1549301742-The_Theory_of_Difference_Schemes__Samarskii

Some variants of the elimination method 39

3. Factorization method. At the final stage other ideas are connected with
   the boundary-value problem

Lyk = Ak Yk-1 - Ck Yk +Bk Yk+l = -Fk, k = 1, 2, ... , N - 1,

(82)

By introducing the transition operator T: T Yk = Yk+ 1 and the iden-

tity operator E: E Yk = Yk and obvious rearranging of the left-hand side of

(82) as a product

where

we find during the course of the elimination of bk = Ck - Ak O'.k that

(83) k = 1, 2, ... , N - 1.

The factorized equation

can be solved using the following algorithm:

• recovery of the function /3k from the equation

or

(84) Ak /3k + Fk

Ck - Ak O'.k '

k = 1, 2, ... , N - 1;

• determination of Yk by the right elimination method. The outcome of
  this procedure is

(85) k=O,l, ... ,N-l,