Java/src/main/java/com/thealgorithms/misc/InverseOfMatrix.java

package com.thealgorithms.misc;

import java.util.Scanner;

/*
 * Wikipedia link : https://en.wikipedia.org/wiki/Invertible_matrix
 *
 * Here we use gauss elimination method to find the inverse of a given matrix.
 * To understand gauss elimination method to find inverse of a matrix:
 * https://www.sangakoo.com/en/unit/inverse-matrix-method-of-gaussian-elimination
 *
 * We can also find the inverse of a matrix
 */
public class InverseOfMatrix {

    public static void main(String[] argv) {
        Scanner input = new Scanner(System.in);
        System.out.println("Enter the matrix size (Square matrix only): ");
        int n = input.nextInt();
        double[][] a = new double[n][n];
        System.out.println("Enter the elements of matrix: ");
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                a[i][j] = input.nextDouble();
            }
        }

        double[][] d = invert(a);
        System.out.println();
        System.out.println("The inverse is: ");
        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < n; ++j) {
                System.out.print(d[i][j] + "  ");
            }
            System.out.println();
        }
        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];
                }
            }
        }
    }
}