Symbolic-numeric interface: A review

A survey of the use of a combination of symbolic and numerical calculations is presented. Symbolic calculations primarily refer to the computer processing of procedures from classical algebra, analysis, and calculus. Numerical calculations refer to both numerical mathematics research and scientific computation. This survey is intended to point out a large number of problem areas where a cooperation of symbolic and numerical methods is likely to bear many fruits. These areas include such classical operations as differentiation and integration, such diverse activities as function approximations and qualitative analysis, and such contemporary topics as finite element calculations and computation complexity. It is contended that other less obvious topics such as the fast Fourier transform, linear algebra, nonlinear analysis and error analysis would also benefit from a synergistic approach

Introduction to the GiNaC Framework for Symbolic Computation within the C++ Programming Language

The traditional split-up into a low level language and a high level language in the design of computer algebra systems may become obsolete with the advent of more versatile computer languages. We describe GiNaC, a special-purpose system that deliberately denies the need for such a distinction. It is entirely written in C++ and the user can interact with it directly in that language. It was designed to provide efficient handling of multivariate polynomials, algebras and special functions that are needed for loop calculations in theoretical quantum field theory. It also bears some potential to become a more general purpose symbolic package

Kira - A Feynman Integral Reduction Program

In this article, we present a new implementation of the Laporta algorithm to reduce scalar multi-loop integrals---appearing in quantum field theoretic calculations---to a set of master integrals. We extend existing approaches by using an additional algorithm based on modular arithmetic to remove linearly dependent equations from the system of equations arising from integration-by-parts and Lorentz identities. Furthermore, the algebraic manipulations required in the back substitution are optimized. We describe in detail the implementation as well as the usage of the program. In addition, we show benchmarks for concrete examples and compare the performance to Reduze 2 and FIRE 5. In our benchmarks we find that Kira is highly competitive with these existing tools.Comment: 37 pages, 3 figure

Symbolic computation: systems and applications

The article presents an overview of symbolic computation systems, their classification-in-history, the most popular CAS, examples of systems and some of their applications. Symbolics versus numeric, enhancement in mathematics, computing nature of CAS, related projects, networks, references are discussed

Launch window analysis of satellites in high eccentricity or large circular orbits

Numerical methods and computer programs for studying the stability and evolution of orbits of large eccentricity are presented. Methods for determining launch windows and target dates are developed. Mathematical models are prepared to analyze the characteristics of specific missions