How to solve a Linear Equation?

Let’s suppose we have to solve this:

x + y + z = 6
2y + 5z = -4
2x + 5y - z = 27

Now we can create matrix from this:

1       1       1               6
0       2       5     and      -4
2       5       -1              27

Now we find Inverse of this matrix to get the solution

Now let’s write the program.

Java Program to Solve Linear Equations:

import java.util.Scanner;
public class Solve_Linear_Equation
{
    public static void main(String args[])
    {
        char []var = {'x', 'y', 'z', 'w'};
        System.out.println("Enter the number of variables in the equations: ");
        Scanner input = new Scanner(System.in);
        int n = input.nextInt();
        System.out.println("Enter the coefficients of each variable for each equations");
        System.out.println("ax + by + cz + ... = d");
        double [][]mat = new double[n][n];
        double [][]constants = new double[n][1];
        //input
        for(int i=0; i<n; i++)
        {
            for(int j=0; j<n; j++)
            {
                mat[i][j] = input.nextDouble();
            }
            constants[i][0] = input.nextDouble();
        }
        //Matrix representation
        for(int i=0; i<n; i++)
        {
            for(int j=0; j<n; j++)
            {
                System.out.print(" "+mat[i][j]);
            }
            System.out.print("  "+ var[i]);
            System.out.print("  =  "+ constants[i][0]);
            System.out.println();
        }
        //inverse of matrix mat[][]
        double inverted_mat[][] = invert(mat);
        System.out.println("The inverse is: ");
        for (int i=0; i<n; ++i)
        {
            for (int j=0; j<n; ++j)
            {
                System.out.print(inverted_mat[i][j]+"  ");
            }
            System.out.println();
        }
        //Multiplication of mat inverse and constants
        double result[][] = new double[n][1];
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j < 1; j++)
            {
                for (int k = 0; k < n; k++)
                {
                    result[i][j] = result[i][j] + inverted_mat[i][k] * constants[k][j];
                }
            }
        }
        System.out.println("The product is:");
        for(int i=0; i<n; i++)
        {
            System.out.println(result[i][0] + " ");
        }
        input.close();
    }
    public static double[][] invert(double a[][])
    {
        int n = a.length;
        double x[][] = new double[n][n];
        double b[][] = new double[n][n];
        int index[] = new int[n];
        for (int i=0; i<n; ++i)
            b[i][i] = 1;
 // Transform the matrix into an upper triangle
        gaussian(a, index);
 // Update the matrix b[i][j] with the ratios stored
        for (int i=0; i<n-1; ++i)
            for (int j=i+1; j<n; ++j)
                for (int k=0; k<n; ++k)
                    b[index[j]][k]
                            -= a[index[j]][i]*b[index[i]][k];
 // Perform backward substitutions
        for (int i=0; i<n; ++i)
        {
            x[n-1][i] = b[index[n-1]][i]/a[index[n-1]][n-1];
            for (int j=n-2; j>=0; --j)
            {
                x[j][i] = b[index[j]][i];
                for (int k=j+1; k<n; ++k)
                {
                    x[j][i] -= a[index[j]][k]*x[k][i];
                }
                x[j][i] /= a[index[j]][j];
            }
        }
        return x;
    }
// Method to carry out the partial-pivoting Gaussian
// elimination.  Here index[] stores pivoting order.
    public static void gaussian(double a[][], int index[])
    {
        int n = index.length;
        double c[] = new double[n];
 // Initialize the index
        for (int i=0; i<n; ++i)
            index[i] = i;
 // Find the rescaling factors, one from each row
        for (int i=0; i<n; ++i)
        {
            double c1 = 0;
            for (int j=0; j<n; ++j)
            {
                double c0 = Math.abs(a[i][j]);
                if (c0 > c1) c1 = c0;
            }
            c[i] = c1;
        }
 // Search the pivoting element from each column
        int k = 0;
        for (int j=0; j<n-1; ++j)
        {
            double pi1 = 0;
            for (int i=j; i<n; ++i)
            {
                double pi0 = Math.abs(a[index[i]][j]);
                pi0 /= c[index[i]];
                if (pi0 > pi1)
                {
                    pi1 = pi0;
                    k = i;
                }
            }
   // Interchange rows according to the pivoting order
            int itmp = index[j];
            index[j] = index[k];
            index[k] = itmp;
            for (int i=j+1; i<n; ++i)
            {
                double pj = a[index[i]][j]/a[index[j]][j];
 // Record pivoting ratios below the diagonal
                a[index[i]][j] = pj;
 // Modify other elements accordingly
                for (int l=j+1; l<n; ++l)
                    a[index[i]][l] -= pj*a[index[j]][l];
            }
        }
    }

OUTPUT:

# javac Solve_Linear_Equation.java
# java Solve_Linear_Equation
Enter the number of variables in the equations:
2
Enter the coefficients of each variable for each equations
ax + by + cz + ... = d
1 2 3
3 2 1
 1.0 2.0  x  =  3.0
 3.0 2.0  y  =  1.0
 The inverse is:
-0.49999999999999994  0.5
 0.7499999999999999  -0.24999999999999997
 The product is:
-0.9999999999999998
 1.9999999999999996