What is Crout matrix decomposition?

In linear algebra, the Crout matrix decomposition is an LU decomposition which decomposes a matrix into a lower triangular matrix , an upper triangular matrix and, although not always needed, a permutation matrix. It was developed by Prescott Durand Crout.

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:

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