Derivation of the Doolittle Factorization.

Consider the matrix product  A = LU where L is lower triangular and U is upper triangular.
Use the rule for finding the element to successively compute the entries in U and L.

Compute the first row of  U  and the first column of  L.

Compute the second row of  U  and the second column of  L.

Compute the third row of  U  and the third column of  L.

Compute the k-th row of  U  and the k-th column of  L.

The above derivation gives rise to the Doolittle subroutine. 

Mathematica Subroutine (Doolittle).

(c) John H. Mathews 2003