Question

Write a program for solving the system Ar-b by using Gauss-Seidel iterative method discussed in class that uses a computer memory for one iterate only. This could be summarized as follows: Input n -- the size of the system, matrix A-(a(i,j), the vectoir b (b (1), . . . , b(n)), the intitial guess x (x(1), ..., x(n)) (you can use x=b) maximum number of iterations --N, tolerance-_ TOL Set k-0, delta-10 while k < N and delta > TOL delta =0.0; norm 0 , 0 ; for i-1:n j-i+1 xneysxnev/a ( i , i); norm max (norm, Ixnewl ) ; delta -max(delta, xnew - x(i)- xnew x(i)|); end i; delta-delta/norm; output delta, the approximate solution x, and the number of iterations k

PLEASE USE MATLAB. The code is basically given to you with the algorithm above.

Answer 1

MATLAB Script (run it as a script, NOT from command window):

close all
clear
clc

n = 2;
a = [16 3; 7 -11];
b = [11 13];
x = [1 1];
N = 1000;
TOL = 1e-6;
[delta, x, k] = gauss_siedel(n,a,b,x,N,TOL)

function [delta, x, k] = gauss_siedel(n,a,b,x,N,TOL)
% n - size of system
% a - matrix
% b - vector [b(1) to b(n)]
% x - initial guess [x(1) to x(n)]
% N - max number of iterations
% TOL - tolerance
  
k = 0; delta = 10;
while k < N && delta > TOL
delta = 0.0;
norm = 0.0;
k = k + 1;
for i = 1:n
xnew = b(i) - sum(a(i,1:i-1).*x(1:i-1)) - sum(a(i,i+1:n).*x(i+1:n));
xnew = xnew / a(i,i);
norm = max(norm, abs(xnew));
delta = max(delta, abs(xnew - x(i)));
x(i) = xnew;
end
delta = delta/norm;
end
end

Output:

delta =
1.4813e-07
x =
0.8122 -0.6650
k =
9