Program for Gauss-Jordan Elimination Method

Difficulty Level : Medium
Last Updated : 21 Apr, 2020

Prerequisite : Gaussian Elimination to Solve Linear Equations

Introduction : The Gauss-Jordan method, also known as Gauss-Jordan elimination method is used to solve a system of linear equations and is a modified version of Gauss Elimination Method.

It is similar and simpler than Gauss Elimination Method as we have to perform 2 different process in Gauss Elimination Method i.e.
1) Formation of upper triangular matrix, and
2) Back substitution

But in case of Gauss-Jordan Elimination Method, we only have to form a reduced row echelon form (diagonal matrix). Below given is the flow-chart of Gauss-Jordan Elimination Method.

Flow Chart of Gauss-Jordan Elimination Method :

Examples :

Input :  2y + z = 4
         x + y + 2z = 6
         2x + y + z = 7

Output :
Final Augumented Matrix is : 
1 0 0 2.2 
0 2 0 2.8 
0 0 -2.5 -3 

Result is : 2.2 1.4 1.2

Explanation : Below given is the explanation of the above example.

Input Augmented Matrix is :

Interchanging R1 and R2, we get

Performing the row operation R3 <- R3 – (2*R1)

Performing the row operations R1 <- R1 – ((1/2)* R2) and R3 <- R3 + ((1/2)*R2)

Performing R1 <- R1 + ((3/5)*R3) and R2 <- R2 + ((2/5)*R3)

Unique Solutions are :

C++

// C++ Implementation for Gauss-Jordan 
// Elimination Method 
#include <bits/stdc++.h> 
using namespace std; 
  
#define M 10 
  
// Function to print the matrix 
void PrintMatrix(float a[][M], int n) 
{ 
    for (int i = 0; i < n; i++) { 
        for (int j = 0; j <= n; j++)  
          cout << a[i][j] << " "; 
        cout << endl; 
    } 
} 
  
// function to reduce matrix to reduced 
// row echelon form. 
int PerformOperation(float a[][M], int n) 
{ 
    int i, j, k = 0, c, flag = 0, m = 0; 
    float pro = 0; 
      
    // Performing elementary operations 
    for (i = 0; i < n; i++) 
    { 
        if (a[i][i] == 0)  
        { 
            c = 1; 
            while ((i + c) < n && a[i + c][i] == 0)  
                c++;             
            if ((i + c) == n) { 
                flag = 1; 
                break; 
            } 
            for (j = i, k = 0; k <= n; k++)  
                swap(a[j][k], a[j+c][k]); 
        } 
  
        for (j = 0; j < n; j++) { 
              
            // Excluding all i == j 
            if (i != j) { 
                  
                // Converting Matrix to reduced row 
                // echelon form(diagonal matrix) 
                float pro = a[j][i] / a[i][i]; 
  
                for (k = 0; k <= n; k++)                  
                    a[j][k] = a[j][k] - (a[i][k]) * pro;                 
            } 
        } 
    } 
    return flag; 
} 
  
// Function to print the desired result  
// if unique solutions exists, otherwise  
// prints no solution or infinite solutions  
// depending upon the input given. 
void PrintResult(float a[][M], int n, int flag) 
{ 
    cout << "Result is : "; 
  
    if (flag == 2)      
      cout << "Infinite Solutions Exists" << endl;     
    else if (flag == 3)      
      cout << "No Solution Exists" << endl; 
      
      
    // Printing the solution by dividing constants by 
    // their respective diagonal elements 
    else { 
        for (int i = 0; i < n; i++)          
            cout << a[i][n] / a[i][i] << " ";         
    } 
} 
  
// To check whether infinite solutions  
// exists or no solution exists 
int CheckConsistency(float a[][M], int n, int flag) 
{ 
    int i, j; 
    float sum; 
      
    // flag == 2 for infinite solution 
    // flag == 3 for No solution 
    flag = 3; 
    for (i = 0; i < n; i++)  
    { 
        sum = 0; 
        for (j = 0; j < n; j++)         
            sum = sum + a[i][j]; 
        if (sum == a[i][j])  
            flag = 2;         
    } 
    return flag; 
} 
  
