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.
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 |
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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