Doolittle Algorithm : LU Decomposition

In numerical analysis and linear algebra, LU decomposition (where ‘LU’ stands for ‘lower upper’, and also called LU factorization) factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. Computers usually solve square systems of linear equations using the LU decomposition, and it is also a key step when inverting a matrix, or computing the determinant of a matrix. The LU decomposition was introduced by mathematician Tadeusz Banachiewicz in 1938.

Let A be a square matrix. An LU factorization refers to the factorization of A, with proper row and/or column orderings or permutations, into two factors, a lower triangular matrix L and an upper triangular matrix U, A=LU.

Doolittle Algorithm :
It is always possible to factor a square matrix into a lower triangular matrix and an upper triangular matrix. That is, [A] = [L][U]

Doolittle’s method provides an alternative way to factor A into an LU decomposition without going through the hassle of Gaussian Elimination.

For a general n×n matrix A, we assume that an LU decomposition exists, and write the form of L and U explicitly. We then systematically solve for the entries in L and U from the equations that result from the multiplications necessary for A=LU.

Example :

Input : 

Output :

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

C++

// CPP Program to decompose a matrix into 
// lower and upper traingular matrix 
#include <bits/stdc++.h> 
using namespace std; 
  
const int MAX = 100; 
  
void luDecomposition(int mat[][MAX], int n) 
{ 
    int lower[n][n], upper[n][n]; 
    memset(lower, 0, sizeof(lower)); 
    memset(upper, 0, sizeof(upper)); 
  
    // Decomposing matrix into Upper and Lower 
    // triangular matrix 
    for (int i = 0; i < n; i++) { 
  
        // Upper Triangular 
        for (int k = i; k < n; k++) { 
  
            // Summation of L(i, j) * U(j, k) 
            int sum = 0; 
            for (int j = 0; j < i; j++) 
                sum += (lower[i][j] * upper[j][k]); 
  
            // Evaluating U(i, k) 
            upper[i][k] = mat[i][k] - sum; 
        } 
  
        // Lower Triangular 
        for (int k = i; k < n; k++) { 
            if (i == k) 
                lower[i][i] = 1; // Diagonal as 1 
            else { 
  
                // Summation of L(k, j) * U(j, i) 
                int sum = 0; 
                for (int j = 0; j < i; j++) 
                    sum += (lower[k][j] * upper[j][i]); 
  
                // Evaluating L(k, i) 
                lower[k][i] = (mat[k][i] - sum) / upper[i][i]; 
            } 
        } 
    } 
  
    // setw is for displaying nicely 
    cout << setw(6) << "      Lower Triangular" 
         << setw(32) << "Upper Triangular" << endl; 
  
    // Displaying the result : 
    for (int i = 0; i < n; i++) { 
        // Lower 
        for (int j = 0; j < n; j++) 
            cout << setw(6) << lower[i][j] << "\t";  
        cout << "\t"; 
  
        // Upper 
        for (int j = 0; j < n; j++) 
            cout << setw(6) << upper[i][j] << "\t"; 
        cout << endl; 
    } 
} 
  
// Driver code 
int main() 
{ 
    int mat[][MAX] = { { 2, -1, -2 }, 
                       { -4, 6, 3 }, 
                       { -4, -2, 8 } }; 
  
    luDecomposition(mat, 3); 
    return 0; 
} 

Java

// Java Program to decompose a matrix into 
// lower and upper traingular matrix 
class GFG  
{ 
      
static int MAX = 100; 
static String s=""; 
static void luDecomposition(int [][]mat, int n) 
{ 
    int [][]lower = new int[n][n]; 
    int [][]upper = new int[n][n]; 
  
    // Decomposing matrix into Upper and Lower 
    // triangular matrix 
    for (int i = 0; i < n; i++) 
    { 
  
        // Upper Triangular 
        for (int k = i; k < n; k++) 
        { 
  
            // Summation of L(i, j) * U(j, k) 
            int sum = 0; 
            for (int j = 0; j < i; j++) 
                sum += (lower[i][j] * upper[j][k]); 
  
            // Evaluating U(i, k) 
            upper[i][k] = mat[i][k] - sum; 
        } 
  
        // Lower Triangular 
        for (int k = i; k < n; k++)  
        { 
            if (i == k) 
                lower[i][i] = 1; // Diagonal as 1 
            else 
            { 
  
