added lecture 13

lecture_12/lecture_12.ipynb

@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 27,
+   "execution_count": 2,
    "metadata": {
     "collapsed": true
    },
@@ -13,7 +13,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 28,
+   "execution_count": 3,
    "metadata": {
     "collapsed": true
    },
@@ -702,7 +702,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 71,
+   "execution_count": 4,
    "metadata": {
     "collapsed": false
    },
@@ -828,7 +828,7 @@
        "\t</g>\n",
        "</g>\n",
        "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"3.00\">\n",
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        "\t</g>\n",
        "\t<g id=\"gnuplot_plot_2a\"><title>backslash</title>\n",
        "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"3.00\">\n",
@@ -837,7 +837,7 @@
        "\t</g>\n",
        "</g>\n",
        "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"3.00\">\n",
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+       "\t<path d=\"M90.0,97.7 L143.8,97.7 M75.1,380.8 L79.8,375.5 L84.4,377.0 L89.1,377.2 L93.7,377.2 L98.4,376.9   L103.0,376.5 L107.7,376.5 L112.3,375.9 L117.0,376.1 L121.6,375.2 L126.2,374.8 L130.9,373.0 L135.5,373.7   L140.2,373.2 L144.8,372.4 L149.5,372.2 L154.1,368.0 L158.8,368.6 L163.4,368.4 L168.0,367.7 L172.7,367.6   L177.3,365.4 L182.0,366.4 L186.6,363.1 L191.3,364.9 L195.9,362.9 L200.6,363.2 L205.2,362.5 L209.9,362.2   L214.5,353.0 L219.1,351.7 L223.8,347.4 L228.4,347.3 L233.1,347.3 L237.7,345.8 L242.4,344.7 L247.0,344.1   L251.7,342.3 L256.3,342.1 L260.9,340.7 L265.6,339.2 L270.2,333.7 L274.9,336.8 L279.5,351.7 L284.2,350.9   L288.8,350.2 L292.9,16.7 M294.0,16.7 L298.1,357.4 L302.8,355.5 L307.4,355.7 L312.0,355.5 L316.7,353.7   L321.3,354.2 L326.0,352.4 L330.6,353.4 L335.3,357.9 L339.9,357.6 L344.6,356.6 L349.2,356.9 L353.8,355.9   L358.5,353.2 L363.1,355.0 L367.8,354.8 L372.4,358.5 L377.1,355.7 L381.7,353.0 L386.4,355.5 L391.0,354.6   L395.7,353.7 L400.3,353.4 L404.9,353.7 L409.6,356.8 L414.2,356.5 L418.9,355.7 L423.5,355.7 L428.2,355.0   L432.8,352.3 L437.5,353.7 L442.1,352.4 L446.7,355.3 L451.4,355.2 L456.0,354.1 L460.7,353.7 L465.3,350.0   L470.0,353.2 L474.6,352.0 L479.3,350.0 L483.9,351.7 L488.6,348.9 L493.2,345.8 L497.8,350.9 L502.5,349.7   L507.1,349.5 L511.8,348.9 L516.4,349.5 L521.1,348.8 L525.7,346.3 L530.4,346.7 L535.0,343.8  \" stroke=\"rgb(  0, 128,   0)\"/></g>\n",
        "\t</g>\n",
        "\t<g id=\"gnuplot_plot_3a\"><title>multiplication</title>\n",
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@@ -846,7 +846,7 @@
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        "</g>\n",
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        "</g>\n",

lecture_13/GS_rel.m

@@ -0,0 +1,36 @@
+function [x,ea,iter] = GS_rel(A,b,lambda,es,maxit)
+% GaussSeidel: Gauss Seidel method
+%   x = GaussSeidel(A,b): Gauss Seidel without relaxation
+% input:
+%   A = coefficient matrix
+%   b = right hand side vector
+%   es = stop criterion (default = 0.00001%)
+%   maxit = max iterations (default = 50)
+% output:
+%   x = solution vector
+if nargin<3,error('at least 2 input arguments required'),end
+if nargin<5|isempty(maxit),maxit=50;end
+if nargin<4|isempty(es),es=0.00001;end
+[m,n] = size(A);
+if m~=n, error('Matrix A must be square'); end
+C = A-diag(diag(A));
+x=zeros(n,1);
+for i = 1:n
+  C(i,1:n) = C(i,1:n)/A(i,i);
+end
+
+d = b./diag(A);
+
+iter = 0;
+while (1)
+  xold = x;
+  for i = 1:n
+    x(i) = d(i)-C(i,:)*x;
+    x(i) = lambda*x(i)+(1-lambda)*xold(i);
+    if x(i) ~= 0
+      ea(i) = abs((x(i) - xold(i))/x(i)) * 100;
+    end
+  end
+  iter = iter+1;
+  if max(ea)<=es | iter >= maxit, break, end
+end

