Linear Algebra :Gauss-Seidel and Rayleigh power method

 

Solving the system of linear equations using Gauss-Seidel method

clc;

close;

clear;

 

a = input("Enter the co-efficient matrix:");

b = input("Enter the constant matrix:");

n = size(a);

oldx = input("Initial guesses:");

x = oldx;

 

for k = 1:20

    fprintf("\n Iteration %d:\t ", k)

    for i = 1:n

        sum = 0;

        for j = 1:n

            if i~=j

                sum = sum+a(i,j) * x(j);

            end

        end

        x(i) = (b(i)-sum)/a(i,i);

        fprintf("x(%d)=%0.4f\t",i,x(i))

    end

    if abs(max(x)-max(oldx))<=0.0001

        break;

    else

 

oldx = x;

    end

end

Note: Enter the coefficient matrix after making matrix diagonally dominant

Example

Use Gauss Siedel method to solve: $$5x+2y+z=1; x+4y+2z=0; x+2y+5z=3$$ considering the initial guess (1,0,3)

Output

Enter the co-efficient matrix:[5 2 1;1 4 2;1 2 5]

Enter the constant matrix:[1;0;3]

Initial guesses:[1;0;3]

 Iteration 1: x(1)=-0.4000 x(2)=-1.4000 x(3)=1.2400 

 Iteration 2: x(1)=0.5120 x(2)=-0.7480 x(3)=0.7968 

 Iteration 3: x(1)=0.3398 x(2)=-0.4834 x(3)=0.7254 

 Iteration 4: x(1)=0.2483 x(2)=-0.4248 x(3)=0.7202 

 Iteration 5: x(1)=0.2259 x(2)=-0.4166 x(3)=0.7215 

 Iteration 6: x(1)=0.2223 x(2)=-0.4163 x(3)=0.7221 

 Iteration 7: x(1)=0.2221 x(2)=-0.4166 x(3)=0.7222 

 Iteration 8: x(1)=0.2222 x(2)=-0.4166 x(3)=0.7222

Rayleigh power method

Program to find the numerically largest eigenvalue and eigenvector using Rayleigh power method

clc;

clear

n = input("Enter the number of varibles:");

a = input("Enter the  matrix:");

x0 = input("Initial guesses:");

 

for k = 1:5

    C=a*x0;

        m=max(abs(C));

        x=(C./m);

if abs((x)-(x0))<=0.0001

break;

else

x0 = x;

end

fprintf("---------------------------------------------\nIteration %d :\n",k);

fprintf("The Eigen vector:\n");

disp(x);

fprintf("The largest Eigen value: %f\n",m);

end

Example

Find the numerically largest eigenvalue and eigenvector of

by taking [1 0 0]' as the initial approximation using Rayleigh power method

Output

Enter the number of varibles:3

Enter the  matrix:[2 0 1;0 2 0; 1 0 2]

Initial guesses:[1; 0 ;0]

---------------------------------------------

Iteration 1 :

The Eigen vector:

    1.0000

         0

    0.5000

The largest Eigen value: 2.000000

---------------------------------------------

Iteration 2 :

The Eigen vector:

    1.0000

         0

    0.8000

The largest Eigen value: 2.500000

---------------------------------------------

Iteration 3 :

The Eigen vector:

    1.0000

         0

    0.9286

The largest Eigen value: 2.800000

---------------------------------------------

Iteration 4 :

The Eigen vector:

    1.0000

         0

    0.9756

The largest Eigen value: 2.928571

---------------------------------------------

Iteration 5 :

The Eigen vector:

    1.0000

         0

    0.9918

The largest Eigen value: 2.975610