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Merge pull request #9 from rcc02007/master
update 3/6/17
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -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); | ||
|
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||
| 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 |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -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 |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -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 |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -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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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -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 |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -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 | ||
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| [x,ea,iters]= Jacobi_rel(A,b,lambda,1e-8); | ||
| end | ||
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