"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.