Numeric Deduction in Symbolic Computation. Application to Normalizing Transformations

TL;DR

This paper develops and implements numeric deduction algorithms for symbolic computation, enabling efficient normalization of Hamiltonian systems with significant memory and computational savings, demonstrated on satellite precession models.

Contribution

It introduces a numeric deduction technique integrated into a computer algebra system for normalization of complex Hamiltonian systems, achieving substantial memory and speed improvements.

Findings

Achieved about 30-fold reduction in memory usage.

Successfully derived a resonant normal form for satellite precession.

Technique is naturally parallelizable, enhancing computational efficiency.

Abstract

Algorithms of numeric (in exact arithmetic) deduction of analytical expressions, proposed and described by Shevchenko and Vasiliev (1993), are developed and implemented in a computer algebra code. This code is built as a superstructure for the computer algebra package by Shevchenko and Sokolsky (1993a) for normalization of Hamiltonian systems of ordinary differential equations, in order that high complexity problems of normalization could be solved. As an example, a resonant normal form of a Hamiltonian describing the hyperboloidal precession of a dynamically symmetric satellite is derived by means of the numeric deduction technique. The technique provides a considerable economy, about 30 times in this particular application, in computer memory consumption. It is naturally parallelizable. Thus the economy of memory consumption is convertible into a gain in computation speed.