TY - GEN

T1 - Schemes of fast evaluation of multivariate monomials for speeding up numerical integration of equations in dynamics

AU - Alesova, I. M.

AU - Babadzanjanz, L. K.

AU - Bregman, A. M.

AU - Bregman, K. M.

AU - Pototskaya, I. Yu

AU - Pupysheva, Yu Yu

AU - Saakyan, A. T.

N1 - Funding Information: The authors acknowledge Saint-Petersburg State University for a research grant 9.37.345.2015. Publisher Copyright: © 2018 Author(s). Copyright: Copyright 2018 Elsevier B.V., All rights reserved.

PY - 2018/7/10

Y1 - 2018/7/10

N2 - Many differential equations of Dynamics (i.e. Celestial Mechanics, Molecular Dynamics, and so on) one can reduce to polynomial form, i.e. to system of differential equations 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 multivariate 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. We consider a scheme of successive multiplications minimizing the total cost of evaluation of multivariate monomials of a system of monomials and the algorithm, which for a given system of third order monomials reduces the original problem to the linear programming problem, and computes such a scheme. It implies that to speed up the process of numerical integration it is natural to minimize the total cost of evaluation of all different monomials in right-hand sides of the differential equations. It turns out that the total evaluation cost of systems of monomials is reduced substantially.

AB - Many differential equations of Dynamics (i.e. Celestial Mechanics, Molecular Dynamics, and so on) one can reduce to polynomial form, i.e. to system of differential equations 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 multivariate 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. We consider a scheme of successive multiplications minimizing the total cost of evaluation of multivariate monomials of a system of monomials and the algorithm, which for a given system of third order monomials reduces the original problem to the linear programming problem, and computes such a scheme. It implies that to speed up the process of numerical integration it is natural to minimize the total cost of evaluation of all different monomials in right-hand sides of the differential equations. It turns out that the total evaluation cost of systems of monomials is reduced substantially.

UR - http://www.scopus.com/inward/record.url?scp=85049954404&partnerID=8YFLogxK

U2 - 10.1063/1.5043752

DO - 10.1063/1.5043752

M3 - Conference contribution

AN - SCOPUS:85049954404

T3 - AIP Conference Proceedings

BT - International Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2017

A2 - Tsitouras, Charalambos

A2 - Simos, Theodore

A2 - Simos, Theodore

A2 - Simos, Theodore

A2 - Simos, Theodore

A2 - Simos, Theodore

PB - American Institute of Physics

T2 - 15th International Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2017

Y2 - 25 September 2017 through 30 September 2017

ER -