Gauss Elimination Method with Partial Pivoting

Goal and Purpose:

Gauss Elimination involves combining equations to eliminate unknowns.

Although it is one of the earliest methods for solving simultaneous

equations, it remains among the most important algorithms in use now a

days and is the basis for linear equation solving on many popular software

packages.

Description of the problem:

In the method of Gauss Elimination the fundamental idea is to add multiples

of one equation to the others in order to eliminate a variable and to continue

this process until only one variable is left. Once this final variable is

determined, its value is substituted back into the other equations in order to

evaluate the remaining unknowns. This method, characterized by step‐by‐step elimination of the variables.

To perform row reduction on a matrix, one uses a sequence of elementary

row operations to modify the matrix until the lower left-hand corner of the

matrix is filled with zeros, as much as possible. There are three types of

elementary row operations:

1) Swapping two rows.

2) Multiplying a row by a non-zero number.

3) Adding a multiple of one row to another row.

The technique of pivoting has been developed to partially avoid

division by zero problem.

Algorithm

1. Start

2. Declare the variables and read the order of the matrix n.

3. Take the coefficients of the linear equation as:

Do for k=1 to n

Do for j=1 to n+1

Read a[k][j] End for j

End for k

4. 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

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

6. 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] = 1/a[k][k] * (a[k][n+1] – sum)

End for k

7. Display the result x[k]

8. Stop

C-Code :

#include<stdio.h>

int main()

{

int i,j,p,n;

double a[10][10],b,c,d,x1,x2,x3,temp,max;

printf("Enter the size of matrix : ");

scanf("%d",&n);

printf("Enter the matrix \n");

for(i=0;i<n;i++)

{

for(j=0;j<=n;j++)

{

scanf("%lf",&a[i][j]);

}

}

max=a[0][0];

if(a[1][0]>=max)

p=1;

if(a[2][0]>=max)

p=2;

for(j=0;j<=n;j++)

{

temp=a[p][j];

a[p][j]=a[0][j];

a[0][j]=temp;

}

printf("AFTER PIVOTOING\n");

for(i=0;i<n;i++)

{

for(j=0;j<=n;j++)

{

printf("%lf ",a[i][j]);

}

printf("\n\n");

}

b=a[1][0];

c=a[2][0];

d=a[0][0];

for(j=0;j<=i;j++)

{

a[1][j]=(a[1][j]-((b/(double)d)*a[0][j]));

a[2][j]=(a[2][j]-((c/(double)d)*a[0][j]));

}

b=a[2][1];

d=a[1][1];

for(j=0;j<=n;j++)

{

a[2][j]=(a[2][j]-((b/(double)d)*a[1][j]));

}

printf("AFTER ELIMINATON\n");

for(i=0;i<n;i++)

{

for(j=0;j<=n;j++)

{

printf("%lf ",a[i][j]);

}

printf("\n\n");

}

x3=(a[2][3]/(double)a[2][2]);

x2=((a[1][3]-(a[1][2]*x3))/(double)a[1][1]);

x1=((a[0][3]-(a[0][1]*x2)-(a[0][2]*x3))/(double)a[0][0]);

printf("the solution are:\n");

printf("x1=%lf\n\n",x1);

printf("x2=%lf\n\n",x2);

printf("x3=%lf\n\n",x3);

return 0; }

Results:

Sample Input:

Size of matrix : 3

Enter Matrix :

4 1 -1 -2

5 1 2 4

6 1 1 6

Output:

After Pivoting:

6 1 1 6

5 1 2 4

4 1 -1 -2

After Elimination:

6 1 1 6

0 0.166667 1.166667 -1

0 0 -4 -4

Conclusion:

Gauss elimination method is one of the earliest methods for solving

simultaneous equations.

Appendices :

Gauss Seidel Method

Goal and purposes:

The main goal and purpose of the program is to solve a system of n

linear simultaneous equation using Gauss Seidel method.

Description of the problem:

Assume that there given a set of n equations:

{A}{X}={B}

suppose there is 3*3 set of equations. If the diagonal elements are all nonzero,

the first equations

can be solved for x1, the second for x2 and the third for x3 to yield

x1= (b1-a12-a13x3)/a11

x2=(b2-a21x1-a23x3)/a22

x3=(b3-a31x1-a32x2)/a33

Now, the solution process can be started by choosing guesses for the x’s.

Algorithm with C - code:

#include<stdio.h>

int main()

{

int i,j;

double a[10][10],x1,x2,x3,prex1,es,b,eax1,ea,eax2,eax3,prex2,prex3;

printf("enter the element of the matrix\n");

for(i=0;i<3;i++)

{

for(j=0;j<=3;j++)

{

scanf("%lf",&a[i][j]);

}

}

printf("initial gueses:\n");

printf("x1=\n");

scanf("%lf",&x1);

printf("x2=\n");

scanf("%lf",&x2);

printf("x3=\n");

scanf("%lf",&x3);

i=1;

prex1=x1;

prex2=x2;

prex3=x3;

while(1)

{

printf("ITERATION NO.%d\n",i);

x1=(((a[0])[3]-(a[0][1]*x2)-(a[0][2]*x3))/(double)a[0][0]);

printf("x1.%d=%lf\n",i,x1);

eax1=((x1-prex1)/(double)x1)*100;

prex1=x1;

printf("eax1=%lf\n",eax1);

x2=(((a[1])[3]-(a[1][0]*x1)-(a[1][2]*x3))/(double)a[1][1]);

printf("x2.%d=%lf\n",i,x2);

eax2=((x2-prex2)/(double)x2)*100;

prex2=x2;

printf("eax2=%lf\n",eax2);

x3=(((a[2])[3]-(a[2][0]*x1)-(a[2][1]*x2))/(double)a[2][2]);

printf("x3.%d=%lf\n",i,x3);

eax3=((x3-prex3)/(double)x3)*100;

prex3=x3;

printf("eax3=%lf\n",eax3);

printf("\n");

if(i==5)

break;

i++;

}

return 0;

}

Results:

Input:

25 5 1 106.8

64 8 1 177.2

144 12 1 279.2

Initial guesses:

1

2

5

Output:

X1 :3.672 X2: -7.851 X3: -155.356

Eax1 :72.7669 Eax2 : 125.474 Eax3 : 103.218

X1 : 12.0564 X2: -54.822 X3 : -798.343

Eax1: 69.5433 Eax2: 85.6948 Eax3: 80.5402

X1: 47.1821 X2: -255.514 X3: -3448.46

Eax1: 74.447 Eax2: 78.5209 Eax3: 76.852

Apendices :

Conclusion:

The Gauss Seidel method is the most commonly used iterative method.