Solution of Algebraic System(LU-Factorization)

We assume that A is non-singular and can be factored into the product

where

It follows that and must satisfy

(10.6)

.

The given system can now be replaced by an equivalent pair of systems

(10.7)

But since L and U are triangular, the solution of these systems are easily obtained

and

The factorization procedure can fail if some vanishes. This difficulty can be avoided by means of the following: