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added lecture 13

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rcc02007 committed Mar 2, 2017
1 parent 9bc0a6b commit ba3c42ceb0b2d70ab66e86df55548563fffe18fc
@@ -2,7 +2,7 @@
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@@ -13,7 +13,7 @@
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@@ -702,7 +702,7 @@
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@@ -828,7 +828,7 @@
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@@ -837,7 +837,7 @@
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@@ -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
@@ -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
@@ -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
@@ -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
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@@ -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


@@ -0,0 +1,59 @@
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