The article focuses on using an algorithm for solving a system of linear equations. We will deal with the matrix of coefficients. Gaussian Elimination does not work on singular matrices (they lead to division by zero).
Input: For N unknowns, input is an augmented matrix of size N x (N+1). One extra column is for Right Hand Side (RHS) mat[N][N+1] = {{3.0, 2.0,-4.0, 3.0}, {2.0, 3.0, 3.0, 15.0}, {5.0, -3, 1.0, 14.0} }; Output: Solution to equations is: 3.000000 1.000000 2.000000 Explanation: Given matrix represents following equations 3.0X1 + 2.0X2 - 4.0X3 = 3.0 2.0X1 + 3.0X2 + 3.0X3 = 15.0 5.0X1 - 3.0X2 + X3 = 14.0 There is a unique solution for given equations, solutions is, X1 = 3.0, X2 = 1.0, X3 = 2.0,
Row echelon form: Matrix is said to be in r.e.f. if the following conditions hold:
- The first non-zero element in each row, called the leading coefficient, is 1.
- Each leading coefficient is in a column to the right of the previous row leading coefficient.
- Rows with all zeros are below rows with at least one non-zero element.
Reduced row echelon form: Matrix is said to be in r.r.e.f. if the following conditions hold –
- All the conditions for r.e.f.
- The leading coefficient in each row is the only non-zero entry in its column.
The algorithm is majorly about performing a sequence of operations on the rows of the matrix. What we would like to keep in mind while performing these operations is that we want to convert the matrix into an upper triangular matrix in row echelon form. The operations can be:
- Swapping two rows
- Multiplying a row by a non-zero scalar
- Adding to one row a multiple of another
The process:
- Forward elimination: reduction to row echelon form. Using it one can tell whether there are no solutions, or unique solution, or infinitely many solutions.
- Back substitution: further reduction to reduced row echelon form.
Algorithm:
- Partial pivoting: Find the kth pivot by swapping rows, to move the entry with the largest absolute value to the pivot position. This imparts computational stability to the algorithm.
- For each row below the pivot, calculate the factor f which makes the kth entry zero, and for every element in the row subtract the fth multiple of the corresponding element in the kth row.
- Repeat above steps for each unknown. We will be left with a partial r.e.f. matrix.
Below is the implementation of above algorithm.
C++
// C++ program to demonstrate working of Guassian Elimination // method #include<bits/stdc++.h> using namespace std; #define N 3 // Number of unknowns // function to reduce matrix to r.e.f. Returns a value to // indicate whether matrix is singular or not int forwardElim( double mat[N][N+1]); // function to calculate the values of the unknowns void backSub( double mat[N][N+1]); // function to get matrix content void gaussianElimination( double mat[N][N+1]) { /* reduction into r.e.f. */ int singular_flag = forwardElim(mat); /* if matrix is singular */ if (singular_flag != -1) { printf ( "Singular Matrix.\n" ); /* if the RHS of equation corresponding to zero row is 0, * system has infinitely many solutions, else inconsistent*/ if (mat[singular_flag][N]) printf ( "Inconsistent System." ); else printf ( "May have infinitely many " "solutions." ); return ; } /* get solution to system and print it using backward substitution */ backSub(mat); } // function for elementary operation of swapping two rows void swap_row( double mat[N][N+1], int i, int j) { //printf("Swapped rows %d and %d\n", i, j); for ( int k=0; k<=N; k++) { double temp = mat[i][k]; mat[i][k] = mat[j][k]; mat[j][k] = temp; } } // function to print matrix content at any stage void print( double mat[N][N+1]) { for ( int i=0; i<N; i++, printf ( "\n" )) for ( int j=0; j<=N; j++) printf ( "%lf " , mat[i][j]); printf ( "\n" ); } // function to reduce matrix to r.e.f. int forwardElim( double mat[N][N+1]) { for ( int k=0; k<N; k++) { // Initialize maximum value and index for pivot int i_max = k; int v_max = mat[i_max][k]; /* find greater amplitude for pivot if any */ for ( int i = k+1; i < N; i++) if ( abs (mat[i][k]) > v_max) v_max = mat[i][k], i_max = i; /* if a prinicipal diagonal element is zero, * it denotes that matrix is singular, and * will lead to a division-by-zero later. */ if (!mat[k][i_max]) return k; // Matrix is singular /* Swap the greatest value row with current row */ if (i_max != k) swap_row(mat, k, i_max); for ( int i=k+1; i<N; i++) { /* factor f to set current row kth element to 0, * and subsequently remaining kth column to 0 */ double f = mat[i][k]/mat[k][k]; /* subtract fth multiple of corresponding kth row element*/ for ( int j=k+1; j<=N; j++) mat[i][j] -= mat[k][j]*f; /* filling lower triangular matrix with zeros*/ mat[i][k] = 0; } //print(mat); //for matrix state } //print(mat); //for matrix state return -1; } // function to calculate the values of the unknowns void backSub( double mat[N][N+1]) { double x[N]; // An array to store solution /* Start calculating from last equation up to the first */ for ( int i = N-1; i >= 0; i--) { /* start with the RHS of the equation */ x[i] = mat[i][N]; /* Initialize j to i+1 since matrix is upper triangular*/ for ( int j=i+1; j<N; j++) { /* subtract all the lhs values * except the coefficient of the variable * whose value is being calculated */ x[i] -= mat[i][j]*x[j]; } /* divide the RHS by the coefficient of the unknown being calculated */ x[i] = x[i]/mat[i][i]; } printf ( "\nSolution for the system:\n" ); for ( int i=0; i<N; i++) printf ( "%lf\n" , x[i]); } // Driver program int main() { /* input matrix */ double mat[N][N+1] = {{3.0, 2.0,-4.0, 3.0}, {2.0, 3.0, 3.0, 15.0}, {5.0, -3, 1.0, 14.0} }; gaussianElimination(mat); return 0; } |
Java
// Java program to demonstrate working of Guassian Elimination // method import java.io.*; class GFG { public static int N = 3 ; // Number of unknowns // function to get matrix content static void gaussianElimination( double mat[][]) { /* reduction into r.e.f. */ int singular_flag = forwardElim(mat); /* if matrix is singular */ if (singular_flag != - 1 ) { System.out.println( "Singular Matrix." ); /* if the RHS of equation corresponding to zero row is 0, * system has infinitely many solutions, else inconsistent*/ if (mat[singular_flag][N] != 0 ) System.out.print( "Inconsistent System." ); else System.out.print( "May have infinitely many solutions." ); return ; } /* get solution to system and print it using backward substitution */ backSub(mat); } // function for elementary operation of swapping two // rows static void swap_row( double mat[][], int i, int j) { // printf("Swapped rows %d and %d\n", i, j); for ( int k = 0 ; k <= N; k++) { double temp = mat[i][k]; mat[i][k] = mat[j][k]; mat[j][k] = temp; } } // function to print matrix content at any stage static void print( double mat[][]) { for ( int i = 0 ; i < N; i++, System.out.println()) for ( int j = 0 ; j <= N; j++) System.out.print(mat[i][j]); System.out.println(); } // function to reduce matrix to r.e.f. static int forwardElim( double mat[][]) { for ( int k = 0 ; k < N; k++) { // Initialize maximum value and index for pivot int i_max = k; int v_max = ( int )mat[i_max][k]; /* find greater amplitude for pivot if any */ for ( int i = k + 1 ; i < N; i++) if (Math.abs(mat[i][k]) > v_max) { v_max = ( int )mat[i][k]; i_max = i; } /* if a prinicipal diagonal element is zero, * it denotes that matrix is singular, and * will lead to a division-by-zero later. */ if (mat[k][i_max] == 0 ) return k; // Matrix is singular /* Swap the greatest value row with current row */ if (i_max != k) swap_row(mat, k, i_max); for ( int i = k + 1 ; i < N; i++) { /* factor f to set current row kth element * to 0, and subsequently remaining kth * column to 0 */ double f = mat[i][k] / mat[k][k]; /* subtract fth multiple of corresponding kth row element*/ for ( int j = k + 1 ; j <= N; j++) mat[i][j] -= mat[k][j] * f; /* filling lower triangular matrix with * zeros*/ mat[i][k] = 0 ; } // print(mat); //for matrix state } // print(mat); //for matrix state return - 1 ; } // function to calculate the values of the unknowns static void backSub( double mat[][]) { double x[] = new double [N]; // An array to store solution /* Start calculating from last equation up to the first */ for ( int i = N - 1 ; i >= 0 ; i--) { /* start with the RHS of the equation */ x[i] = mat[i][N]; /* Initialize j to i+1 since matrix is upper triangular*/ for ( int j = i + 1 ; j < N; j++) { /* subtract all the lhs values * except the coefficient of the variable * whose value is being calculated */ x[i] -= mat[i][j] * x[j]; } /* divide the RHS by the coefficient of the unknown being calculated */ x[i] = x[i] / mat[i][i]; } System.out.println(); System.out.println( "Solution for the system:" ); for ( int i = 0 ; i < N; i++) { System.out.format( "%.6f" , x[i]); System.out.println(); } } // Driver program public static void main(String[] args) { /* input matrix */ double mat[][] = { { 3.0 , 2.0 , - 4.0 , 3.0 }, { 2.0 , 3.0 , 3.0 , 15.0 }, { 5.0 , - 3 , 1.0 , 14.0 } }; gaussianElimination(mat); } } // This code is contributed by Dharanendra L V. |
PHP
<?php // PHP program to demonstrate working // of Guassian Elimination method $N = 3; // Number of unknowns // function to get matrix content function gaussianElimination( $mat ) { global $N ; /* reduction into r.e.f. */ $singular_flag = forwardElim( $mat ); /* if matrix is singular */ if ( $singular_flag != -1) { print ( "Singular Matrix.\n" ); /* if the RHS of equation corresponding to zero row is 0, * system has infinitely many solutions, else inconsistent*/ if ( $mat [ $singular_flag ][ $N ]) print ( "Inconsistent System." ); else print ( "May have infinitely many solutions." ); return ; } /* get solution to system and print it using backward substitution */ backSub( $mat ); } // function for elementary operation // of swapping two rows function swap_row(& $mat , $i , $j ) { global $N ; //printf("Swapped rows %d and %d\n", i, j); for ( $k = 0; $k <= $N ; $k ++) { $temp = $mat [ $i ][ $k ]; $mat [ $i ][ $k ] = $mat [ $j ][ $k ]; $mat [ $j ][ $k ] = $temp ; } } // function to print matrix content at any stage function print1( $mat ) { global $N ; for ( $i =0; $i < $N ; $i ++, print ( "\n" )) for ( $j =0; $j <= $N ; $j ++) print ( $mat [ $i ][ $j ]); print ( "\n" ); } // function to reduce matrix to r.e.f. function forwardElim(& $mat ) { global $N ; for ( $k =0; $k < $N ; $k ++) { // Initialize maximum value and index for pivot $i_max = $k ; $v_max = $mat [ $i_max ][ $k ]; /* find greater amplitude for pivot if any */ for ( $i = $k +1; $i < $N ; $i ++) if ( abs ( $mat [ $i ][ $k ]) > $v_max ) { $v_max = $mat [ $i ][ $k ]; $i_max = $i ; } /* if a prinicipal diagonal element is zero, * it denotes that matrix is singular, and * will lead to a division-by-zero later. */ if (! $mat [ $k ][ $i_max ]) return $k ; // Matrix is singular /* Swap the greatest value row with current row */ if ( $i_max != $k ) swap_row( $mat , $k , $i_max ); for ( $i = $k + 1; $i < $N ; $i ++) { /* factor f to set current row kth element to 0, * and subsequently remaining kth column to 0 */ $f = $mat [ $i ][ $k ]/ $mat [ $k ][ $k ]; /* subtract fth multiple of corresponding kth row element*/ for ( $j = $k + 1; $j <= $N ; $j ++) $mat [ $i ][ $j ] -= $mat [ $k ][ $j ] * $f ; /* filling lower triangular matrix with zeros*/ $mat [ $i ][ $k ] = 0; } //print(mat); //for matrix state } //print(mat); //for matrix state return -1; } // function to calculate the values of the unknowns function backSub(& $mat ) { global $N ; $x = array_fill (0, $N , 0); // An array to store solution /* Start calculating from last equation up to the first */ for ( $i = $N - 1; $i >= 0; $i --) { /* start with the RHS of the equation */ $x [ $i ] = $mat [ $i ][ $N ]; /* Initialize j to i+1 since matrix is upper triangular*/ for ( $j = $i + 1; $j < $N ; $j ++) { /* subtract all the lhs values * except the coefficient of the variable * whose value is being calculated */ $x [ $i ] -= $mat [ $i ][ $j ] * $x [ $j ]; } /* divide the RHS by the coefficient of the unknown being calculated */ $x [ $i ] = $x [ $i ] / $mat [ $i ][ $i ]; } print ( "\nSolution for the system:\n" ); for ( $i = 0; $i < $N ; $i ++) print (number_format( strval ( $x [ $i ]), 6). "\n" ); } // Driver program /* input matrix */ $mat = array ( array (3.0, 2.0,-4.0, 3.0), array (2.0, 3.0, 3.0, 15.0), array (5.0, -3, 1.0, 14.0)); gaussianElimination( $mat ); // This code is contributed by mits ?> |
