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MatrixDeterminantLaplaceExpansion.java
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MatrixDeterminantLaplaceExpansion.java
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/**
* This is an implementation of finding the determinant of an nxn matrix using Laplace/cofactor
* expansion. Although this method is mathematically beautiful, it is computationally intensive and
* not practical for matrices beyond the size of 7-8.
*
* <p>Time Complexity: ~O((n+2)!)
*
* @author William Fiset, [email protected]
*/
package com.williamfiset.algorithms.linearalgebra;
/**
* This is an implementation of finding the determinant of an nxn matrix using Laplace/cofactor
* expansion. Although this method is mathematically beautiful, it is computationally intensive and
* not practical for matrices beyond the size of 7-8.
*
* <p>Time Complexity: ~O((n+2)!)
*
* @author William Fiset, [email protected]
*/
/**
* This is an implementation of finding the determinant of an nxn matrix using Laplace/cofactor
* expansion. Although this method is mathematically beautiful, it is computationally intensive and
* not practical for matrices beyond the size of 7-8.
*
* <p>Time Complexity: ~O((n+2)!)
*
* @author William Fiset, [email protected]
*/
public class MatrixDeterminantLaplaceExpansion {
// Define a small value of epsilon to compare double values
static final double EPS = 0.00000001;
public static void main(String[] args) {
double[][] m = {{6}};
System.out.println(determinant(m)); // 6
m =
new double[][] {
{1, 2},
{3, 4}
};
System.out.println(determinant(m)); // -2
m =
new double[][] {
{1, -2, 3},
{4, -5, 6},
{7, -8, 10}
};
System.out.println(determinant(m)); // 3
m =
new double[][] {
{1, -2, 3, 7},
{4, -5, 6, 2},
{7, -8, 10, 3},
{-8, 10, 3, 2}
};
System.out.println(determinant(m)); // -252
m =
new double[][] {
{1, -2, 3, 7},
{4, -5, 6, 2},
{7, -8, 10, 3},
{-8, 10, 3, 2}
};
System.out.println(determinant(m)); // -252
m =
new double[][] {
{1, -2, 3, 7, 12},
{4, -5, 6, 2, 4},
{7, -8, 10, 3, 1},
{-8, 10, 8, 3, 2},
{5, 5, 5, 5, 5}
};
System.out.println(determinant(m)); // -27435
m = new double[][]{
{1 , 3 , 5 , 9},
{1 , 3 , 1 , 7},
{4 , 3 , 9 , 7},
{5 , 2 , 0 , 9},
}; // determinant(mat1) = -376 , mat(4 * 4)
System.out.println(determinant(m));
m = new double[][]{
{1 , 3 , 5 , 4},
{2 , 3 , 1 , 3},
{4 , 3 , 9 , 7},
{5 , 2 , 6 , 9},
}; // determinant(mat2) = -152 , mat(4 * 4)
System.out.println(determinant(m));
m = new double[][]{
{4 , 7 , 2 , 3},
{1 , 3 , 1 , 2},
{2 , 5 , 3 , 4},
{1 , 4 , 2 , 3},
}; // determinant(mat3) = -3 , mat(4 * 4)
System.out.println(determinant(m));
m = new double[][]{
{1 , 0 , 0 , 0 , 0 , 2},
{0 , 1 , 0 , 0 , 2 , 0},
{0 , 0 , 1 , 2 , 0 , 0},
{0 , 0 , 2 , 1 , 0 , 0},
{0 , 2 , 0 , 0 , 1 , 0},
{2 , 0 , 0 , 0 , 0 , 1},
}; // determinant(mat4) = -27 , mat(6 * 6)
System.out.println(determinant(m));
m = new double[][]{
{1 , 1 , 9 , 3 , 1 , 2 , 3},
{9 , 1 , 8 , 4 , 2 , 3 , 1},
{3 , 2 , 7 , 2 , 9 , 5 , 5},
{4 , 6 , 2 , 1 , 7 , 9 , 6},
{5 , 3 , 1 , 3 , 1 , 5 , 3},
{2 , 7 , 9 , 5 , 0 , 1 , 2},
{2 , 1 , 3 , 8 , 9 , 1 , 4}
}; // determinant(mat5) = 66704 mat(7 * 7)
System.out.println(determinant(m));
m = new double[][]{
{1 , 1 , 9 , 3 , 1 , 2 , 3 , 9},
{9 , 1 , 8 , 4 , 2 , 3 , 1 , 8},
