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FAST EVALUATION OF MULTIVARIATE MONOMIALS FOR SPEEDING UP NUMERICAL INTEGRATION IN SPACE DYNAMICS

L. K. Babadzanjanz

Abstract

Many differential equations of Dynamics (i.e. Celestial Mechanics, Molecular Dynam-ics, and so on) one can reduce to polynomial form, i.e. to system of differential equa-tions with polynomial (in unknowns) right-hand sides. It implies that at every step of numerical integration of these equations, one needs to evaluate many different multivar-iate monomials for many values of variables, and that is why minimizing the evaluation cost of a system of monomials in the right-hand sides is an important problem. In Alesova, Babadzanjanz, et al., ?Schemes of Fast Evaluation of Multivariate Monomials for Speeding up Numerical Integration of Equations in Dynamics? (AIP, volume 1978, issue 1, 2018) we considered a scheme of successive multiplications minimizing the to-tal cost of evaluation of multivariate monomials of a system of monomials and the algo-rithm, which for a given system of third order monomials reduces the original problem to the linear programming problem, and computes such a scheme. Then we proposed the algorithm and the corresponding Mathematica program that, given an arbitrary sys-tem of multivariate cubic monomials constructs the linear programming problem men-tioned. We have also presented the results of corresponding numerical experiments and have shown that the total evaluation cost of systems of monomials has reduced substan-tially. An important requirement for the process of solving differential equations in Dynamics is high accuracy at large time intervals. One of effective tools for obtaining such solu-tions is the Taylor series method. In Alesova, Babadzanjanz, et al., ?High-Precision Numerical Integration of Equations in Dynamics? (AIP, volume 1959, issue 1, 2018) we considered the equations of the N-body problem in various polynomial forms (with and without additional third order polynomial perturbations). This allowed us to obtain ef-fective algorithms for finding the Taylor coefficients, a priori error estimates at each step of integration, and an optimal choice of the order of the approximation used. More-over, we considered a number of corresponding numerical experiments, which showed the effectiveness of the Taylor series method implementation presented. In present work, we generalize the results mentioned above on the case of systems of multivariate fifth order monomials (using in corresponding numerical experiments dif-ferential equations of the N-body problem with additional fifth order polynomial pertur-bations).

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