                // Summation of L(k, j) * U(j, i) 
                int sum = 0; 
                for (int j = 0; j < i; j++) 
                    sum += (lower[k][j] * upper[j][i]); 
  
                // Evaluating L(k, i) 
                lower[k][i] = (mat[k][i] - sum) / upper[i][i]; 
            } 
        } 
    } 
  
    // setw is for displaying nicely 
    System.out.println(setw(2) + "     Lower Triangular"
        + setw(10) + "Upper Triangular"); 
  
    // Displaying the result : 
    for (int i = 0; i < n; i++)  
    { 
        // Lower 
        for (int j = 0; j < n; j++) 
            System.out.print(setw(4) + lower[i][j] + "\t");  
        System.out.print("\t"); 
  
        // Upper 
        for (int j = 0; j < n; j++) 
            System.out.print(setw(4) + upper[i][j] + "\t"); 
        System.out.print("\n"); 
    } 
} 
static String setw(int noOfSpace) 
{ 
    s=""; 
    for(int i = 0;i<noOfSpace;i++) 
        s+=" "; 
    return s; 
} 
  
// Driver code 
public static void main(String arr[]) 
{ 
    int mat[][] = { { 2, -1, -2 }, 
                    { -4, 6, 3 }, 
                    { -4, -2, 8 } }; 
  
    luDecomposition(mat, 3); 
} 
} 
  
/* This code contributed by PrinciRaj1992 */

Python3

# Python3 Program to decompose  
# a matrix into lower and 
# upper traingular matrix 
MAX = 100; 
  
def luDecomposition(mat, n): 
  
    lower = [[0 for x in range(n)]  
                for y in range(n)]; 
    upper = [[0 for x in range(n)]  
                for y in range(n)]; 
                  
    # Decomposing matrix into Upper  
    # and Lower triangular matrix 
    for i in range(n): 
  
        # Upper Triangular 
        for k in range(i, n):  
  
            # Summation of L(i, j) * U(j, k) 
            sum = 0; 
            for j in range(i): 
                sum += (lower[i][j] * upper[j][k]); 
  
            # Evaluating U(i, k) 
            upper[i][k] = mat[i][k] - sum; 
  
        # Lower Triangular 
        for k in range(i, n): 
            if (i == k): 
                lower[i][i] = 1; # Diagonal as 1 
            else: 
  
                # Summation of L(k, j) * U(j, i) 
                sum = 0; 
                for j in range(i): 
                    sum += (lower[k][j] * upper[j][i]); 
  
                # Evaluating L(k, i) 
                lower[k][i] = int((mat[k][i] - sum) /
                                       upper[i][i]); 
  
    # setw is for displaying nicely 
    print("Lower Triangular\t\tUpper Triangular"); 
  
    # Displaying the result : 
    for i in range(n): 
          
        # Lower 
        for j in range(n): 
            print(lower[i][j], end = "\t");  
        print("", end = "\t"); 
  
        # Upper 
        for j in range(n): 
            print(upper[i][j], end = "\t"); 
        print(""); 
  
# Driver code 
mat = [[2, -1, -2], 
       [-4, 6, 3], 
       [-4, -2, 8]]; 
  
luDecomposition(mat, 3); 
  
# This code is contributed by mits 

C#

// C# Program to decompose a matrix into 
// lower and upper traingular matrix 
using System; 
      
class GFG  
{ 
      
static int MAX = 100; 
static String s=""; 
static void luDecomposition(int [,]mat, int n) 
{ 
    int [,]lower = new int[n, n]; 
    int [,]upper = new int[n, n]; 
  
    // Decomposing matrix into Upper and Lower 
    // triangular matrix 
    for (int i = 0; i < n; i++) 
    { 
  
        // Upper Triangular 
        for (int k = i; k < n; k++) 
        { 
  
            // Summation of L(i, j) * U(j, k) 
            int sum = 0; 
            for (int j = 0; j < i; j++) 
                sum += (lower[i,j] * upper[j,k]); 
  
            // Evaluating U(i, k) 
            upper[i, k] = mat[i, k] - sum; 
        } 
  
        // Lower Triangular 
        for (int k = i; k < n; k++)  
        { 
            if (i == k) 
                lower[i, i] = 1; // Diagonal as 1 
            else
            { 
  
