Program for Gauss-Jordan Elimination Method

25 Nov, 2021
Prerequisite : Gaussian Elimination to Solve Linear EquationsIntroduction : 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 substitutionBut 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 Augmented 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++
Java
C#
Javascript
// C++ Implementation for Gauss-Jordan// Elimination Method#include <bits/stdc++.h>using namespace std; #define M 10 // Function to print the matrixvoid 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 existsint 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 codeint 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 Augmented Matrix is : " << endl;    PrintMatrix(a, n);    cout << endl;         // Printing Solutions(if exist)    PrintResult(a, n, flag);     return 0;}
// Java Implementation for Gauss-Jordan// Elimination Methodclass GFG {     static int M = 10; // Function to print the matrixstatic 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 existsstatic 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 codepublic 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 Augmented Matrix is : ");    PrintMatrix(a, n);    System.out.println("");         // Printing Solutions(if exist)    PrintResult(a, n, flag);}} /* This code contributed by PrinciRaj1992 */
// C# Implementation for Gauss-Jordan// Elimination Methodusing System;using System.Collections.Generic; class GFG{static int M = 10; // Function to print the matrixstatic 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 existsstatic 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 codepublic 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 Augmented Matrix is : ");    PrintMatrix(a, n);    Console.WriteLine("");         // Printing Solutions(if exist)    PrintResult(a, n, flag);}} // This code is contributed by 29AjayKumar
<script> // JavaScript Implementation for Gauss-Jordan// Elimination Method let M = 10; // Function to print the matrixfunction PrintMatrix(a,n){    for (let i = 0; i < n; i++)    {        for (let j = 0; j <= n; j++)            document.write(a[i][j] + " ");        document.write("<br>");    }} // function to reduce matrix to reduced// row echelon form.function PerformOperation(a,n){    let i, j, k = 0, c, flag = 0, m = 0;    let 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++)            {                let 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)                let 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.function PrintResult(a,n,flag){    document.write("Result is : ");       if (flag == 2)            document.write("Infinite Solutions Exists<br>");    else if (flag == 3)            document.write("No Solution Exists<br>");                  // Printing the solution by dividing constants by    // their respective diagonal elements    else {        for (let i = 0; i < n; i++)                    document.write(a[i][n] / a[i][i] +" ");        }} // To check whether infinite solutions// exists or no solution existsfunction CheckConsistency(a,n,flag){    let i, j;    let 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 codelet a=[[ 0, 2, 1, 4 ],                    [ 1, 1, 2, 6 ],                    [ 2, 1, 1, 7 ]];// Order of Matrix(n)let n = 3, flag = 0; // Performing Matrix transformationflag = PerformOperation(a, n); if (flag == 1)        flag = CheckConsistency(a, n, flag); // Printing Final Matrixdocument.write("Final Augmented Matrix is : <br>");PrintMatrix(a, n);document.write("<br>"); // Printing Solutions(if exist)PrintResult(a, n, flag);   // This code is contributed by rag2127 </script>
Final Augmented 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.
 
princiraj1992
29AjayKumar
jeroenoorschot
rag2127
sumitgumber28
Mathematical
Matrix
Mathematical
Matrix
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