1549301742-The_Theory_of_Difference_Schemes__Samarskii

Other iterative 1nethods 743

where w is the exact solution to the equation Rw

resolving operator subject to the relations

rk and T,n is the

( 46) T,;, = T,n'

Here the dependence q upon 1n is 0111ittecl. This should cause no confusion.

vVith the initial value w(O) = 0 in view, we find that

Upon substituting this expression into the equation Rw = rk we obtain the

equation Bwk = rk with the 111embers

·w k = ,,,(m) ~ ,

thereby completing the task of searching for the correction. By the same

token, Yk+l = Yk - Tk+l wk.

Still using the framework of the general theory, it is straightforward

to verify that the operator B is self-adjoint ( B = B*) with the aid of the

relations R = R* and r;, = T,,,. We shall need yet the constants fl and f 2

of the energetic equivalence of the operators Band A. Because of (46), we

thus have

(1 - q) E < E - T,n < (1 + q) E,

where y = R^112 x. From such reasoning it see1ns clear that

(1 - q) B < R < (1 + q) E,
0 0
g1vmg f 1 1 - q and f 2 = 1 + q and implying that fl = c 1 (1-q) and
f 2 = C1 ( l + q).
For Chebyshev's scheme with such an operator we obtain

(^1) c2(1+q)^2
n.o(f) = (^2) c In -
1 (^1 - q)
,
f
since
C-'-, - C1 (1 - q)
c., (1 + q)