Cholesky Decomposition : Matrix Decomposition

In linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions. One of them is Cholesky Decomposition.

The Cholesky decomposition or Cholesky factorization is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. The Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations.

The Cholesky decomposition of a Hermitian positive-definite matrix A is a decomposition of the form A = [L][L]T, where L is a lower triangular matrix with real and positive diagonal entries, and LT denotes the conjugate transpose of L. Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a unique Cholesky decomposition. Every symmetric, positive definite matrix A can be decomposed into a product of a unique lower triangular matrix L and its transpose: A = L LT

The following formulas are obtained by solving above lower triangular matrix and its transpose. These are the basis of Cholesky Decomposition Algorithm :  Example :

Input : Output : Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Below is the implementation of Cholesky Decomposition.

C++

 // CPP program to decompose a matrix using // Cholesky Decomposition #include using namespace std;    const int MAX = 100;    void Cholesky_Decomposition(int matrix[][MAX],                                        int n) {     int lower[n][n];     memset(lower, 0, sizeof(lower));        // Decomposing a matrix into Lower Triangular     for (int i = 0; i < n; i++) {         for (int j = 0; j <= i; j++) {             int sum = 0;                if (j == i) // summation for diagnols             {                 for (int k = 0; k < j; k++)                     sum += pow(lower[j][k], 2);                 lower[j][j] = sqrt(matrix[j][j] -                                         sum);             } else {                    // Evaluating L(i, j) using L(j, j)                 for (int k = 0; k < j; k++)                     sum += (lower[i][k] * lower[j][k]);                 lower[i][j] = (matrix[i][j] - sum) /                                       lower[j][j];             }         }     }        // Displaying Lower Triangular and its Transpose     cout << setw(6) << " Lower Triangular"            << setw(30) << "Transpose" << endl;     for (int i = 0; i < n; i++) {                    // Lower Triangular         for (int j = 0; j < n; j++)             cout << setw(6) << lower[i][j] << "\t";         cout << "\t";                    // Transpose of Lower Triangular         for (int j = 0; j < n; j++)             cout << setw(6) << lower[j][i] << "\t";         cout << endl;     } }    // Driver Code int main() {     int n = 3;     int matrix[][MAX] = { { 4, 12, -16 },                         { 12, 37, -43 },                         { -16, -43, 98 } };     Cholesky_Decomposition(matrix, n);     return 0; }

Java

 // Java program to decompose  // a matrix using Cholesky // Decomposition    class GFG {        //static int MAX = 100; static void Cholesky_Decomposition(int[][] matrix,                                 int n) {     int[][] lower = new int[n][n];        // Decomposing a matrix      // into Lower Triangular     for (int i = 0; i < n; i++)     {         for (int j = 0; j <= i; j++)         {             int sum = 0;                // summation for diagnols             if (j == i)              {                 for (int k = 0; k < j; k++)                     sum += (int)Math.pow(lower[j][k], 2);                 lower[j][j] = (int)Math.sqrt(matrix[j][j] -                                                     sum);             }                             else             {                    // Evaluating L(i, j)                  // using L(j, j)                 for (int k = 0; k < j; k++)                     sum += (lower[i][k] *                              lower[j][k]);                 lower[i][j] = (matrix[i][j] - sum) /                                     lower[j][j];             }         }     }        // Displaying Lower      // Triangular and its Transpose     System.out.println(" Lower Triangular\t Transpose");     for (int i = 0; i < n; i++)      {                    // Lower Triangular         for (int j = 0; j < n; j++)             System.out.print(lower[i][j] + "\t");         System.out.print("");                    // Transpose of          // Lower Triangular         for (int j = 0; j < n; j++)             System.out.print(lower[j][i] + "\t");         System.out.println();     } }    // Driver Code public static void main(String[] args) {     int n = 3;     int[][] matrix = new int[][]{{4, 12, -16},                     {12, 37, -43},                     {-16, -43, 98}};                            Cholesky_Decomposition(matrix, n); } }    // This code is contributed by mits

Python3

 # Python3 program to decompose  # a matrix using Cholesky  # Decomposition import math MAX = 100;    def Cholesky_Decomposition(matrix, n):        lower = [[0 for x in range(n + 1)]                  for y in range(n + 1)];        # Decomposing a matrix     # into Lower Triangular     for i in range(n):          for j in range(i + 1):              sum1 = 0;                # sum1mation for diagnols             if (j == i):                  for k in range(j):                     sum1 += pow(lower[j][k], 2);                 lower[j][j] = int(math.sqrt(matrix[j][j] - sum1));             else:                                    # Evaluating L(i, j)                 # using L(j, j)                 for k in range(j):                     sum1 += (lower[i][k] *lower[j][k]);                 if(lower[j][j] > 0):                     lower[i][j] = int((matrix[i][j] - sum1) /                                                 lower[j][j]);        # Displaying Lower Triangular     # and its Transpose     print("Lower Triangular\t\tTranspose");     for i in range(n):                     # Lower Triangular         for j in range(n):             print(lower[i][j], end = "\t");         print("", end = "\t");                    # Transpose of         # Lower Triangular         for j in range(n):             print(lower[j][i], end = "\t");         print("");    # Driver Code n = 3; matrix = [[4, 12, -16],           [12, 37, -43],           [-16, -43, 98]]; Cholesky_Decomposition(matrix, n);    # This code is contributed by mits

C#

 // C# program to decompose  // a matrix using Cholesky // Decomposition using System;    class GFG {        //static int MAX = 100; static void Cholesky_Decomposition(int[,] matrix,                                    int n) {     int[,] lower = new int[n, n];        // Decomposing a matrix      // into Lower Triangular     for (int i = 0; i < n; i++)     {         for (int j = 0; j <= i; j++)         {             int sum = 0;                // summation for diagnols             if (j == i)              {                 for (int k = 0; k < j; k++)                     sum += (int)Math.Pow(lower[j, k], 2);                 lower[j, j] = (int)Math.Sqrt(matrix[j, j] -                                                       sum);             }                             else             {                    // Evaluating L(i, j)                  // using L(j, j)                 for (int k = 0; k < j; k++)                     sum += (lower[i, k] *                              lower[j, k]);                 lower[i, j] = (matrix[i,j] - sum) /                                        lower[j, j];             }         }     }        // Displaying Lower      // Triangular and its Transpose     Console.WriteLine("  Lower Triangular\t   Transpose");     for (int i = 0; i < n; i++)      {                    // Lower Triangular         for (int j = 0; j < n; j++)             Console.Write(lower[i, j] + "\t");         Console.Write("");                    // Transpose of          // Lower Triangular         for (int j = 0; j < n; j++)             Console.Write(lower[j, i] + "\t");         Console.WriteLine();     } }    // Driver Code static int Main() {     int n = 3;     int[,] matrix = {{4, 12, -16},                      {12, 37, -43},                      {-16, -43, 98}};                             Cholesky_Decomposition(matrix, n);     return 0; } }    // This code is contributed by mits

PHP



Output:

Lower Triangular                   Transpose
2         0         0             2         6        -8
6         1         0             0         1         5
-8         5         3             0         0         3

References: Wikipedia – Cholesky decomposition
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