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Cholesky Decomposition : Matrix Decomposition
  • Last Updated : 21 May, 2021

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 :
 

L_{j,j}=\sqrt {A_{j,j}-\sum_{k=0}^{j-1}(L_{j,k})^{2}}     [Tex]L_{i,j}=\frac{1}{L_{j,j}}(A_{i,j}-\sum_{k=0}^{j-1}L_{i,k}L_{j,k})     [/Tex]
 

Example



 

Input : 

Output :

 

 

 

 

 

Below is the implementation of Cholesky Decomposition. 

 

C++




// CPP program to decompose a matrix using
// Cholesky Decomposition
#include <bits/stdc++.h>
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




<?php
// PHP program to decompose
// a matrix using Cholesky
// Decomposition
$MAX = 100;
 
function Cholesky_Decomposition($matrix, $n)
{
    $lower;
    for ($i = 0; $i <= $n; $i++)
    for ($j = 0; $j <= $n; $j++)
    $lower[$i][$j] = 0;
 
    // Decomposing a matrix
    // into Lower Triangular
    for ($i = 0; $i < $n; $i++)
    {
        for ($j = 0; $j <= $i; $j++)
        {
            $sum = 0;
 
            // summation for diagnols
            if ($j == $i)
            {
                for ($k = 0; $k < $j; $k++)
                    $sum += pow($lower[$j][$k], 2);
                $lower[$j][$j] = sqrt($matrix[$j][$j] -
                                      $sum);
            }
            else
            {
                // Evaluating L(i, j)
                // using L(j, j)
                for ($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
    echo "     Lower Triangular" .
           str_pad("Transpose", 30, " ",
                   STR_PAD_BOTH) . "\n";
    for ($i = 0; $i < $n; $i++)
    {
         
        // Lower Triangular
        for ($j = 0; $j < $n; $j++)
            echo str_pad($lower[$i][$j], 6,
                         " ", STR_PAD_BOTH)."\t";
        echo "\t";
         
        // Transpose of
        // Lower Triangular
        for ($j = 0; $j < $n; $j++)
            echo str_pad($lower[$j][$i], 6,
                         " ", STR_PAD_BOTH)."\t";
        echo "\n";
    }
}
 
// Driver Code
$n = 3;
$matrix = array(array(4, 12, -16),
                array(12, 37, -43),
                array(-16, -43, 98));
Cholesky_Decomposition($matrix, $n);
 
// This code is contributed by vt_m.
?>

Javascript




<script>
// javascript program to decompose
// a matrix using Cholesky
// Decomposition
   // function MAX = 100;
function Cholesky_Decomposition(matrix,n)
{
    var lower = Array(n).fill(0).map(x => Array(n).fill(0));
 
    // Decomposing a matrix
    // into Lower Triangular
    for (var i = 0; i < n; i++) {
        for (var j = 0; j <= i; j++) {
            var sum = 0;
 
            // summation for diagnols
            if (j == i) {
                for (var k = 0; k < j; k++)
                    sum += parseInt(Math.pow(lower[j][k],
                                         2));
                lower[j][j] = parseInt(Math.sqrt(
                    matrix[j][j] - sum));
            }
 
            else {
 
                // Evaluating L(i, j)
                // using L(j, j)
                for (var 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
    document.write(" Lower Triangular     Transpose<br>");
    for (var i = 0; i < n; i++) {
 
        // Lower Triangular
        for (var j = 0; j < n; j++)
            document.write(lower[i][j] + "        ");
         
 
        // Transpose of
        // Lower Triangular
        for (var j = 0; j < n; j++)
            document.write(lower[j][i] + "        ");
        document.write('<br>');
    }
}
 
// Driver Code
var n = 3;
var matrix = [[ 4, 12, -16 ],
                               [ 12, 37, -43 ],
                               [ -16, -43, 98 ] ];
 
Cholesky_Decomposition(matrix, n);
 
// This code contributed by Princi Singh
</script>

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 
This article is contributed by Shubham Rana. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 

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