%%Matlab code for solving
clear all
close all
%Matrix form of given linear equation
A=[9 1 1 1; 1 8 1 1; 1 1 7 1; 1 1 1 6];
b=[75;54;43;34];
%displaying A and b matrix
fprintf('Matrix A of given linear equation.\n')
disp(A)
fprintf('Matrix b vector.\n')
disp(b)
%initial guess
x0=[0;0;0;0];
%error convergence
conv=10^-12;
[x]=gauss_eliminationn(A,b);
fprintf('Solution Matrix for Gauss Elimination is x=\n ')
disp(x)
[x,it]=Jacobi_method(A,b,x0,conv);
fprintf('Solution Matrix for Jacobi method is x=\n ')
disp(x)
fprintf('Total number of iterations for Jacobi method is
%d.\n',it)
[x,it]=Gauss_method(A,b,x0,conv);
fprintf('\nSolution Matrix for Gauss Siedel is x=\n ')
disp(x)
fprintf('Total number of iterations for Gauss Siedel method is
%d.\n',it)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Function for Gauss Elimination
function [x]=gauss_eliminationn(A,b)
%A is the coefficient matrix and b is the result matrix for
Ax=b
%x is the solution matrix
%example A=[15 6 8 11;6 6 5 3;8 5 7 6;11 3 6 9];
b=[40;20;26;29];
%so by x=gauss_eliminationn(A,b), we get x=[1 1 1 1]T;
%below is the algorithm for Gaussian elimination
A=[A,b];%A is the matrix with A and b
sz_A=size(A);
ss_A=sz_A(1,1);
%Loop for Gaussian elimination matrix formation
for A matrix
for i=1:ss_A-1
A1=A(i:end,i);
b=find(A1==max(A1));
b=b+(i-1);
A2=A(b,:);
A(b,:)=A(i,:);
A(i,:)=A2;
m=max(A1);
for
j=i+1:ss_A
a=A(j,i)/m;
A(j,:)=A(j,:)-a*A(i,:);
end
end
%Final A matrix after Gaussian elimination
B=A;
sz=size(A);
ss=sz(1,1);
x(ss)=A(end,end)/A(end,end-1);
%loop for backward substitution of Gaussian
elimination matrix for finding x
for ii=ss-1:-1:1
sum=0;
for jj=ii+1:ss
sum=sum+A(ii,jj)*x(jj);
end
x(ii)=(A(ii,end)-sum)/A(ii,ii);
end
x=x';
end
%Function for Jacobi method
function [x,it]=Jacobi_method(A,b,x0,conv)
%Jacobi method
ea=1;it=0;
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
x1(i)=(b(i)-s)/A(i,i);
end
if it==1
ea=10;
else
ea = norm(x1 - x0);
end
x0=x1;
end
x=x0';
end
%Function for Gauss Siedel method
function [x,it]=Gauss_method(A,b,x0,conv)
%Gauss Siedel method
ea=1;it=0;
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==1
ea=10;
else
ea = norm(x1 - x0);
end
x1=x0;
end
x=x0;
end
%%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%%
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