                // Summation of L(k, j) * U(j, i) 
                int sum = 0; 
                for (int j = 0; j < i; j++) 
                    sum += (lower[k, j] * upper[j, i]); 
  
                // Evaluating L(k, i) 
                lower[k, i] = (mat[k, i] - sum) / upper[i, i]; 
            } 
        } 
    } 
  
    // setw is for displaying nicely 
    Console.WriteLine(setw(2) + "     Lower Triangular"
        + setw(10) + "Upper Triangular"); 
  
    // Displaying the result : 
    for (int i = 0; i < n; i++)  
    { 
        // Lower 
        for (int j = 0; j < n; j++) 
            Console.Write(setw(4) + lower[i,j] + "\t");  
        Console.Write("\t"); 
  
        // Upper 
        for (int j = 0; j < n; j++) 
            Console.Write(setw(4) + upper[i,j] + "\t"); 
        Console.Write("\n"); 
    } 
} 
static String setw(int noOfSpace) 
{ 
    s=""; 
    for(int i = 0;i<noOfSpace;i++) 
        s+=" "; 
    return s; 
} 
  
// Driver code 
public static void Main(String []arr) 
{ 
    int [,]mat = { { 2, -1, -2 }, 
                    { -4, 6, 3 }, 
                    { -4, -2, 8 } }; 
  
    luDecomposition(mat, 3); 
} 
} 
  
// This code is contributed by Princi Singh 

PHP

<?php 
// PHP Program to decompose  
// a matrix into lower and 
// upper traingular matrix 
$MAX = 100; 
  
function luDecomposition($mat, $n) 
{ 
    $lower; 
    $upper; 
    for($i = 0; $i < $n; $i++) 
    for($j = 0; $j < $n; $j++) 
    { 
        $lower[$i][$j]= 0; 
        $upper[$i][$j]= 0; 
    } 
    // Decomposing matrix  
    // into Upper and Lower 
    // triangular matrix 
    for ($i = 0; $i < $n; $i++)  
    { 
  
        // Upper Triangular 
        for ($k = $i; $k < $n; $k++)  
        { 
  
            // Summation of  
            // L(i, j) * U(j, k) 
            $sum = 0; 
            for ($j = 0; $j < $i; $j++) 
                $sum += ($lower[$i][$j] *  
                         $upper[$j][$k]); 
  
            // Evaluating U(i, k) 
            $upper[$i][$k] = $mat[$i][$k] - $sum; 
        } 
  
        // Lower Triangular 
        for ($k = $i; $k < $n; $k++)  
        { 
            if ($i == $k) 
                $lower[$i][$i] = 1; // Diagonal as 1 
            else
            { 
  
                // Summation of L(k, j) * U(j, i) 
                $sum = 0; 
                for ($j = 0; $j < $i; $j++) 
                    $sum += ($lower[$k][$j] *  
                             $upper[$j][$i]); 
  
                // Evaluating L(k, i) 
                $lower[$k][$i] = (int)(($mat[$k][$i] -  
                                $sum) / $upper[$i][$i]); 
            } 
        } 
    } 
  
    // setw is for  
    // displaying nicely 
    echo "\t\tLower Triangular"; 
    echo "\t\t\tUpper Triangular\n"; 
  
    // Displaying the result : 
    for ($i = 0; $i < $n; $i++)  
    { 
        // Lower 
        for ($j = 0; $j < $n; $j++) 
            echo "\t" . $lower[$i][$j] . "\t";  
        echo "\t"; 
  
        // Upper 
        for ($j = 0; $j < $n; $j++) 
        echo $upper[$i][$j] . "\t"; 
        echo "\n"; 
    } 
} 
  
// Driver code 
$mat = array(array(2, -1, -2), 
             array(-4, 6, 3), 
             array(-4, -2, 8)); 
  
luDecomposition($mat, 3); 
  
// This code is contributed by mits 
?> 

Output:

      Lower Triangular                Upper Triangular
     1         0         0             2        -1        -2    
    -2         1         0             0         4        -1    
    -2        -1         1             0         0         3

This article is contributed by Shubham Rana. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.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.

My Personal Notes

Suggest a Topic

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

C++

Java

Python3

C#

PHP

Recommended Posts: