1549301742-The_Theory_of_Difference_Schemes__Samarskii

Direct n1ethods^647

Having recovered from the systern (6) the vectors Yi with even nutnbers,

we succeed in finding these with odd numbers fron1 the following equations:

ri(O)y.-p(OJ lJ 1 - j + y. 1+1 + y. 1-1, j = 1, 3, 5, ... , N2 - 1.

The same procedure works albeit with obvious modifications for the vectors

Y1 subject to the system (6), whose subscripts j a.re oddly (but unevently)

even, etc. As a final resull. we obtain a. proper system of equations with

regard to a.ll the unknowns:

(7)

J-.^2 k-1 , 3.^2 k-1 , 5.^2 k-1 ,. .. , N 2 ^2 k-1 ,

k = n, n - 1, ... , 2, 1,

Yo= Fa,

where c(kl a.ncl FJk) niust satisfy the recurrence formulas

(8) k=l,2, ... ,n-1, C(O) = C ,

pCkJ = p(k-1) + c<k-lJpU-1) + F(k-1)

J j-2k-l , 1 1+::»·-11

}. = 2k 1 2 ' 2k 1 3 ' 2k ,. .. 1 Jv2 " - qk L 1 k = 1,2,. .. ,n-1.

The dec0111position algorithm necessitates perfonning the fa.ctoriza.-

tion of the opera.tor C(k) of the special structure

zk

C(k)= IT(C-μ1E),

(2l-1)7f

(9) μz = 2 cos 2k+1 ,

l=l

ma.king it possible to reduce the further inversion of the opera.tor CU) to

successive inversion of difference opera.tors by the elimination method. In

what follows one sin1ple equation

serves t.o rnotiva.te what. is done. \,Yit.h representation (9) in viPw, the

sought function v = v(^2 ' J \Nill a.ppea.r as the ont.conw of successivP solution

of three-point difference equations

(C-μ 1 E)v(1l=<p, (C-μ 1 E)v(ll=v(l-ll, l=2,3, ... ,2k,