1549301742-The_Theory_of_Difference_Schemes__Samarskii

Direct 1nethocls^649

showing the new vectors to be sensible ones. Summarizing, the numerical

solution of problem (2) can be done using only two operation: inversion of

the operators c(k-J) and summation of relevant vectors in the process of

calculation of the right-hand sides of these equations. Thus, the computa-

tional procedures include the following steps:

• specifications of the initial values p)^0 l and q)^0 l so that

qJ (O) = F. J , PJ (O) = Q. , J = 1, 2, ... , N 2 - l;

• for all k = 1, '2, ... , n - 1 solution of (,h(c equations

and calculations of the vectors PJk) and q)kl by the recurrence for-

nrnlas

PJ CkJ _ - Pj Ck-lJ +^5 ck-1J j ,

for all j = 2k, 2 · 2k, 3 · 2k, ... , N 2 - 2k;

• the solution of the equations

Yn=Fo, YN 2 =FN 2 ,

for deterrnination of the unknown vectors by the formula

__ <.k-1) + sC.k-lJ

Yi P1 J

f or · a^11 J. -- 2k-l ,^3 · 2k-l ,^5 · 2k-l , ... , N 2 - 2k-l , k -- n, n -^1 , ... ,.^1

During the course of the decomposition method the users will perform
Q = O(N 1 N 2 log 2 N 2 ) arithmetic operations with the extra storage about
1.5N, where N is the total number of the unknowns. Some modification of
the preceding algorithm with insignificant prolongations in time may be of
assistance in mastering the last difficulty involved.