Unit - 4

Numerical Solution of linear simultaneous equations

Question and answer

1. 2X+Y=4

X-3Y=9

X+4Y=-5

The augmented matrix is.,

       2     1      4

       1    -3     9 r1  r2

       1    4     -5

        1   -3   9

        2     1    4-2r1+r2 & -r1+r3

        1     4    -5

         1     -3    9

         0      7   -14   ( 1/7) r2  &( 1/7) r3

         0       7   -14

       1    -3    9

        0   1    -2-1r2+r3

       0     1     -2

           1     0      3

           0     1     -2

0      0      0

Here the final matrix we has reduced to echelon form and it also represents system of equations with x=3,y=-2.

2.     X+y+z=5

2+3y=5z=8

4x+5z=2

So for the given above set of linear equations the Gauss-Jordan method represents as the following

1    1    15

2     3    58

4     0     52

The augmented matrix is.,

1     1     1

2      3    5

4      0     5

Now we reduce the above matrix in gauss Jordan form.

2    3    5r2      r1

1     1    1r1       r2

4     0    5

1    2     4r2 - 2r1

1    1     1

4     0     5

1     2    4

1     1     1 r3 – r1

3     -2     1

So here represents the guass Jordan method by having a unique number one diagonally in the matrix.

3.     X+y+z=3

2x+3y=7z=0

X+3y-2z=17

The given set of linear equations can be written as.,

1    1   1  3

2     3   7  0

1     3   -2  17

The augmented matrix o

For the given system of equations is.,

1    1   1

2    3    7

1     3    -2

1      1     1

7       3      2c1c2

-2      3     1

1     1       1

5       1       0r2             r2 - 2r1

-2      3       1

So ,here the Gauss-Jordan method completes by making all the diagonal elements to unit number.

4.     Use Jacobi’s method to solve the system of equations:

Since

So, we express the unknown with large coefficient in terms of other coefficients.

          (1)

Let the initial approximation be

2.35606

0.91666

1.932936

0.831912

3.016873

1.969654

3.010217

1.986010

1.988631

0.915055

1.986532

0.911609

1.985792

0.911547

1.98576

0.911698

Since the approximation in ninth and tenth iteration is same up to three decimal places, hence the solution of the given equations is

5.     Solve by Jacobi’s Method, the equations

Given equation can be rewrite in the form

     … (i)

..(ii)

..(iii)

Let the initial approximation be

Putting these values on the right of the equation (i), (ii) and (iii) and so we get

Putting these values on the right of the equation (i), (ii) and (iii) and so we get

Putting these values on the right of the equation (i), (ii) and (iii) and so we get

0.90025

Putting these values on the right of the equation (i), (ii) and (iii) and so we get

Putting these values on the right of the equation (i), (ii) and (iii) and so we get

Hence   solution approximately is

6.     Use Jacobi’s method to solve the system of the equations

Rewrite the given equations

        (1)

Let the initial approximation be

1.2

1.3

0.9

1.03

0.9946

0.9934

1.0015

Hence the solution of the above equation correct to two decimal places is

7.     Use Gauss –Seidel Iteration method to solve the system of equations

Since

So, we express the unknown of larger coefficient in terms of the unknowns with smaller coefficients.

Rewrite the above system of equations

      (1)

Let the initial approximation be

3.14814

2.43217

2.42571

2.4260

Hence the solution correct to three decimal places is

8.     Solve the following system of equations

By Gauss-Seidel method.

Rewrite the given system of equations as

      (1)

Le t the initial approximation be

Thus the required solution is

9.     Solve the following equations by Gauss-Seidel Method

Rewrite the above system of equations

          (1)

Let the initial approximation be

Hence the required solution is