The Crout factorization is similar to the Doolittle factorization except that    instead of  .  

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 would lead rise to the following Crout subroutine.

Example 1 (b).  Find the A = LU  factorization for the following matrix using the Crout subroutine.

Solution.

The matrix has been factored it is similar to the Doolittle factorization. But this time the elements on the diagonal of  U  are .

There is more computing effort required for the Crout method than the Doolittle method.  For this reason we prefer the Doolittle method.

(c) John H. Mathews