited)
1,022
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  1. %main
  2. %задание 1
  3. a=-pi; b=pi; ns=[5 10 15 20 25 30 35 50 70 100 200 500 1000]; lambda=3/(10*pi);
  4. K=@(x,s) 1./(0.64*cos((x+s)/2).*cos((x+s)/2)-1);
  5. f=@(x) 25-16*sin(x).*sin(x);
  6. u=@(x) 17/2+(128/17)*cos(2*x);
  7. for i=1:length(ns)
  8.     x=linspace(a,b,ns(i));
  9.     h(i)=x(2)-x(1);
  10.     [y,x]=fredII_trapez(K,f,lambda,a,b,ns(i));
  11.     u_=u(x)';
  12.     err(i)=norm(y-u_,inf)/norm(u_,inf);
  13. end
  14. loglog(h,err)
  15. title('loglog(h,err)');
  16. xlabel('h');
  17. ylabel('err');
  18.  
  19. %задание 2
  20. %main
  21. % a=0; b=1; lambda = 1/2 ;n=1000;
  22. % f=@(x) (5/6)*x;
  23. % K=@(x,s) x*s;
  24. % x=linspace(a,b,n);
  25. % u=@(x) x;
  26. % tic;
  27. % [y,x]=fredII_trapez(K,f,lambda,a,b,n);
  28. % toc;
  29. % % tic;
  30. % % [y,x]=fredII_trapez_upd(K,f,lambda,a,b,n);
  31. % % toc;
  32. % plot(x,y,'ro',x,u(x),'g');
  33. % xlabel('x');
  34. % ylabel('y');
  35. % u_=u(x)';
  36. % err=norm(y-u_,inf)/norm(u_,inf);
  37.  
  38. для задания 3
  39.  
  40. function [y,x] = fredII_trapez_upd(K,f,lambda,a,b,n)
  41. x=linspace(a,b,n);
  42. h=x(2)-x(1);
  43. w=[1/2,ones(1,n-2), 1/2]*h*lambda;
  44. %A=-K(x,x')*diag(w')+eye(n);
  45. A=-K(x,x')*spdiags(w',0,n,n)+speye(n);
  46. F=f(x);
  47. y=A\F'
  48. end
  49.  
  50. % %задание 4
  51. % %main
  52. % a=0; b=1; ns=[5 10 15 20 25 30 50 100]; lambda=-1;
  53. % K=@(x,s) x.*exp(x.*s);
  54. % f=@(x) exp(x);
  55. % for i=1:length(ns)
  56. %     x=linspace(a,b,ns(i));
  57. %     h(i)=x(2)-x(1);
  58. %     [y1,x]=fredII_trapez(K,f,lambda,a,b,ns(i));
  59. %     [y2,x]=fredII_simpson(K,f,lambda,a,b,ns(i));
  60. %     u=@(x) ones(1,numel(x));
  61. %     u_=u(x)';
  62. %     err1(i)=norm(y1-u_,inf)/norm(u_,inf);
  63. %     err2(i)=norm(y2-u_,inf)/norm(u_,inf);
  64. % end
  65. % %plot(h,err1,'r');
  66. % loglog(h,err1,'r')
  67. % hold on
  68. % %plot(h,err2,'g');
  69. % loglog(h,err2,'g')
  70. % title('red - trapez, green - simpson');
  71. % xlabel('h');
  72. % ylabel('err');
  73.  
  74. function [y,x] = fredII_simpson(K,f,lambda,a,b,n)
  75. x=linspace(a,b,n)';
  76. h=x(2)-x(1);
  77. if mod(n,2)==1
  78.     w=[1/3,2/3*(1+mod(1:(n-2),2)),1/3]*h*lambda;
  79. else
  80.     w=[1/2,5/6,2/3*(1+mod(1:(n-3),2)),1/3]*h*lambda;
  81. end
  82. A=zeros(n);
  83. for i=1:n
  84.     A(:,i)=-w(i)*K(x,x(i));
  85.     A(i,i)=A(i,i)+1;
  86. end
  87. F=f(x);
  88. y=A\F;
  89. end
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