MATLAB Codes for Jacobi method
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The document describes a lab report from a student, Saloni Singhal, focusing on the implementation of the Gauss-Jacobi method in MATLAB for solving matrix equations. It covers the theory of diagonal dominance, provides script files and function codes, and discusses the conditions under which the iterative method is applicable, specifically stressing the importance of convergence. The report also includes references to relevant resources and highlights potential caveats regarding the method's effectiveness with ill-conditioned systems.
![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](https://image.slidesharecdn.com/20463983-230523144909-6e892d52/85/MATLAB-Codes-for-Jacobi-method-4-320.jpg)
![Script File Contd.
%check diag dominance
for i=1:n
[a b]=max(aug(i,1:n));
if a~=aug(i,i)
fprintf("divergent")
exit
end
end
%assign tolerance
error=1;
p=0;
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)];
xx=[xold(1:i-1) xold(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](https://image.slidesharecdn.com/20463983-230523144909-6e892d52/85/MATLAB-Codes-for-Jacobi-method-5-320.jpg)
![Function
File
function [x,convergence] = seidelA,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
temp=aug(i,:);
aug(i,:)=aug(j,:);
aug(j,:)=temp;
break
end
end
end](https://image.slidesharecdn.com/20463983-230523144909-6e892d52/85/MATLAB-Codes-for-Jacobi-method-6-320.jpg)
![Program 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
%setting tolerance limit for terminating iterations
error=1;
p=0;
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)];
%xold takes initial values calculated in successive
iterations
xx=[xold(1:i-1) xold(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](https://image.slidesharecdn.com/20463983-230523144909-6e892d52/85/MATLAB-Codes-for-Jacobi-method-7-320.jpg)
MATLAB Codes for Jacobi method
- 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 Jacobi method. 2. Write both the script file and function for the program created.
- 3. Theory Diagonal Dominance: • Iterative Method can be applied when diagonal dominance exist that is when absolute value of its leading diagonal elements is greater than or equal to the sum of absolute value of the remaining element. • It is a method of solving a matrix equation that has no zeros along its main diagonal. Each diagonal element is solved for, and an approximate value plugged in. The process is then iterated until it converges. • Applied to non square matrix also as opposed to the direct methods. • 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 dominance for i=1:n [a b]=max(aug(i,1:n)); if a~=aug(i,i) fprintf("divergent") exit end end %assign tolerance error=1; p=0; 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)]; xx=[xold(1:i-1) xold(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
- 6. Function File function [x,convergence] = seidelA,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 temp=aug(i,:); aug(i,:)=aug(j,:); aug(j,:)=temp; break end end end
- 7. Program 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 %setting tolerance limit for terminating iterations error=1; p=0; 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)]; %xold takes initial values calculated in successive iterations xx=[xold(1:i-1) xold(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
- 8. Cases: (Convergent and Divergent Matrix)
- 9. Caveats • Iterative Method can only be used to give solutions when the system of linear equation is convergent or diagonally dominant. • The Jacobi iterative method works fine with well-conditioned linear systems. If the linear system is ill- conditioned, it is most probably that the Jacobi method will fail to converge.
- 10. References • https://mathworld.wolfram.com/JacobiMethod .html#:~:text=The%20Jacobi%20method%20i s%20a,then%20iterated%20until%20it%20co nverges. • Lecture notes of Prof. S.C. Pandey Sir • MATLAB Documentation