Question

in MATLAB

3. Write a function called gauss_seidel that inputs an n x n matrix, A, a column vector, b, an initial guess xo), an error tolerance e, and a maximum number of iterations, and output an approximate solution obtained using the Gauss-Seidel method, the error and the number of iterations. The header should look like (x, err, N] = gauss_seidel (A, b, x0, tol, Nmax). Use the method to find approximate solutions to the linear system -2 1 0 0 0 1 2 10 -2 0 0 0 2 -4 0 0 0 0 0 2 1 0 0 0 -3 -7 con do

Answer 1

`Hey,

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clc
clear all
close all
format long
A=[-2,1,0,0,0;2,10,-2,0,0;0,2,-4,0,0;0,0,0,2,1;0,0,0,-3,-7];
b=[-5;3;3;-7;-8];
[x,err,N]=gauss_seidel(A,b,[1;1;1;1;1],1e-5,100)
function [x,err,N] = gauss_seidel(A,d,x0,tol,Imax)
% gauss_seidel: implements the Jacobi iterative method
% for solving a system of linear equations.
% Input: A = square coefficient matrix (nxn)
% d = right hand side vector (nx1)
% x0 = initial guess (nx1) (default = zeros)
% tol = stop criterion (default = 10^-6)
% Imax = maximum iterations (default = 100)
% Output: Solution vector.
if nargin<5, Imax=100; end
if nargin<4, tol=10^-6; end
[m,n]=size(A);
if m~=n, error('Matrix A must be square'); end
if nargin<3, x=zeros(n,1); else x=x0; end
C=A;
for i=1:n
C(i,i)=0;
end
for i=1:n
C(i,1:n)=C(i,1:n)/A(i,i);
end
b=zeros(1,n);
for i=1:n
b(i)=d(i)/A(i,i);
end
iter=0;
while true
x_old=x;
for i=1:n
x(i)=b(i)-C(i,:)*x;
if x(i)~=0
err(i)=abs((x(i)-x_old(i))/x(i));
end
end
iter=iter+1;
if iter>=Imax || norm((x-x_old)./x)<tol
err=norm((x-x_old)./x);
N=iter;
return
end
end
end

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