Vl Prefacewhen only a priori estimates of the algorithm quality are available. The
same applies equally well to the total nmnber of logic operations regardless
of how computers will be chosen. Therefore, in theoretical investigations it
is fairly common to compare computer algorithms by means of the number
of arithmetic operations Q 0 (c). An economical algorithm having the mini-
mal number of operations is sought in the class of possible algorithms (for
instance, with the sa1ne asymptotics for Qo(c) as E -+ 0). This is one of
the main proble1ns in the theory of numerical 1nethods.
At the initial stage of such an analysis of algorithn1s it is usually sup-
posed that a computing process is ideal, that is, computations are carried
out with an infinite number of significant digits. But any computer nrnkes
calculations with a finite speed and a finite munber of digits. Not all num-
bers are accessible to computers; there are computer null and computer
infinity. For instance, abnormal tennination occurs when computer infinity
arises during the course of execution. A cotnputing proces,; 1nay become
unstable, thus causing difficulties. In such cases rounding errors may ac-
cumulate to a considerable extent so that the algorith1n will be useless in
practical applications (several examples of unstable algorithms are avail-
able in Chapter 2). A real, that is, a proper computer algorithm should
be stable and should not allow excessively large intermediate values lead-
ing to abnormal termination. From such reasoning it seems clear that an
ideal computer algorithm 1nay be optimal with respect to the number Q 0 (c)
and quite unfit for computers. These requirements necessitate seeking an
opti1nal real c01nputer algorith1n.
Any numerical experiment is not a one-time calculation by standard
formulas. First and foremost, it is the c01nputation of a nun1ber of possi-
bilities for various mathe1natical models. For instance, it is required to find
the opti1nal conditions for a che1nical process, that is, the conditions under
which the reaction is completed inost rapidly. A solution of this problem
depends on a number of parameters (for instance, te1nperature, pressure,
co1nposi ti on of the reacting mixture, etc.). In order to find the op tin1al
workable conditions, it is necessary to carry out computations for different
values of those parameters, thereby exhausting all possibilities. Of course,
son1e situations exist in which an algorithm is to be used only several ti1nes
or even once.
The statement of the proble1n of finding an optimal algorithm depends
on how it is to be applied (for individual variants or a great nmnber of
variants).
Here various issues surrounding progran1n1ing, organization and re-
alization of computing are of little interest and will be excluded fro1n
further consideration. It is worth noting, however, that progra1n1ning