1. The document describes creating an M-file to implement the Gauss-Seidel method to solve systems of linear equations. 2. It provides the theory behind Gauss-Seidel, example script and function files to implement the method, and concludes that Gauss-Seidel converges faster than Jacobi iteration and can be applied to non-square matrices. 3. Some caveats are that iterative methods like Gauss-Seidel only work for convergent systems that exhibit diagonal dominance.

1. PRACTICAL
Name- Saloni Singhal
M.Sc. (Statistics) II-Sem.
Roll No: 2046398
Course- MATH-409 L
Numerical Analysis Lab
Submitted To: Dr. S.C. Pandey

2. OBJECTIVE
1. Create an M-file to implement
Gauss-Seidel method.
1. Write both the script file and
function for the program created.

3. Theory
Gauss Seidel is a modification of jacobi iteration
where as soon as the approximation of unknown is
found it is immediately used in the step. Rest all
the procedure remains same.
• The method is easily derived by examining each
of
the Xn equations

4. Script File
% defines matrix A
A = [2 1 1; 3 5 2;2 1 4];
% defines vector b
b = [5;15;8]
% solves linear system (i.e. solves Ax=b
for x)
[m n]=size(A);
x=zeros(n,1);
if m~=n
fprintf("incorrect size")
exit
end
aug=[A b];
%diagonal dominance
for i=1:n-1
for j=i:n
[a b]=max(aug(j,1:n));
if b==i
temp=aug(i,:);
aug(i,:)=aug(j,:);
aug(j,:)=temp;
break
end
end
end

5. Script File Contd.
%check diag dom
for i=1:n
[a b]=max(aug(i,1:n));
if a~=aug(i,i)
fprintf("divergent")
exit
end
end
error=1;
p=0;
while error>=0.01
xold=x;
p=p+1;
fprintf("x%d:",p);
xold
for i=1:n
augx=[aug(i,1:i-1) aug(i,i+1:n)];
xx=[x(1:i-1) x(i+1:n)];
x(i)=(aug(i,n+1)-dot(augx,xx))/aug(i,i);
end
xold=abs(x-xold);
error=max(xold);
end
x
error
Replaced with updated
value in approximation

6. Function File
function [x,convergence] = jacobi(A,b)
[m n]=size(A);
x=zeros(n,1);
if m~=n
fprintf("incorrect size")
exit
end
aug=[A b];
%converting into strictly or partially diagonal dominant
for i=1:n-1
for j=i:n
[a b]=max(aug(j,1:n));
if b==i

7. temp=aug(i,:);
aug(i,:)=aug(j,:);
aug(j,:)=temp;
break
end
end
end
%check diag dom
for i=1:n
[a b]=max(aug(i,1:n));
if a~=aug(i,i)
fprintf("divergent")
exit
end
end
%setting tolerance limit for terminating
iterations
error=1;
p=0;
Program Contd.

8. while error>=0.01
xold=x;
p=p+1;
fprintf("x%d:",p);
xold
%approximations
for i=1:n
augx=[aug(i,1:i-1) aug(i,i+1:n)];
%x takes the newly updated value obtained after every
approximation
xx=[x(1:i-1) x(i+1:n)];
x(i)=(aug(i,n+1)-dot(augx,xx))/aug(i,i);
end
xold=abs(x-xold);
error=max(xold);
end
x
error
end
Program Contd.

9. Cases/Exam
ple

10. Conclusion
• The results show that Gauss-Seidel method is more
efficient than Jacobi method by considering
maximum number of iteration required to converge
and accuracy of the result.
• Gauss Seidel takes lesser number of iterations as it
converges faster.
• Applied to non square matrix also as opposed to the
direct methods.

11. Caveats
• Iterative Method can only be used to
give solutions when the system of
linear equation is convergent.
• Iterative Method can only be used to
give solutions when strict or partial
diagonal dominance exists.