// Driver code 
int main() 
{ 
    float a[M][M] = {{ 0, 2, 1, 4 },  
                     { 1, 1, 2, 6 },  
                     { 2, 1, 1, 7 }}; 
                       
    // Order of Matrix(n) 
    int n = 3, flag = 0; 
      
    // Performing Matrix transformation 
    flag = PerformOperation(a, n); 
      
    if (flag == 1)      
        flag = CheckConsistency(a, n, flag);     
  
    // Printing Final Matrix 
    cout << "Final Augumented Matrix is : " << endl; 
    PrintMatrix(a, n); 
    cout << endl; 
      
    // Printing Solutions(if exist) 
    PrintResult(a, n, flag); 
  
    return 0; 
} 

Java

// Java Implementation for Gauss-Jordan 
// Elimination Method 
class GFG { 
      
static int M = 10; 
  
// Function to print the matrix 
static void PrintMatrix(float a[][], int n) 
{ 
    for (int i = 0; i < n; i++)  
    { 
        for (int j = 0; j <= n; j++)  
            System.out.print(a[i][j] + " "); 
        System.out.println(); 
    } 
} 
  
// function to reduce matrix to reduced 
// row echelon form. 
static int PerformOperation(float a[][], int n) 
{ 
    int i, j, k = 0, c, flag = 0, m = 0; 
    float pro = 0; 
      
    // Performing elementary operations 
    for (i = 0; i < n; i++) 
    { 
        if (a[i][i] == 0)  
        { 
            c = 1; 
            while ((i + c) < n && a[i + c][i] == 0)  
                c++;          
            if ((i + c) == n)  
            { 
                flag = 1; 
                break; 
            } 
            for (j = i, k = 0; k <= n; k++)  
            { 
                float temp =a[j][k]; 
                a[j][k] = a[j+c][k]; 
                a[j+c][k] = temp; 
            } 
        } 
  
        for (j = 0; j < n; j++)  
        { 
              
            // Excluding all i == j 
            if (i != j)  
            { 
                  
                // Converting Matrix to reduced row 
                // echelon form(diagonal matrix) 
                float p = a[j][i] / a[i][i]; 
  
                for (k = 0; k <= n; k++)                  
                    a[j][k] = a[j][k] - (a[i][k]) * p;              
            } 
        } 
    } 
    return flag; 
} 
  
// Function to print the desired result  
// if unique solutions exists, otherwise  
// prints no solution or infinite solutions  
// depending upon the input given. 
static void PrintResult(float a[][], int n, int flag) 
{ 
    System.out.print("Result is : "); 
  
    if (flag == 2)      
    System.out.println("Infinite Solutions Exists");  
    else if (flag == 3)      
    System.out.println("No Solution Exists"); 
      
      
    // Printing the solution by dividing constants by 
    // their respective diagonal elements 
    else { 
        for (int i = 0; i < n; i++)          
            System.out.print(a[i][n] / a[i][i] +" ");      
    } 
} 
  
// To check whether infinite solutions  
// exists or no solution exists 
static int CheckConsistency(float a[][], int n, int flag) 
{ 
    int i, j; 
    float sum; 
      
    // flag == 2 for infinite solution 
    // flag == 3 for No solution 
    flag = 3; 
    for (i = 0; i < n; i++)  
    { 
        sum = 0; 
        for (j = 0; j < n; j++)      
            sum = sum + a[i][j]; 
        if (sum == a[i][j])  
            flag = 2;      
    } 
    return flag; 
} 
  
// Driver code 
public static void main(String[] args)  
{ 
    float a[][] = {{ 0, 2, 1, 4 },  
                    { 1, 1, 2, 6 },  
                    { 2, 1, 1, 7 }}; 
                      
    // Order of Matrix(n) 
    int n = 3, flag = 0; 
      
    // Performing Matrix transformation 
    flag = PerformOperation(a, n); 
      
    if (flag == 1)      
        flag = CheckConsistency(a, n, flag);  
  
    // Printing Final Matrix 
    System.out.println("Final Augumented Matrix is : "); 
    PrintMatrix(a, n); 
    System.out.println(""); 
      
    // Printing Solutions(if exist) 
    PrintResult(a, n, flag); 
} 
} 
  
/* This code contributed by PrinciRaj1992 */

C#

// C# Implementation for Gauss-Jordan 
// Elimination Method 
using System; 
using System.Collections.Generic; 
  
class GFG 
{ 
static int M = 10; 
  
