Derivation and example of the PreCholesky Factorization.

This method is illustrated only to show that the Crout method can be modified to obtain computed values similar to the Cholesky factorization.
A more efficient method is the Cholesky subroutine for matrices that are real, symmetric and positive definite.

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 leads to the following PreCholesky subroutine.

Implementation of the PreCholesky subroutine will produce an LU factorization where .

Exercise 1 (c).  Find the A = LU  factorization for the matrix    using the PreCholesky subroutine.

Solution.

But what is the purpose of a having this third method ?

    In the special case that  A  is known to be real, symmetric and positive definite there is an advantage to the Cholesky method.  If  A  is real, symmetric and positive definite then  U is the transpose of  L  and both do not need to be computed. Hence there will be a tremendous saving of computing effort if this property is considered.  BUT the Cholesky can only be used if  A  is real, symmetric and positive definite.

(c) John H. Mathews 2003