GiNaCRA: A C++ library for real algebraic computations

Lecture Notes in Computer Science, 2010

There is a growing interest in numeric-algebraic techniques in the computer algebra community as such techniques can speed up many applications. This paper is concerned with one such approach called Exact Numeric Computation (ENC). The ENC approach to algebraic number computation is based on iterative verified approximations, combined with constructive zero bounds. This paper describes Core 2, the latest version of the Core Library, a package designed for applications such as non-linear computational geometry. The adaptive complexity of ENC combined with filters makes such libraries practical. Core 2 smoothly integrates our algebraic ENC subsystem with transcendental functions with ε-accurate comparisons. This paper describes how the design of Core 2 addresses key software issues such as modularity, extensibility, efficiency in the a setting that combines algebraic and transcendental elements. Our redesign preserves the original goals of the Core Library, namely, to provide a simple and natural interface for ENC computation to support rapid prototyping and exploration. We present examples, experimental results, and timings for our new system, released as Core Library 2.0.

This report describes our work on implementation of effective numerical routines for polynomials and polynomial matrices in the MATHEMATICA software. Such operations are recalled during the controller design process if the so called polynomial or algebraic design methods are employed. This research is also motivated by the fact that MATHEMATICA developers pay attention to control engineers needs and produce the Control Systems Professional package for use with MATH-EMATICA and, as we believe, a set of routines for algebraic approach could conveniently complement the existing bunch of programs primarily intended for state-space representations.

Lecture Notes in Computer Science, 2005

We propose exact, complete and efficient methods for 2 problems: First, the real solving of systems of two bivariate rational polynomials of arbitrary degree. This means isolating all common real solutions in rational rectangles and calculating the respective multiplicities. Second, the computation of the sign of bivariate polynomials evaluated at two algebraic numbers of arbitrary degree. Our main motivation comes from nonlinear computational geometry and computer-aided design, where bivariate polynomials lie at the inner loop of many algorithms. The methods employed are based on Sturm-Habicht sequences, univariate resultants and rational univariate representation. We have implemented them very carefully, using advanced object-oriented programming techniques, so as to achieve high practical performance. The algorithms are integrated in the public-domain C++ software library synaps, and their efficiency is illustrated by 9 experiments against existing implementations. Our code is faster in most cases; sometimes it is even faster than numerical approaches.

2006

Expression packages constitute a key technique for achieving the fundamental goals of Exact Geometric Computation (EGC). In this paper, we address the redesign of the Expression Package in the Core Library, addressing the following issues:

Theoretical Computer Science, 2008

We present exact and complete algorithms based on precomputed Sturm-Habicht sequences, discriminants and invariants, that classify, isolate with rational points and compare the real roots of polynomials of degree up to 4. We have closed formulas for all isolating points. Moreover we combine these results with a simple version of rational univariate representation so as to isolate and compute the multiplicity of all common real roots of a bivariate system of integer polynomials of total degree ≤ 2. We present our implementation within synaps and we perform experimentation and comparison with all available software. Our package is 2-10 times faster, even when compared to inexact software or to sofware with intrinsic filtering.

2005

We present algorithmic and complexity results concerning computations with one and two real algebraic numbers, as well as real solving of univariate polynomials and bivariate polynomial systems with integer coefficients using Sturm-Habicht sequences.

Journal of Symbolic Computation, 1988

This paper explains how computer algebra (Reduce) was used to analyse the expressions resulting from the complexity analysis of a family of algorithms for the isolation of real roots of polynomials. The expressions depend on sufficiently many parameters, and are sufficiently complex, that manual analysis would have been almost impossible. In addition, by analysing constant factors as well as the 0 form of the expressions, we obtain more information about the relative costs of algorithms with the same 0 complexity.

