"COMPUTER ALGEBRA AT KELDYSH INSTITUTE"
G.B.Efimov, I.B.Tshenkov, E.Yu.Zueva
Keldysh Institute of Applied Mathematics RAS
Moskow, 125047, Miusskaya sq.4, Russia
efimov@applmat.msk.su
Keldysh Institute of Applied Mathematics of Russian Academia of
Science (RAS) was founded by M.V.Keldysh and A.N.Tichonov for
solving difficult scientific problems of national importance,
such as nuclear physics, cybernetics, space mechanics and others.
At the Institute experts on different areas - mathematicians,
physicists, mechanicians, computer scientists - were working
together, in close contacts with each others. Such famous
scientists as K.I.Babenko, I.M.Gelfand, A.A.Liapunov, I.B.Zeldo-
vich, D.E.Okhozimsky, A.A.Samarsky, V.S.Yablonsky, M.R.Shura-
Bura, A.N.Miamlin, S.P.Kurdiumov, T.M.Eneev were among them.
Idea of Computer Algebra became attractive for experts on Applied
Mathematics due to existence of numerous difficult problems to be
solved with it, as well as due to enthusiasm caused by success
of early computers. First experiments were made as early as the
beginning of sixties. Manipulations with trigonometric and power
series were implemented with soviet computer "Strela" by Z.Vlasova
and I.Zadyhaylo (non-published). In 1964 D.E.Okhozimsky proposed
to built the solution of cosmodynamics problem in the form of two
asymptotic power series near two singularities. These
assimptotics were conjugated by common numerical solution in
the regular area. G.Efimov realized this approach for simplest
Poisson series (1970).
>From 1963 M.L.Lidov with his group did numerous experiments
concerned CA application to sputnik dynamics problems. Method
united both analytical and numerical approaches was proposed.
For elliptic orbits and distortions of different sorts,
analytical approach was used for Hamilton disturbing function
H* building. Then, coordinate transformation and calculation
of right parts of disturbed motion equations, in every step
of integration, is done via Hamilton function
differentiation. This approach provides high accuracy method
of motion calculation and allows to avoid labor-consuming
calculations. Unfortunately, requirements to CA systems to be
used in this scheme were rather high, and available CA
systems were not capable to satisfy them. Thus, these very
interesting experiments didn't produce practically usable
integrated (analytically-numerical) system.
>From 1970 A.P.Markeev used CA for Gamilton's systems normalization
and periodic solution stability analysis. The next steps in this
direction were done by A.G.Sokolsky. This work was later continued
in MAI and ITA RAS by the same scientific school.
V.A.Saryshev and S.A.Gutnik used CA for the problem of sputnik
equilibrium stability (1984).
G.B.Efimov created a system for derivation the equations of
motion and used it in Robotics applications.
V.F.Tourchin created an original computer language REFAL (1969)
based on new principle of programming - associative text
processing, without directly addressed control. Computer
Algebra was among potential areas of REFAL applications.
However, first REFAL realization was rather "scientific" then
practical, since it was isolated and not compatible with
"ordinary" soft - numerical packages, library support,
memory allocation and so on. Additional efforts of many
people required to do REFAL modifications practically
usable, in particular for Computer Algebra Applications. There
were REFAL-LISP comparison by L.K.Eisymont, the project by
I.B.Zadyhailo and A.N.Miamlin of specialized REFAL-computer
creation with REFAL approach realized on hardware level, and
others. I.B.Tshenkov and M.Yu. Shashkov developed specialized
Computer Algebra REFAL system (DISLAN) used for operators
differential scheme creation.
The result of the last work was positively evaluated by
A.A.Samarsky, leader in soviet mathematical modeling.
Samarsky supported organization of Computer Algebra
Conference in Gorky (Nijniy Novgorod) in 1984. This Conference
was the first meeting of Computer Algebra experts from all Soviet
Union. The results of about 20 years research, as well as future
plans and perspective directions, were discussed. Keldysh
Institute presented a number of papers devoted to CA systems,
mechanical applications, and two reviews on CA applications - on
Applied Mathematics and Mechanics.
To generalize the experience of common work of mathematicians,
mechanicians and programmers, some classification work was done
by G.B.Efimov and M.V.Grosheva. There were reviews of CA systems
and CA applications for mechanical problems. Tables of CA systems
features were presented for users. These reviews provided
convenient tool for CA systems comparison and selection for
potential users - experts in applied areas.
Last years, "authorized" systems created especially for
solving some concrete problems and used mostly by authors
themselves, became out of dated. Instead, general purpose
systems of common usage such as MATHEMATIKA, MAPLE, REDUCE and
programs written for them became popular.
Most important works of last years at Keldysh Institute are the
following. Several Hydrodynamics problems were resolved by I.B.
Tshenkov and Ya.M. Kajdan with aid of REFAL-based CA program and
by M.Yu.Shashkov and L.Platonova in REDUCE. A.D.Bruno and his
followers obtained interesting results. In particular,
V.F.Edneral realized some algorithms of normalization in Hamilton
systems. S.Yu. Sadov with Vahedov investigated stability of
motion for celestial mechanics problems.
The work was carried out under support of RFFI grant N 96-15
-97229, N 98-01-00941.