Numerical Method - Old Questions

Comparison between Gauss Elimination method and Gauss Jordan method

• In Gauss- elimination method, the augmented matrix is reduced to Echelon form using row operations and then back substitution is applied to get values of unknowns, occurring in a system of linear equations.
• In Gauss-Jordan method, the augmented matrix is reduced to reduced Echelon form using elementary row operations to obtain values of unknowns of a system of linear equations.
• For small system it more convenient to use Gauss Jordan method.
• Gauss Jordan requires more computational work than Gauss Elimination.

Given system of equation,

    2x+3y+4z=5

    3x+4y+5z=6

    4x+5y+6z=7

The augmented matrix of given system is;

Applying, 

Applying, 

Now, using backward substitution;

3z = 0

i.e. z = 0

Finding the value of y using the value of z

Again, finding the value of x using the value of y and z

Hence the value of x, y, z are -2, 3, 0 respectively.

Algorithm for Gauss elimination method

1. Start

2.  Read the order of matrix ‘n’ and take the coefficients of linear equation.

3.  Do for k=1 to n-1

     Do for i=k+1 to n

     Do for j=k+1 to n+1

     a[i][j]=a[i][j]-a[i][k]/a[k][k]*a[k][j]

     end for j

     end for i

     end for k

4. Compute x[n]=a[n][n+1]/a[n][n]

5. Do for k=n-1 to 1

       Sum=0

    Do for j=k+1 to n

     Sum=Sum+a[k][j]*x[j]

    end for j

      x[k]=(a[k][n+1]-Sum)/a[k][k]

     end for k

6. Display the result x[k]

7. Stop