GE computes an factorization of , where is unit lower triangular and is upper triangular.

** Theorem:** * There exists a unique factorization
of if and only if is
non-singular for . If is singular for some
then the factorization may exist, but if so it
is not unique.*

** Proof:** By induction (from Golub and Van Loan, 1992).

The effect of partial or complete pivoting is equivalent to multiplying
the matrix by a permuation matrix . GE with partial pivoting (GEPP)
applied to is equivalent to GE without pivoting applied to the
row-permuted matrix . This holds for all **n**: GEPP computes a
factorization .

Computing an factorization is equivalent to solving the equations:

If these non-linear equations are examined in the right order, they can be
easily solved. For added generality let
, , and consider an
factorization with and ( is lower
triangular). Suppose we know the first **k - 1** columns of and the
first **k - 1** rows of . Setting ,

We can solve for the underlined elements in the **k**th row of and
then the **k**th column of . This process is called * Doolittle's
method*. It computes the factorization, assuming that the
factorization exists.

** Cost:** flops.

Doolittle's method is mathematically equivalent to GE without pivoting, for we have,

Doolittle's method is well suited to calculations by hand or with a desk calculator, because it obviates the need to store the intermediate quantities . It is also attractive when we can accumulate inner products in extended precision. It is straightforward to incorporate partial pivoting into Doolittle's method. However, complete pivoting cannot be incorporated without changing the method.

Thu Jan 29 05:58:31 EST 1998