Jacobi method

Jacobi method of solution of system of equations

Jacobi method of solution of system of equations

We can express linear equations in matrix form A*X=B ...... (i)

where A is called Coefficient matrix and B Right hand side vector.

Now, decomposing the matrix A into its diagonal and unondiagonal component,

we get:

A = D + R, where

Further, the system of linear equations (i) can be expressed as:

(D+R)*X = B

or D*X + R*X = B

or X = inv(D)*B – inv(D)*R*X –(ii)

In Gauss-Seidel method, the equation (ii) is solved iteratively by solving the left hand value of x and then using previously found x on right hand side. Mathematically, the iteration process in Gauss-Seidel method can be expressed as:

X(n+1) = inv(D)*B – inv(D)*R*X(n) or inv(D)*(B – R*X(n) )

Note : The matrix should be diagonally dominant matrix.

Example : Solve using Jacobi method:

10x1 -x2+ 2x3 = 6

-x1 + 11x2 - x3 +3x4 = 25

2x1 - x2+ 10x3 -x4 = -11

3x2 - x3 + 8x4 = 15

Solution: The equation can be written as, A*X = B where

Decomposing A = D+R

Enter Coefficient Matrix A :[10 -1 2 0;-1 11 -1 3; 2 -1 10 -1;0 3 -1 8]

Enter RHS Matrix B :[6 25 -11 15]'

Enter no of iteration : 20

D =

10 0 0 0

0 11 0 0

0 0 10 0

0 0 0 8

R =

0 -1 2 0

-1 0 -1 3

2 -1 0 -1

0 3 -1 0

X =

Columns 1 through 6

0 0.6000 1.0473 0.9326 1.0152 0.9890

0 2.2727 1.7159 2.0533 1.9537 2.0114

0 -1.1000 -0.8052 -1.0493 -0.9681 -1.0103

0 1.8750 0.8852 1.1309 0.9738 1.0214

Columns 7 through 12

1.0032 0.9981 1.0006 0.9997 1.0001 0.9999

1.9922 2.0023 1.9987 2.0004 1.9998 2.0001

-0.9945 -1.0020 -0.9990 -1.0004 -0.9998 -1.0001

0.9944 1.0036 0.9989 1.0006 0.9998 1.0001

Columns 13 through 18

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

2.0000 2.0000 2.0000 2.0000 2.0000 2.0000

-1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Columns 19 through 20

1.0000 1.0000

2.0000 2.0000

-1.0000 -1.0000

1.0000 1.0000

Here, we see that we get solution from 13th iteration onwards.