TY - GEN

T1 - Fast evaluation of multivariate monomials for speeding up numerical integration in space dynamics

AU - Babadzanjanz , L.K.

AU - Pototskaya , I.Y.

AU - Pupysheva , Y.Y.

AU - Saakyan , A.T.

N1 - Conference code: 19

PY - 2019

Y1 - 2019

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. 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 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. Then we proposed the algorithm and the corresponding Mathematica program that, given an arbitrary system of multivariate cubic monomials constructs the linear programming problem mentioned. We have also presented the results of corresponding numerical experiments and have shown that the total evaluation cost of systems of monomials has reduced substantially. 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 solutions 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 effective 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. Moreover, 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 differential equations of the N-body problem with additional fifth order polynomial perturbations).

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. 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 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. Then we proposed the algorithm and the corresponding Mathematica program that, given an arbitrary system of multivariate cubic monomials constructs the linear programming problem mentioned. We have also presented the results of corresponding numerical experiments and have shown that the total evaluation cost of systems of monomials has reduced substantially. 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 solutions 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 effective 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. Moreover, 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 differential equations of the N-body problem with additional fifth order polynomial perturbations).

KW - Dynamics

KW - Multivariate monomials

KW - Numerical integration

KW - Taylor series method

UR - http://www.scopus.com/inward/record.url?scp=85073364062&partnerID=8YFLogxK

U2 - 10.5593/sgem2019/6.2/S28.082

DO - 10.5593/sgem2019/6.2/S28.082

M3 - Conference contribution

T3 - International Multidisciplinary Scientific GeoConference-SGEM

SP - 647

EP - 654

BT - 19th International Multidisciplinary Scientific Geoconference, SGEM 2019; Albena; Bulgaria; 30 June 2019 до 6 July 2019

T2 - 19th International Multidisciplinary Scientific Geoconference, SGEM 2019

Y2 - 9 December 2019 through 11 December 2019

ER -