Two-layer iteration sche1nes 667h SIS SCP1/10^200 32
1/50^5000 160
1/100^20000 320Fron1 here it seems clear that much more iterations are needed in SIS than
in SCP which is, generally speaking, preferable.
However, some progress may be achieved in reducing the total number
of operations by making the well-founded choice of the initial approxin1a-
tion.- On computational stability of iterative inethods. Until recent years the
iterative method with optimal set of Chebyshev's parameters was of little
use in numerical solution of grid equations. This can be explained by real
facts that various sequences turn out to be nonequivalent in computational
procedures.
At the initial stage such of such an analysis of algorithms it is usu-
ally supposed that a con1puting process is ideal, that is, computations are
carried out with an infinite nun1ber of significant digits. But any computer
inakes calculations with a finite speed and a finite number of digits. Not all
numbers are accessible to computers, there are computer null and computer
infinity. For instance, abnormal termination occurs when computer infinity
arises during the course of execution. A computing process may become
unstable, thus causing difficulties. In such cases rounding errors may ac-
cumulate to a considerable extent so that the a.lgorithrn will be useless in
practical applications
For example, for doing so with the set of Chebyshev's parameters tk
in increasing order
( 41)2k -1
tk =cos k = 1,2, ... ,n,
2n7f ,or in reverse order
( 42)2k - 1
tk = - cos k=l,2, ... ,n,
2n7f ,abnormal terminations may occur for sufficiently small~ in connection with
growing intermediate values Yk for k < n. Such a danger is caused by a
nonmonotone character of the approximation Yk t.o u, since the norm of the