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.
Terms of U matrix are given by:
And the terms for L matrix:
Example :
Input :
Output :
Implementation:
C++
// C++ Program to decompose a matrix into// lower and upper triangular 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 codeint 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 triangular matrixclass 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 triangular matrixMAX = 100def 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 codemat = [[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 triangular matrixusing 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 triangular 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?> |
Javascript
<script>// Javascript Program to decompose a matrix// into lower and upper triangular matrix // function MAX = 100;var s = "";function luDecomposition(mat, n){ var lower = Array(n).fill(0).map( x => Array(n).fill(0)); var upper = Array(n).fill(0).map( x => Array(n).fill(0)); // Decomposing matrix into Upper and // Lower triangular matrix for(var i = 0; i < n; i++) { // Upper Triangular for(var k = i; k < n; k++) { // Summation of L(i, j) * U(j, k) var sum = 0; for(var 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(var k = i; k < n; k++) { if (i == k) // Diagonal as 1 lower[i][i] = 1; else { // Summation of L(k, j) * U(j, i) var sum = 0; for(var j = 0; j < i; j++) sum += (lower[k][j] * upper[j][i]); // Evaluating L(k, i) lower[k][i] = parseInt((mat[k][i] - sum) / upper[i][i]); } } } // Setw is for displaying nicely document.write(setw(2) + "Lower Triangular" + setw(10) + "Upper Triangular<br>"); // Displaying the result : for(var i = 0; i < n; i++) { // Lower for(var j = 0; j < n; j++) document.write(setw(4) + lower[i][j] + "\t"); document.write(setw(10)); // Upper for(var j = 0; j < n; j++) document.write(setw(4) + upper[i][j] + "\t"); document.write("<br>"); }}function setw(noOfSpace) { var s = ""; for(i = 0; i < noOfSpace; i++) s += " "; return s;}// Driver codevar mat = [ [ 2, -1, -2 ], [ -4, 6, 3 ], [ -4, -2, 8 ] ];luDecomposition(mat, 3);// This code is contributed by Amit Katiyar </script> |
Output
Lower Triangular Upper Triangular
1 0 0 2 -1 -2
-2 1 0 0 4 -1
-2 -1 1 0 0 3
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