The Dope Knight Rises

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
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@@ -828,7 +828,7 @@
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        "\t</g>\n",
        "\t<g id=\"gnuplot_plot_2a\"><title>backslash</title>\n",
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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/LU_naive.m

@@ -0,0 +1,27 @@
+function [L, U] = LU_naive(A)
+% GaussNaive: naive Gauss elimination
+%   x = GaussNaive(A,b): Gauss elimination without pivoting.
+% input:
+%   A = coefficient matrix
+%   y = right hand side vector
+% output:
+%   x = solution vector
+[m,n] = size(A);
+if m~=n, error('Matrix A must be square'); end
+nb = n;
+L=diag(ones(n,1));
+U=A;
+% forward elimination
+for k = 1:n-1
+  for i = k+1:n
+    fik = U(i,k)/U(k,k);
+    L(i,k)=fik;
+    U(i,k:nb) = U(i,k:nb)-fik*U(k,k:nb);
+  end
+end
+%% back substitution
+%x = zeros(n,1);
+%x(n) = Aug(n,nb)/Aug(n,n);
+%for i = n-1:-1:1
+%  x(i) = (Aug(i,nb)-Aug(i,i+1:n)*x(i+1:n))/Aug(i,i);
+%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
+
+