{3 , 2 , 7 , 2 , 9 , 5 , 5 , 7},
{4 , 6 , 2 , 1 , 7 , 9 , 6 , 6},
{5 , 3 , 1 , 3 , 1 , 5 , 3 , 5},
{2 , 7 , 9 , 5 , 0 , 1 , 2 , 4},
{2 , 1 , 3 , 8 , 9 , 1 , 4 , 3},
{6 , 1 , 6 , 7 , 9 , 1 , 4 , 2}
}; // determinant(mat6) = -39240 , mat(8 * 8)
System.out.println(determinant(m));
m = new double[][]{
{1 , 1 , 9 , 3 , 1 , 2 , 3 , 9 , 1},
{9 , 1 , 8 , 4 , 2 , 3 , 1 , 8 , 2},
{3 , 2 , 7 , 2 , 9 , 5 , 5 , 7 , 3},
{4 , 6 , 2 , 1 , 7 , 9 , 6 , 6 , 4},
{5 , 3 , 1 , 3 , 1 , 5 , 3 , 5 , 5},
{2 , 7 , 9 , 5 , 0 , 1 , 2 , 4 , 6},
{2 , 1 , 3 , 8 , 9 , 1 , 4 , 3 , 7},
{6 , 1 , 6 , 7 , 9 , 1 , 4 , 2 , 8},
{9 , 8 , 7 , 4 , 3 , 3 , 4 , 2 , 9}
}; // determinant(mat7) = 1910870 , mat( 9 * 9)
System.out.println(determinant(m));
m = new double[][]{
{1 , 2 , 4 , 8 , 6 , 3 , 4 , 8 , 0 , 2},
{2 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 1},
{5 , 2 , 3 , 4 , 8 , 9 , 1 , 9 , 8 , 3},
{1 , 1 , 1 , 6 , 4 , 2 , 5 , 9 , 8 , 7},
{9 , 5 , 0 , 1 , 2 , 0 , 6 , 0 , 0 , 0},
{8 , 4 , 0 , 1 , 2 , 3 , 4 , 5 , 8 , 4},
{7 , 3 , 3 , 6 , 7 , 8 , 9 , 1 , 7 , 3},
{1 , 2 , 4 , 0 , 0 , 0 , 0 , 3 , 5 , 2},
{1 , 1 , 0 , 4 , 5 , 0 , 0 , 4 , 2 , 1},
{1 , 0 , 0 , 0 , 9 , 0 , 0 , 1 , 1 , 6}
}; // determinant(mat0) = 17265530 (1.726553E7)
System.out.println(determinant(m));
}
// Given an n*n matrix, this method finds the determinant using Laplace/cofactor expansion.
// Time Complexity: ~O((n+2)!)
public static double determinant(double[][] matrix) {
final int n = matrix.length;
// Use closed form for 1x1 determinant
if (n == 1) return matrix[0][0];
// Use closed form for 2x2 determinant
if (n == 2) return matrix[0][0] * matrix[1][1] - matrix[0][1] * matrix[1][0];
// For 3x3 matrices and up use Laplace/cofactor expansion
return laplace(matrix);
}
// This method uses cofactor expansion to compute the determinant
// of a matrix. Unfortunately, this method is very slow and uses
// A LOT of memory, hence it is not too practical for large matrices.
private static double laplace(double[][] m) {
final int n = m.length;
// Base case is 3x3 determinant
if (n == 3) {
/*
* Used as a temporary variables to make calculation easy
* | a b c |
* | d e f |
* | g h i |
*/
double a = m[0][0], b = m[0][1], c = m[0][2];
double d = m[1][0], e = m[1][1], f = m[1][2];
double g = m[2][0], h = m[2][1], i = m[2][2];
return a * (e * i - f * h) - b * (d * i - f * g) + c * (d * h - e * g);
}
int det = 0;
for (int i = 0; i < n; i++) {
double c = m[0][i];
if (c > EPS){
int sign = ((i & 1) == 0) ? +1 : -1;
det += sign * m[0][i] * laplace( constructMatrix(m ,0 , i) );
}
}
return det;
}
// Constructs a matrix one dimension smaller than the last by
// excluding the top row and some selected column. This
// method ends up consuming a lot of space we called recursively multiple times
// since it allocates memory for a new matrix.
private static double[][] constructMatrix(double[][] mat , int excludingRow , int excludingCol){
int n = mat.length;
double[][] newMatrix = new double[n - 1][n - 1];
int rPtr = -1;
for (int i = 0; i < n; i++) {
if (i == excludingRow)continue;
++rPtr;
int cPtr = -1;
for (int j = 0; j < n; j++) {
if (j == excludingCol)continue;
newMatrix[rPtr][++cPtr] = mat[i][j];
} // end of inner loop
} // end of outer loop
return newMatrix;
} // end of createSubMatrix
}