Gaussian Elimination to Solve Linear Equations

Difficulty Level : Medium
Last Updated : 15 Apr, 2021

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,

matrix11

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.

matrix1

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:

process

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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