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 :

## 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; }

## 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

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