12 Prelhninariesas well as ID'i+1I < 1 are ensured by ICl'il < 1. The assumption 10' 11 = lx 11 <
1 implies ICl'il < 1 for all i = 1, 2, ... , n. The lower estimate holds true for
the denominator in formula (15):
I l - Cl' N x2 I > 1 - I Ct N I · I x2 I > 1 - I x2 I > 0 ,
because either lx 2 I < 1 or ICl'NI < 1.
Thus, we have proved that under conditions (16) the right elimination
algorithm is correct, tneaning nonzero denominators in formulae (11), (12)
and (15). So, under conditions (16) problem (9) has a unique solution given
by formulae (10)-(15).
It is necessary to point out that calculations by these formulae may
induce accumulation of rounding errors arising in arithmetic operations. As
': .. res2:1lt :::e actually solve the same E_roblem but with perturbed coefficients
Ai, Bi, Ci, x 1 , x 2 and right parts Fi, 'jj 1 , 'jj 2. If N is sufficiently large, the
growth of rounding errors may cause large deviations of the computational
solution fj; from the proper solution Yi.
The trivial example shows how instability may arise in the process of
calculations of Yi by the formula Yi+l = q Yi, q > 1. One expects that, for
any Yo, there exists a nu1nber n 0 such that overflow occur for Yn = qn Yo, n =
n 0 , thus causing a abnormal tennination. An i1nportant obstacle in dealing
with this problem is that Yi satisfies the equation Yi+l = q Yi + 77 with a
rounding error 77. Indeed, for the error fJ Yi = Yi - Yi the equation is valid:
fJ Yi+l = q fJ Yi + 17, fJ Yo = 17,
from which it follows that the value. qi - 1
fJ Yi = q' 77 + q - 1 77 , q > 1 ,
increases exponentially along with increasing i.
Returning to the right elimination method, we show that the condi-
tions ICl'il < 1 guarantee that the error fJ Yi+l = Yi+l - Yi+l arising when
computing Yi does not increase. Indeed, the equations
Yi = Cl'i+l Yi+l + /Ji+l , Yi = Cl'i+l Yi+l + /Ji+l
imply that
(j Yi = Cl'i+l (j Yi+l ) lfJ Yi I < 1Cl'i+1 I. lfJ Yi+1 I ,
meaning lfJ Yi I < lfJ Yi+1 I because ID'i+1 I < l.
Taking into account perturbations of the coefficients Cl'i+l and /Ji+l,
another conclusion can be drawn that the accuracy E in determination of
the solution Yi of problem (9) is
max lfJ Yil ~ E 0 N^2 ,
l<i<N
where E 0 is a rounding error and N is the total number of grid nodes.