Week 6 - Understanding linear systems

PROBLEM STATEMENT:

cofficient matrix:

`A=[[5,1, 2],[-3,9,4],[1,2,-7]]`

solution matrix

`x=[[x1],[x2],[x3]]`

right side matrix

`B=[[10],[-14],[33]]`

we have a 3 gernal linear sytems the equations

AX=B

`X=[A^-]B`

Eigen values of given matrix=`[A-lamdaI]`

`A=[[5,1, 2],[-3,9,4],[1,2,-7]]`-`lamda[[1,0,0],[0,1,0],[0,0,1]]`

we get the equtions A

ITERATIVE MATRIX

JACOBI:

TJAC=`-(D^-1)*(L+U)`

GAUSSIDEAL:

`GAUS=`-(D+L)^-1)*U`

SOR:

`TSOR=-(D+omegaL)^-1*((1-omega)*L)*U`

MATLAB CODE:

clc clear allclose all%Defining the matrix A L= [0 0 0;-3 0 0;1 2 0];U =[0 1 2;0 0 4;0 0 0];D=[5 0 0 ;0 9 0;0 0 -7];A= L + U + D;B=[10;-14;33]; % iterative methods% jacobiTjac=-inv(D)*(L+U);% spctral radiusspctral_radius_jacobi=max(real(eig(Tjac)));% define tolarance & errortolarance=1e-5;error=9e9;xold=[0;0;0];%iterative method[error_jacobi,jacobi_iter,Spectral_radius_jacobi]=jacobi_method(L,U,D,B)[error_gauss,gauss_iter,Spectral_radius_gauss]=gauss_method(L,U,D,B)[error_SOR,SOR_iter,Spectral_radius_SOR]=SOR_method(L,U,D,B)figure(1)plot(1:jacobi_iter-1,error_jacobi)title_text = sprintf('Jacobi Iteration Number = %d', jacobi_iter);title(title_text)xlabel('Iteration')ylabel('error')figure(2)plot(1:gauss_iter-1,error_gauss)title_text = sprintf('gauss Iteration Number = %d', gauss_iter);title(title_text)xlabel('Iteration')ylabel('error')figure(3)plot(1:SOR_iter-1,error_SOR)title_text = sprintf('SOR Iteration Number = %d', SOR_iter);title(title_text)xlabel('Iteration')ylabel('error')

FUNCTION : JACOBI

function [error_jacobi,jacobi_iter,Spectral_radius_jacobi]=jacobi_method(L,U,D,B)%Defining the matrix A L= [0 0 0;-3 0 0;1 2 0];U =[0 1 2;0 0 4;0 0 0];D=[5 0 0 ;0 9 0;0 0 -7];A= L + U + D;B=[10;-14;33]; % define tolarance & error_jacobierror_jacobi=9e9;tolerance=1e-5;xold=[0;0;0];jacobi_iter=1;c=inv(D)*B;% Iterative matrixTjac=-inv(D)*((L+U));% spectral radiusSpectral_radius_jacobi=max(abs(real(eig(Tjac))));while (error_jacobi>tolerance)    x=Tjac*xold +(c);    error_jacobi(jacobi_iter) = max(abs(x-xold));    xold=x;    jacobi_iter=jacobi_iter+1;endend

FUNCTION GAUASS:

function [error_gauss,gauss_iter,Spectral_radius_gauss]=gauss_method(L,U,D,B)%Defining the matrix A L= [0 0 0;-3 0 0;1 2 0];U =[0 1 2;0 0 4;0 0 0];D=[5 0 0 ;0 9 0;0 0 -7];A= L + U + D;B=[10;-14;33]; % define tolarance & error_gausserror_gauss=9e9;tolerance=1e-5;xold=[0;0;0];gauss_iter=1;c=inv(D+L)*B;% Iterative matrixTgauss=-inv(D+L)*U;% spectral radiusEigen_value=eig(Tgauss);Spectral_radius_gauss=max(abs(real(Eigen_value)));while (error_gauss>tolerance)    x=Tgauss*xold +(c);    error_gauss(gauss_iter) = max(abs(x-xold));    xold=x;    gauss_iter=gauss_iter+1;endend    

FUNCTION SOR:

function [error_SOR,SOR_iter,Spectral_radius_SOR]=SOR_method(L,U,D,B)%Defining the matrix A L= [0 0 0;-3 0 0;1 2 0];U =[0 1 2;0 0 4;0 0 0];D=[5 0 0 ;0 9 0;0 0 -7];A= L + U + D;B=[10;-14;33]; % define tolarance & error_sorerror_SOR=9e9;tolerance=1e-5;xold=[0;0;0];SOR_iter=1;omega=1.1;c=inv(D+omega*L)*B;% Iterative matrixTSOR=-inv(D+(omega*L))*(((1-omega)*L)+U);% spectral radiusSpectral_radius_SOR=max(abs(real(eig(TSOR))));while (error_SOR>tolerance)    x=TSOR*xold + (c);    error_SOR(SOR_iter) = max(abs(x-xold));    xold=x;    SOR_iter=SOR_iter+1;endend

RESULT

RESULT FOR GAUSS:

RESULT FOR SOR:

SPACRTRAL RADIUS: