9-TH WORKSHOP ON COMPUTER ALGEBRA

DUBNA 2005

May 17-18, 2005, Dubna

The 9-th two-day workshop on Computer Algebra will be held on May 17-18, 2005. This worshop is a traditional joint meeting of Moscow computer algebra seminar with the seminar on computational and applied mathematics of the Laboratory of Information Technologies of the Joint Institute for Nuclear Research. It is the continuation of a series of workshops which started in 1997 by Compuer Science Faculty of MSU, Skobeltsyn Institute of Nuclear Physics of MSU and Joint Institute for Nuclear Research. The workshop is conceived with the purpose of presenting topics of current interest and providing a stimulating environment for scientific discussion on new developments in computer algebra.

The participation will take place BY REQUESTS only. Participants have to fill the following forms (including abstracts and personal data) in Russian and English (see bellow). The requests should be sent to Alla Bogolubskaya (abogol at jinr.ru), NOT LATER than May 11.

The duration of reports is 20 mins including discussion. Invited 30 mins reports will take place also. Participants who plan to visit Dubna for one day only should inform Alla Bogolubskaya about preferable date of the visit.

Current hotel prices now are 745 rub/night - a single room.

The transport timetable is at the site: http://www.dubna.ru/transport.php

Most interesting reports will be published (after a review treatment) in "Programmirovanie" Journal. The full version of papers (better in Russian) should be passed to Sergei Abramov in during of the seminar (NOT LATER!) in printed form (as a paper) AND as a file in PostScript or PDF format.

First Name: ______________________

Second Name: _____________________

Family Name: _____________________

Affiliation: ____________________

Position: ________________________

Abstract (including the title and a list of authors but without a list of references).

LIST OF ABSTRACTS

 Symbolic Solutions of Inhomogeneous Linear Ordinary Differential Equations Anna Ryabenko (Dorodnicyn Computing Centre RAS) For a given l.o.d.e. Ly(x)=f(x) with polynomial coefficients and a rational right hand side f(x), we determine a set of points x=a such that there can be a power series solution y(x)=\sum_{n=0}^\infty c_n (x-a)^n with hypergeometric coefficients, i.e. c_{n+1}/c_n is a rational in n for large enought n. Inverse Problem for Discrete Elliptic Equation with Prescribed Symmetry Conditions S.I.Serdyukova (JINR, Dubna) We developed a numerical and analytical algorithm for reconstructing the two-dimensional discrete elliptic equation of a part of spectrum and prescribed symmetry conditions. The right problem is solved in rectangle M \times N with zero-boundary conditions. If given symmetry conditions are satisfied the eigenfunctions can be prolonged from the rectangle on the whole plain with reserving continuity of the first derivatives. The problem is reduced to reconstruction of a symmetric five-diagonal matrix. We proved that when prescribed symmetry conditions are satisfied the considered block three-diagonal matrix and all its blocks are persymmery. Such matrix has L1$the possibilities are even richer: for any$s,t\in \zs_+ \cup \{\infty\}$there exists an$H$-system with$\dim V_1 =s$and$\dim V_2=t$. Additionally we revisit the Sato-Ore theorem and show that, contrary to some interpretations in the literature, this theorem does not imply that any solution of an$H$-system is of the form $$R(n_1,\dots ,n_d) \frac{\prod_{i=1}^p \Gamma(a_{i1} n_1 +\dots + a_{id}n_d+ \alpha_i)} {\prod_{j=1}^q \Gamma(b_{j1} n_1 +\dots + b_{jd}n_d+ \beta_j)} \,u_1^{n_1}\cdots u_d^{n_d},$$ where$R\in \cs(x_1,\dots ,x_d)$,$a_{ik},b_{jk} \in \zs$, and$\alpha_i,\beta _j \in \cs$. We give an appropriate formulation of the Ore-Sato theorem on possible form of solutions of$H$-systems. Gosper's Algorithm, Accurate Summation, and Discrete Newton-Leibniz Formula S.A. Abramov (Dorodnicyn Computing Centre RAS), M. Petkovsek (University of Ljubljana, Slovenia) Sufficient conditions are given for validity of the discrete Newton-Leibniz formula when the indefinite sum is obtained either by Gosper's algorithm or by Accurate Summation algorithm. It is shown that sometimes a polynomial can be factored from the summand in such a way that the safe summation range is increased. Applications of BK Method of LPDOs Factorization Shemyakova Katya (1.RISC, Austria; 2. MSU, Dept. of Mechanics and Mathematics) The problem of factorization of LPDOs is well-known and a lot of deep results are already obtained (Tsarev and its bib). However, mostly the results were not constructive and play the role of pure existence theorems. The last year the first algorithm for factorization of LPDO of arbitrary order is presented (Grigoriev, Schwarz). Immediately BK-algorithm appeared which is based on the ideas of the first one but in another approach. In resent paper we study problems related with the factorization of LPDO such that the number of factorizations and the commutativity of factors, capabilities of the method itself. There is also the first version of MAPLE-based implementation of BK-algorithm. The code is not presented but all the results starting with Sect.3 up to the end are obtained partially with its help. Optimal Algorithm to Compute Essential Values of Hypergeometric Type Functions Alexey Bogolubsky (MSU, Dept. of Mechanics and Mathematics), Sergey Skorokhodov (Dorodnicyn Computing Centre RAS) The new method is proposed to compute the values of big set of functions at their singular points. Method is based on using of complete asymptotical expansions by Hurwitz functions \zeta(s, \alpha) for efficient summation of Dirichlet-type series. Application of the method for computation of practically important function p_F_p-1 ([a]; [b]; 1) is being considered in detail; the comparison with existing methods and their program realizations has been carried out. One of the real challenges is to make efficient choice of \alpha parameter (which is influencing convergence speed greatly); to do it, one must solve strongly nonlinear optimization problem. Computer analytical calculations play key role on both stages of developing and usage of the method. Besides, stochastic approach is also applied efficiently: genetic alghorithm is used to solve one of the subproblems. The whole of the methods developed allows to achieve significant perfomance increase in comparison with traditional techniques. Analysis of the Solution Component Stability of Linear Differential or Recurrent Equations System with Maple G.V. Plotnikov (Dept. of Computational Mathematics and Cybernetics, MSU) The complex of the MAPLE-procedures based on Abramov-Bronstein's algorithm is described. This complex allows to consider components of the solution of the linear differential or recurrent equations system (to examine only a part of unknowns). Application of this algorithm in stability of solution components is also analyzed. Cases of constant and variable coefficients are examined. Experimental Study of Efficiency of Involutive Criteria V.P. Gerdt, D.A. Yanovich (JINR, Dubna) In this paper we present results of computer experiments to study effectiveness of involutive criteria for avoiding some useless prolongations in construction of polynomial Janet bases. These bases are typical representative of involutive bases. Our experimental study shows that the role of criteria in the involutive approach is definitely weaker than that in Buchberger's algorithm. Symbolic Algorithm for Decomposition of Evolution Operator for Time-Dependent Schroedinger Equation V.P. Gerdt, A.A. Gusev, V.A. Rostovtsev, T.V. Tupikova, S.I. Vinitsky (JINR, Dubna), Y. Uwano (Hakodate University, Japan) An symbolic algorithm based on unitary evolution operator decomposition to generate a multi-layer implicit schemes for solving the time dependent Schroedinger equation is elaborated. In framework of the algorithm a gauge transformation for extraction of symmetric operators for construction chonomical evolution schemes is realised also. The efficiency of the generated schemes is demonstrated on some integrable models. CONSTRUCTIVITY OF INVOLUTIVE DIVISIONS Alexander Semenov (MSU, Dept. of Mechanics and Mathematics) Constructivity of involutive divisions plays a significant role in the theory of involutive bases, as this property is sufficient for existence of the minimal involutive basis of an arbitrary polynomial ideal The contribution contains new examples of constructive and non-constructive pairwise involutive divisions. Analysis Methods of Multiple Eigenvalues of Spheroidal Wave Functions Skorokhodov S.L. (Dorodnicyn Computing Centre RAS), Khristoforov D.V. (MSU, Dept. of Mechanics and Mathematics) Spheroidal wave functions are eigenfunctions of special differential operator of second order. They play an important role in many applied problems of diffraction, astrophysics, etc. When the spheroid slenderness parameter$c$is real or pure imaginary, the numerable set of different real eigenvalues$\lambda_n(c)$corresponds to different spheroidal functions. Otherwise, when the differential operator is not self-adjoint, the values of$\lambda_n(c)$can coincide. This case originates in the complex plane$c$the singular points$c_b$--- square-root branch points of eigenvalues. Several modern methods are proposed for the structure of these singularities analysis. By means of effective symbolic evaluations the explicit power series expansions$\lambda_n=\lambda_n(c)$were obtained and the character of singularities on the boundary of convergence disk was studied. The strong cancellation of digits appearing during$c_b$evalution was overcome. Application of Pad\`e approximants and its generalization --- Hermite-Pade approximants -- allowed to compute branch points$c_b$and eigenvalues$\lambda_n(c_b)\$ with high precision and computation speed. Discrete Relations On Abstract Simplicial Complexes V. V. Kornyak (JINR, Dubna) A system of discrete relations is, on the one hand, generalization of a cellular automaton, or, on the other hand, a set-theoretic analog of a system of polynomial equations. Methods of investigation of systems of discrete relations based on set-theoretic and topological constructions are considered. The proposed approach is implemented as a C program. The results of application of the approach to some binary cellular automata are presented. In the binary case a cellular automaton can be represented as a system of polynomial relations. The standard tool for investigation of polynomial relations is the Groebner basis method. We compare our approach with this method. Symbolic-numeric program for solving two-dimensional Schroedinger equation I.N.Belyaeva, N.A. Chekanov (Belgorod State University, Belgorod), A.A. Gusev, V.A. Rostovtsev (JINR, Dubna), Yu.A.Ukolov (Belgorod State University, Belgorod), Y. Uwano (Hakodate University, Hakodate, Japan) and S.I. Vinitsky (JINR, Dubna) The symbolic-numeric program for solving of the two-dimensional Schroedinger equation. The rresponding algorithm of this program using a conventional pseudocode is described too. As example, the energy spectrum and wave functions for generalized Henon-Heiles Hamiltonian were obtained. Weak-admissible Monomial Orderings and the Universal Groebner Base N.N.Vasiliev (PDMI RAS, St.Petersburg ) We consider non admissible monomial orderings which are only adjusted to usual division of monomials. We suggest some ways to define Groebner bases for such kind of orderings. We prove also the finiteness of universal base of an ideal defined for all weak admissible orderings.

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