C#
// C# program to demonstrate working // of Guassian Elimination method using System; class GFG{ // Number of unknowns public static int N = 3; // Function to get matrix content static void gaussianElimination( double [,]mat) { /* reduction into r.e.f. */ int singular_flag = forwardElim(mat); /* if matrix is singular */ if (singular_flag != -1) { Console.WriteLine( "Singular Matrix." ); /* if the RHS of equation corresponding to zero row is 0, * system has infinitely many solutions, else inconsistent*/ if (mat[singular_flag,N] != 0) Console.Write( "Inconsistent System." ); else Console.Write( "May have infinitely " + "many solutions." ); return ; } /* get solution to system and print it using backward substitution */ backSub(mat); } // Function for elementary operation of swapping two // rows static void swap_row( double [,]mat, int i, int j) { // printf("Swapped rows %d and %d\n", i, j); for ( int k = 0; k <= N; k++) { double temp = mat[i, k]; mat[i, k] = mat[j, k]; mat[j, k] = temp; } } // Function to print matrix content at any stage static void print( double [,]mat) { for ( int i = 0; i < N; i++, Console.WriteLine()) for ( int j = 0; j <= N; j++) Console.Write(mat[i, j]); Console.WriteLine(); } // Function to reduce matrix to r.e.f. static int forwardElim( double [,]mat) { for ( int k = 0; k < N; k++) { // Initialize maximum value and index for pivot int i_max = k; int v_max = ( int )mat[i_max, k]; /* find greater amplitude for pivot if any */ for ( int i = k + 1; i < N; i++) { if (Math.Abs(mat[i, k]) > v_max) { v_max = ( int )mat[i, k]; i_max = i; } /* If a prinicipal diagonal element is zero, * it denotes that matrix is singular, and * will lead to a division-by-zero later. */ if (mat[k, i_max] == 0) return k; // Matrix is singular /* Swap the greatest value row with current row */ if (i_max != k) swap_row(mat, k, i_max); for ( int i = k + 1; i < N; i++) { /* factor f to set current row kth element * to 0, and subsequently remaining kth * column to 0 */ double f = mat[i, k] / mat[k, k]; /* subtract fth multiple of corresponding kth row element*/ for ( int j = k + 1; j <= N; j++) mat[i, j] -= mat[k, j] * f; /* filling lower triangular matrix with * zeros*/ mat[i, k] = 0; } } // print(mat); //for matrix state } // print(mat); //for matrix state return -1; } // Function to calculate the values of the unknowns static void backSub( double [,]mat) { // An array to store solution double []x = new double [N]; /* Start calculating from last equation up to the first */ for ( int i = N - 1; i >= 0; i--) { /* start with the RHS of the equation */ x[i] = mat[i,N]; /* Initialize j to i+1 since matrix is upper triangular*/ for ( int j = i + 1; j < N; j++) { /* subtract all the lhs values * except the coefficient of the variable * whose value is being calculated */ x[i] -= mat[i,j] * x[j]; } /* divide the RHS by the coefficient of the unknown being calculated */ x[i] = x[i] / mat[i,i]; } Console.WriteLine(); Console.WriteLine( "Solution for the system:" ); for ( int i = 0; i < N; i++) { Console.Write( "{0:F6}" , x[i]); Console.WriteLine(); } } // Driver code public static void Main(String[] args) { /* input matrix */ double [,]mat = { { 3.0, 2.0, -4.0, 3.0 }, { 2.0, 3.0, 3.0, 15.0 }, { 5.0, -3, 1.0, 14.0 } }; gaussianElimination(mat); } } // This code is contributed by shikhasingrajput |
Output:
Solution for the system: 3.000000 1.000000 2.000000
Illustration:
Time Complexity: Since for each pivot we traverse the part to its right for each row below it, O(n)*(O(n)*O(n)) = O(n3).
We can also apply Gaussian Elimination for calculating:
- Rank of a matrix
- Determinant of a matrix
- Inverse of an invertible square matrix
This article is contributed by Yash Varyani. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
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