[Now twelve years old, but still worth a look.] Exact symbolic computation with polynomials and matrices over polynomial rings has wide applicability to many fields. By "exact symbolic" we mean computation with polynomials whose coefficients are integers (of any size), rational numbers, or finite fields, as opposed to coefficients that are "floats" of a certain precision. Such computation is part of most computer algebra systems ("CA systems"). Over the last dozen years several large CA systems have become widely available, such as Axiom, Derive, Macsyma, Magma, Maple, Mathematica, and Reduce. They tend to have great breadth, be produced by profit-making companies, and be relatively expensive. However, most if not all of these systems have difficulty computing with the polynomials and matrices that arise in actual research. Real problems tend to produce large polynomials and large matrices that the general CA systems cannot handle. In the last few years several smaller CA systems focused on polynomials have been produced at universities by individual researchers or small teams. They run on Macs, PCs, and workstations. They are freeware or shareware. Several claim to be much more efficient than the large systems at exact polynomial computations. The list of these systems includes CoCoA, Fermat, MuPAD, Pari-GP, and Singular.

Computers, Materials & Continua, 2021

In this research article, we construct a family of derivative free simultaneous numerical schemes to approximate all real zero of non-linear polynomial equation. We make a comparative analysis of the newly constructed numerical schemes with a well-known existing simultaneous method for determining all the distinct real zeros of polynomial equations using computer algebra system Mat Lab. Lower bound of convergence of simultaneous schemes is calculated using Mathematica. Global convergence property of the numerical schemes is presented by taking random starting initial approximation and their convergence history are graphically presented. Some real life engineering applications along with some higher degree polynomials are considered as numerical test problems to show performance and efficiency of the derivative free family of numerical methods with comparison of an existing method of same order in literature. Local computational order of convergence, CPU time, graph of computational order of convergence and residual error graphs elaborate efficiency, robustness and authentication of the suggested family of numerical methods in its domain.

Lecture Notes in Computer Science, 2012

We present SMT-RAT, a C++ toolbox offering theory solver modules for the development of SMT solvers for nonlinear real arithmetic (NRA). NRA is an important but hard-to-solve theory and only fragments of it can be handled by some of the currently available SMT solvers. Our toolbox contains modules implementing the virtual substitution method, the cylindrical algebraic decomposition method, a Gröbner bases simplifier and a general simplifier. These modules can be combined according to a user-defined strategy in order to exploit their advantages.

Lecture Notes in Computer Science, 2014

The Basic Polynomial Algebra Subprograms (BPAS) provides arithmetic operations (multiplication, division, root isolation, etc.) for univariate and multivariate polynomials over common types of coefficients (prime fields, complex rational numbers, rational functions, etc.). The code is mainly written in CilkPlus [10] targeting multicore processors. The current distribution focuses on dense polynomials and the sparse case is work in progress. A strong emphasis is put on adaptive algorithms as the library aims at supporting a wide variety of situations in terms of problem sizes and available computing resources.

Lecture Notes in Computer Science, 2010

BIBECHANA, 2012

Real numbers are something which are associated with the practical life. This number system is one dimensional. Situations arise when the real numbers fail to provide a solution. Perhaps the Italian mathematician Gerolamo Cardano is the first known mathematician who pointed out the necessity of imaginary and complex numbers. Complex numbers are now a vital part of sciences and are used in various branches of engineering, technology, electromagnetism, quantum theory, chaos theory etc. A complex number constitutes a real number along with an imaginary number that lies on the quadrature axis and gives an additional dimension to the number system. Therefore any computation based on complex numbers, is usually complex because both the real and imaginary parts of the number are to be simultaneously dealt with. Modern scientific calculators are capable of performing on a wide range of functions on complex numbers in their COMP and CMPLX modes with an equal ease as with the real numbers. In...

1992

Characterizing real algebraic numbers by a sign-sequence (according to Thom's lemma), using a variant of Ben Or Kozan and Reif algorithm for reducing the solving of systems of polynomial inequalities to the problem of counting real zeroes satisfying one polynomial inequality, and using a multivariate Sturm theory generalizing Hermite quadratic forms method for counting real zeroes, we prove that the computations on real algebraic numbers are in NC.

Communication Papers of the 2019 Federated Conference on Computer Science and Information Systems, 2019

This paper is a continuation of the discussion undertaken in paper [31]. We present in the current paper the corrected, and also given in a slightly changed form, Vandermonde formulae for the roots of some quintic polynomials considered in J.P. Tignol's monograph [26]. The proofs of selected trigonometric identities from [31] are given and some new identities have been generated by the occasion, which also can be used for testing our Langrange algorithm for the case of cubic polynomials. Moreover, we present here the decomposition of polynomials belonging to some two-parameter family of polynomials related to the Chebyshev polynomials of the first kind.