// Function to print the matrix 
static void PrintMatrix(float [,]a, int n) 
{ 
    for (int i = 0; i < n; i++)  
    { 
        for (int j = 0; j <= n; j++)  
            Console.Write(a[i, j] + " "); 
        Console.WriteLine(); 
    } 
} 
  
// function to reduce matrix to reduced 
// row echelon form. 
static int PerformOperation(float [,]a, int n) 
{ 
    int i, j, k = 0, c, flag = 0; 
      
    // Performing elementary operations 
    for (i = 0; i < n; i++) 
    { 
        if (a[i, i] == 0)  
        { 
            c = 1; 
            while ((i + c) < n && a[i + c, i] == 0)  
                c++;          
            if ((i + c) == n)  
            { 
                flag = 1; 
                break; 
            } 
            for (j = i, k = 0; k <= n; k++)  
            { 
                float temp = a[j, k]; 
                a[j, k] = a[j + c, k]; 
                a[j + c, k] = temp; 
            } 
        } 
  
        for (j = 0; j < n; j++)  
        { 
              
            // Excluding all i == j 
            if (i != j)  
            { 
                  
                // Converting Matrix to reduced row 
                // echelon form(diagonal matrix) 
                float p = a[j, i] / a[i, i]; 
  
                for (k = 0; k <= n; k++)                  
                    a[j, k] = a[j, k] - (a[i, k]) * p;              
            } 
        } 
    } 
    return flag; 
} 
  
// Function to print the desired result  
// if unique solutions exists, otherwise  
// prints no solution or infinite solutions  
// depending upon the input given. 
static void PrintResult(float [,]a, 
                        int n, int flag) 
{ 
    Console.Write("Result is : "); 
  
    if (flag == 2)      
    Console.WriteLine("Infinite Solutions Exists");  
    else if (flag == 3)      
    Console.WriteLine("No Solution Exists"); 
      
    // Printing the solution by dividing  
    // constants by their respective 
    // diagonal elements 
    else 
    { 
        for (int i = 0; i < n; i++)          
            Console.Write(a[i, n] / a[i, i] + " ");      
    } 
} 
  
// To check whether infinite solutions  
// exists or no solution exists 
static int CheckConsistency(float [,]a,  
                            int n, int flag) 
{ 
    int i, j; 
    float sum; 
      
    // flag == 2 for infinite solution 
    // flag == 3 for No solution 
    flag = 3; 
    for (i = 0; i < n; i++)  
    { 
        sum = 0; 
        for (j = 0; j < n; j++)      
            sum = sum + a[i, j]; 
        if (sum == a[i, j])  
            flag = 2;      
    } 
    return flag; 
} 
  
// Driver code 
public static void Main(String[] args)  
{ 
    float [,]a = {{ 0, 2, 1, 4 },  
                  { 1, 1, 2, 6 },  
                  { 2, 1, 1, 7 }}; 
                      
    // Order of Matrix(n) 
    int n = 3, flag = 0; 
      
    // Performing Matrix transformation 
    flag = PerformOperation(a, n); 
      
    if (flag == 1)      
        flag = CheckConsistency(a, n, flag);  
  
    // Printing Final Matrix 
    Console.WriteLine("Final Augumented Matrix is : "); 
    PrintMatrix(a, n); 
    Console.WriteLine(""); 
      
    // Printing Solutions(if exist) 
    PrintResult(a, n, flag); 
} 
} 
  
// This code is contributed by 29AjayKumar 

Output:

Final Augumented Matrix is : 
1 0 0 2.2 
0 2 0 2.8 
0 0 -2.5 -3 

Result is : 2.2 1.4 1.2

Applications :

Solving System of Linear Equations : Gauss-Jordan Elimination Method can be used for finding the solution of a systems of linear equations which is applied throughout the mathematics.
Finding Determinant : The Gaussian Elimination can be applied to a square matrix in order to find determinant of the matrix.
Finding Inverse of Matrix : The Gauss-Jordan Elimination method can be used in determining the inverse of a square matrix.
Finding Ranks and Bases : Using reduced row echelon form, the ranks as well as bases of square matrices can be computed by Gaussian elimination method.

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