Gauss Elimination with Partial Pivoting C++

Table of contents

Intro: Gauss Elimination with Partial Pivoting

Gauss Elimination with Partial Pivoting is a direct method to solve the system of linear equations.

In this method, we use Partial Pivoting i.e. you have to find the pivot element which is the highest value in the first column & interchange this pivot row with the first row.

Then you can use the normal Gauss Elimination method to transform the Augmented Matrix into the upper triangular matrix.

Why Partial Pivoting?

Gauss elimination has a limitation where it fails to show exact solution.

Division by Zero
Round Off Errors

So to overcome this problem we use Partial Pivoting with Gauss elimination.

Ex. Find the Solution of following Linear Equations using Gauss Elimination with Partial Pivoting?

x + y + z = 6
x – y + z = 2
2x – y + 3z = 9

Step 1:- Write the given System of Equations in the form of AX=b, i.e. Matrix Form.

Where,
A = Coefficient Matrix,
X = variables (Column Matrix),
B = constants (Column Matrix.

Step 2:- Find Augmented Matrix C = [ Ab ]

Step 3: Find the Pivot Element.

Select Largest Absolute Value from 1^st Column.

2. Interchange Pivot Row with 1^st Row.

3. Now 1^st element ( Pivot element ) should be 1.

Step 4: Transform into Upper Triangular Matrix Form ( Echelon ).

Echelon: Upper Triangular Matrix with Diagonal Elements 1 or Non-zero.

Repeat Step 3 in Sub Matrix. Find New Pivot Element.

2. Interchange R2 with R3 ( pivot row).

Step 5: Using Back Substitution Find x,y,z.

C++ Program Partial Pivoting

 //gauss elimination Partial Pivoting
	//techindetail.com

#include<iostream>
#include<stdio.h>
#include<conio.h>
#include<math.h>

using namespace std;

int main()
{

    int n,i,j,k,temp;
    float a[10][10],c,d[10]={0};

    cout<<"Enter No of equations ? ";
    cin>>n;
    cout<<"Coefficient of all Variables : \n";
    for(i=0;i<n;i++)
    {
        cout<<"equation: "<<i+1<< "   ";
        for(j=0;j<=n;j++)
            cin>>a[i][j];
    }
	// partial pivoting
	
    for(i=n-1;i>0;i--)        
    {
        if(a[i-1][0]<a[i][0])
            for(j=0;j<=n;j++)
            {
                c=a[i][j];
                a[i][j]=a[i-1][j];
                a[i-1][j]=c;
            }
    }
    //DISPLAY MATRIX
    
    for(i=0;i<n;i++)
    {
        for(j=0;j<=n;j++)
            printf("%6.1f",a[i][j]);
        printf("\n");
    }
    
    //changing to upper triangular matrix
    //Forward elimination process
    
    for(k=0;k<n-1;k++)
        for(i=k;i<n-1;i++)
        {
            c= (a[i+1][k]/a[k][k]) ;

            for(j=0;j<=n;j++)
                a[i+1][j]-=c*a[k][j];
        }

     // DISPLAYING UPPER TRIANGULAr MATRIX

    printf("\n\n");
    for(i=0;i<n;i++)
    {
        for(j=0;j<=n;j++)
            printf("%6.1f",a[i][j]);

        printf("\n");
	}
	
    //Backward Substitution method

    for(i=n-1;i>=0;i--)
    {
        c=0;
        for(j=i;j<=n-1;j++)
            c=c+a[i][j]*d[j];

        d[i]=(a[i][n]-c)/a[i][i];
    }

	// DISPLAY
    
    for(i=0;i<n;i++)
    cout<<d[i]<<endl;


    getch();
    return 0;
    
    //techindetail.com
}
Code language: C++ (cpp)

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