Crout matrix decomposition

Short description: Type of matrix factorization

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In linear algebra, the Crout matrix decomposition is an LU decomposition which decomposes a matrix into a lower triangular matrix (L), an upper triangular matrix (U) and, although not always needed, a permutation matrix (P). It was developed by Prescott Durand Crout. [1]

The Crout matrix decomposition algorithm differs slightly from the Doolittle method. Doolittle's method returns a unit lower triangular matrix and an upper triangular matrix, while the Crout method returns a lower triangular matrix and a unit upper triangular matrix.

So, if a matrix decomposition of a matrix A is such that:

A = LDU

being L a unit lower triangular matrix, D a diagonal matrix and U a unit upper triangular matrix, then Doolittle's method produces

A = L(DU)

and Crout's method produces

A = (LD)U.

## Implementations

C implementation:

void crout(double const **A, double **L, double **U, int n) {
int i, j, k;
double sum = 0;

for (i = 0; i < n; i++) {
U[i][i] = 1;

for (j = 0; j < n; j++) {
for (i = j; i < n; i++) {
sum = 0;
for (k = 0; k < j; k++) {
sum = sum + L[i][k] * U[k][j];

L[i][j] = A[i][j] - sum;

for (i = j; i < n; i++) {
sum = 0;
for(k = 0; k < j; k++) {
sum = sum + L[j][k] * U[k][i];

if (L[j][j] == 0) {
printf("det(L) close to 0!\n Can't divide by 0...\n");
exit(EXIT_FAILURE);

U[j][i] = (A[j][i] - sum) / L[j][j];

Octave/Matlab implementation:

function [L, U] = LUdecompCrout(A)

[R, C] = size(A);
for i = 1:R
L(i, 1) = A(i, 1);
U(i, i) = 1;
end
for j = 2:R
U(1, j) = A(1, j) / L(1, 1);
end
for i = 2:R
for j = 2:i
L(i, j) = A(i, j) - L(i, 1:j - 1) * U(1:j - 1, j);
end

for j = i + 1:R
U(i, j) = (A(i, j) - L(i, 1:i - 1) * U(1:i - 1, j)) / L(i, i);
end
end
end

## References

1. ↑ Press, William H. (2007). Numerical Recipes 3rd Edition: The Art of Scientific Computing. Cambridge University Press. pp. 50–52. ISBN 9780521880688.

* Implementation using functions  In Matlab

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