1549301742-The_Theory_of_Difference_Schemes__Samarskii

652 Methods for Solving Grid Equations

where Yj and Fj are vectors of the same order Mj, Cj is a square matrix

of size Mj x M 1 , Aj is a rectangular matrix of size M1 x Mj-1 and Bj is a

rectangular matrix of size Mj x M1+1 · As usual, we may attempt a solution

of this problem in the form

( 19) Y j = oc i + 1 Yi+ 1 + /3 1 + 1 , j = N - 1 , N - 2,... , 1, 0 ,

where °'j is a rectangular Mj-J x Mj-1natrix and f3j is an Nlj_ 1 -dimensional

vector. Following established practice with Gaussian elimination, we derive

frorn ( 18)-( 19) the recurrence relations for finding oc j and 131 both:

oc j + 1 = ( Cj - A j oc i )-^1 B 1 , j = 1, 2,... , N - 1 ,

°'1= C'-1B 0 o,

Yj = °'j+l Yj+I + f3 1 + 1 , j = 1V -1,N - 2, ... , 1,0.

For a complete and rigorous treat1nent, it is required that

and, moreover, at least one of these inequaliLies should be stricL. This
provides the sufficient background for the stability of the matrix elimination
method with respect to random errors, meaning II °'j II < 1, j = 1, 2, ... , N.
In the case (3), which interests us, the members become Aj = Bj = E,
Ci = C for 1 < j < N - 1 and Bo= AN = 0, Co= CN = E, by means of
which the ensuing formulas can be written in simplified form: