Question

1. [12 marks] In the following parts of this question, write a MATLAB code to solve a linear system A b (A is a square nonsingular matrix) using Jacobi and Gauss-Seidel algorithms. Do not use the built-in Matlab functions for solving linear systems (a) Write a Matlab function called Jacobi that consumes a square n x n matrix A, and an n x 1 vector b, and uses the Jacobi technique to solve the system Ax-b, starting with the zero vector as the initial guess. Your function should stop when norm-2 of the difference between two successive iterates is less than 106,or if your function has performed 1000 iterations without reaching the termination condition. Your function must produce the value of x when the function terminates, along with the number of iterations performed to that point (b) Write a Matlab function called Gauss-Seidel, which meets all the same specifications as the function Jacobi described in part a), except thatt uses the Gauss-Seidel technique to solve Ax-b c) Consider the following n × n system. for values of n > 1: T&. Solve this system using the Jacobi and Gauss-Seidel functions for the following values of n : 5,250, 1250. For this part of the problem, you may calculate the exact solution as xe- A/b, to use in error calculation below Produce a table of results, where each row contains the following information » the value of n e the relative error in the Jacobi solution compared to xe, for n * itj, the number of iterations required by Jacobi for n e the relative error in the Gauss-Seidel solution g compared to re, for n » itg, the number of iterations required by Gauss-Seidel for n (d) Compare these techniques in terms of number of iterations and runtimes. To measure runtimes accurately, run both few times (say 10), disregarding the time for the first run, and take the average time (you may use tic,toc function to measure runtime

Answer 1

SMatlab code for Gauss Siedel Method and Jacobi method clear all close all %Loop for table for n iterations nn-[5 250 1250: fprintf ( Table for Jacobi and Gauss iterations\nn') fprintf(tnterr_jacobltn jacobiterr_gauss\tn gauss n n') for i-l:3 n=nn ( i ) ; treating A matrix A-ones (n,n) for ii-1:n A(ii,ii)n; end creating B vector b-ones (n,l)i %initial x and convergence x0-zeros (n,1) conv=10 ^-6 ; %Jacobi method [x j,it_j]-Jacobi method (A, b, x0, conv) %Gauss Siedel method [x g,it_gl-Gauss_method(A, b, x0,conv) %exact solution x ext-Ab; errorj-norm(x_j-x ext); error_g-norm (x_g-x_ext); fprint f ( . \t2d, \tte, \t2d, \tte, \t2d, \n' , n, error-j, it-), error-g, it-g) end measuring runtime n-50 A=ones ( n, n ) ; for ii-l:n end %creating B vector b-ones (n,1); initial x and convergence x0-zeros (n,l); conv-10-6 %loop for 10 time running for i-1:10 tic; [ x-j, it-j ] :Jacobi-method ( A, b , x 0 , conv ) ; t_jac(i) toc; tic; [x g,it_gl-Gauss_method(A, b, x0, conv) t_gau ( i )-toc ; end

break end if it--1 else end ea-10; ea norm(x1 - x0'inf' end x-x0 end Table for Jacobi and Gauss iterations n err jacob n_jacob err_gauss n_gauss 5, 1.949448e-07, 63, 5.361279e-08, 10, 250,5.757154e-04, 1000, 3.142972e-08, 1, 1250, 6.354979e-03, 1000, 3.293975e-08, 1, Mean Runtime for Jacobi method for n-50 is 0.010920 Mean Runtime for Gauss Siedel method for for n-50 is 0.000494. Published with MATLAB® R2018a

%%Matlab code for Gauss Siedel Method and Jacobi method
clear all
close all
%Loop for table for n iterations
nn=[5 250 1250];
fprintf('Table for Jacobi and Gauss iterations\n\n')
fprintf('\tn\terr_jacob\tn_jacob\terr_gauss\tn_gauss\n\n')
for i=1:3
    n=nn(i);
    %creating A matrix
    A=ones(n,n);
    for ii=1:n
        A(ii,ii)=n;
    end
    %creating B vector
    b=ones(n,1);
    %initial x and convergence
    x0=zeros(n,1);
    conv=10^-6;
    %Jacobi method
    [x_j,it_j]=Jacobi_method(A,b,x0,conv);
    %Gauss Siedel method
    [x_g,it_g]=Gauss_method(A,b,x0,conv);
    %exact solution
    x_ext=A\b;
    error_j=norm(x_j-x_ext);
    error_g=norm(x_g-x_ext);
    fprintf('\t%d,\t%e,\t%d,\t%e,\t%d,\n',n,error_j,it_j,error_g,it_g)
end
  

%measuring runtime
    n=50;
    A=ones(n,n);
    for ii=1:n
        A(ii,ii)=n;
    end
    %creating B vector
    b=ones(n,1);
    %initial x and convergence
    x0=zeros(n,1);
    conv=10^-6;
    %loop for 10 time running
for i=1:10
    tic;
    [x_j,it_j]=Jacobi_method(A,b,x0,conv);
    t_jac(i)=toc;
  
    tic;
    [x_g,it_g]=Gauss_method(A,b,x0,conv);
    t_gau(i)=toc;
end
  
fprintf('\n\nMean Runtime for Jacobi method for n=%d is %f.\n',n,mean(t_jac))
fprintf('Mean Runtime for Gauss Siedel method for for n=%d is %f.\n',n,mean(t_gau))
  

%Function for Jacobi method
function [x,it]=Jacobi_method(A,b,x0,conv)
%Jacobi method
ea=1;it=0;x1=zeros(length(b),1);
while ea>=conv
    it=it+1;
    for i=1:length(b)
        s=0;
        for j=1:length(b)
            if i~=j
                s=s+A(i,j)*x0(j);
            end
          
        end
         x2(i)=(b(i)-s)/A(i,i);
    end
    x0=x2;
     if it==1000
        break
     end
    if it==1
        ea=10;
    else
        ea = norm(x1 - x0', 'inf');
    end
    x1=x0;
end
x=x0';
end

%Function for Gauss Siedel method
function [x,it]=Gauss_method(A,b,x0,conv)
%Jacobi method
ea=1;it=0;x1=zeros(length(b),1);
while ea>=conv
    it=it+1;
    for i=1:length(b)
        s=0;
        for j=1:length(b)
            if i~=j
                s=s+A(i,j)*x0(j);
            end
          
        end
         x0(i)=(b(i)-s)/A(i,i);
    end
     if it==1000
        break
     end
    if it==1
        ea=10;
    else
        ea = norm(x1 - x0', 'inf');
    end
    x1=x0;
end
x=x0';
end


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