lecture_13/GaussSeidel.m

@@ -0,0 +1,35 @@
+function x = GaussSeidel(A,b,es,maxit)
+% GaussSeidel: Gauss Seidel method
+%   x = GaussSeidel(A,b): Gauss Seidel without relaxation
+% input:
+%   A = coefficient matrix
+%   b = right hand side vector
+%   es = stop criterion (default = 0.00001%)
+%   maxit = max iterations (default = 50)
+% output:
+%   x = solution vector
+if nargin<2,error('at least 2 input arguments required'),end
+if nargin<4|isempty(maxit),maxit=50;end
+if nargin<3|isempty(es),es=0.00001;end
+[m,n] = size(A);
+if m~=n, error('Matrix A must be square'); end
+C = A-diag(diag(A));
+x=zeros(n,1);
+for i = 1:n
+  C(i,1:n) = C(i,1:n)/A(i,i);
+end
+
+d = b./diag(A);
+
+iter = 0;
+while (1)
+  xold = x;
+  for i = 1:n
+    x(i) = d(i)-C(i,:)*x;
+    if x(i) ~= 0
+      ea(i) = abs((x(i) - xold(i))/x(i)) * 100;
+    end
+  end
+  iter = iter+1;
+  if max(ea)<=es | iter >= maxit, break, end
+end

lecture_13/Jacobi.m

@@ -0,0 +1,39 @@
+function x = Jacobi(A,b,es,maxit)
+% GaussSeidel: Gauss Seidel method
+%   x = GaussSeidel(A,b): Gauss Seidel without relaxation
+% input:
+%   A = coefficient matrix
+%   b = right hand side vector
+%   es = stop criterion (default = 0.00001%)
+%   maxit = max iterations (default = 50)
+% output:
+%   x = solution vector
+if nargin<2,error('at least 2 input arguments required'),end
+if nargin<4|isempty(maxit),maxit=50;end
+if nargin<3|isempty(es),es=0.00001;end
+[m,n] = size(A);
+if m~=n, error('Matrix A must be square'); end
+C = A-diag(diag(A));
+x=zeros(n,1);
+for i = 1:n
+  C(i,1:n) = C(i,1:n)/A(i,i);
+end
+
+d = b./diag(A);
+
+iter = 0;
+while (1)
+  xold = x;
+  x = d-C*x;
+  % if any values of x are zero, we add 1 to denominator so error is well-behaved
+  i_zero=find(x==0);
+  i=find(x~=0);
+  if length(i_zero)>0
+    ea(i_zero)=abs((x-xold)./(1+x)*100);
+    ea(i) = abs((x(i) - xold(i))./x(i)) * 100;
+  else
+    ea = abs((x - xold)./x) * 100;
+  end
+  iter = iter+1;
+  if max(ea)<=es | iter >= maxit, break, end
+end

lecture_13/Jacobi_rel.m

@@ -0,0 +1,41 @@
+function [x,ea,iter]= Jacobi_rel(A,b,lambda,es,maxit)
+% GaussSeidel: Gauss Seidel method
+%   x = GaussSeidel(A,b): Gauss Seidel without relaxation
+% input:
+%   A = coefficient matrix
+%   b = right hand side vector
+%   es = stop criterion (default = 0.00001%)
+%   maxit = max iterations (default = 50)
+% output:
+%   x = solution vector
+if nargin<3,error('at least 2 input arguments required'),end
+if nargin<5|isempty(maxit),maxit=50;end
+if nargin<4|isempty(es),es=0.00001;end
+[m,n] = size(A);
+if m~=n, error('Matrix A must be square'); end
+C = A-diag(diag(A));
+x=zeros(n,1);
+for i = 1:n
+  C(i,1:n) = C(i,1:n)/A(i,i);
+end
+
+d = b./diag(A);
+
+iter = 0;
+while (1)
+  xold = x;
+  x = d-C*x;
+  % Add relaxation parameter lambda to current iteration
+  x = lambda*x+(1-lambda)*xold;
+  % if any values of x are zero, we add 1 to denominator so error is well-behaved
+  i_zero=find(x==0);
+  i=find(x~=0);
+  if length(i_zero)>0
+    ea(i_zero)=abs((x-xold)./(1+x)*100);
+    ea(i) = abs((x(i) - xold(i))./x(i)) * 100;
+  else
+    ea = abs((x - xold)./x) * 100;
+  end
+  iter = iter+1;
+  if max(ea)<=es | iter >= maxit, break, end
+end

lecture_13/lambda_fcn.m

@@ -0,0 +1,8 @@
+function iters = lambda_fcn(A,b,lambda)
+  % function to minimize the number of iterations for a given Ax=b solution
+  % using default Jacobi_rel parameters of es=0.00001 and maxit=50
+
+  [x,ea,iters]= Jacobi_rel(A,b,lambda,1e-8);
+end
+
+

lecture_13/lecture_13.aux

@@ -0,0 +1,59 @@
+\relax 
+\providecommand\hyper@newdestlabel[2]{}
+\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
+\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
+\global\let\oldcontentsline\contentsline
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