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A Symbolic-Numeric Software Package for the Computation of the GCD of Several Polynomials
This survey is intended to present a package of algorithms for the computation of exact or approximate GCDs of sets of several polynomials and the evaluation of the quality of the produced solutions. These algorithms are designed to operate in symbolic-numeric computational environments. The key of their effectiveness is the appropriate selection of the right type of operations (symbolic or numeric) for the individual parts of the algorithms. Symbolic processing is used to improve on the conditioning of the input data and handle an ill-conditioned sub-problem and numeric tools are used in accelerating certain parts of an algorithm. A sort description of the basic algorithms of the package is presented by using the symbolic-numeric programming code of Maple
Symbolic-numeric algorithms for univariate polynomials
Thesis (Ph. D. in Science)--University of Tsukuba, (B), no. 2485, 2010.3.25 Includes bibliographical referencesNote to the re-typeset version: This is re-typeset version of the original dissertation. While I have maintained the original contents without changing any words and/or formulas in the main body, I have added the following information: 1. Copyright notice of corresponding articles in each chapter; 2. Digital Object Identifiers (DOI) or URLs of references as many as possible.Please note that the number of pages is slightly increased in the present edition from that of the original edition, possibly by changes of page style parameters.200
Subresultants and Generic Monomial Bases
Given n polynomials in n variables of respective degrees d_1,...,d_n, and a
set of monomials of cardinality d_1...d_n, we give an explicit
subresultant-based polynomial expression in the coefficients of the input
polynomials whose non-vanishing is a necessary and sufficient condition for
this set of monomials to be a basis of the ring of polynomials in n variables
modulo the ideal generated by the system of polynomials. This approach allows
us to clarify the algorithms for the Bezout construction of the resultant.Comment: 22 pages, uses elsart.cls. Revised version accepted for publication
in the Journal of Symbolic Computatio
Cyclic Resultants
We characterize polynomials having the same set of nonzero cyclic resultants.
Generically, for a polynomial of degree , there are exactly
distinct degree polynomials with the same set of cyclic resultants as .
However, in the generic monic case, degree polynomials are uniquely
determined by their cyclic resultants. Moreover, two reciprocal
(``palindromic'') polynomials giving rise to the same set of nonzero cyclic
resultants are equal. In the process, we also prove a unique factorization
result in semigroup algebras involving products of binomials. Finally, we
discuss how our results yield algorithms for explicit reconstruction of
polynomials from their cyclic resultants.Comment: 16 pages, Journal of Symbolic Computation, print version with errata
incorporate
Software Engineering and Complexity in Effective Algebraic Geometry
We introduce the notion of a robust parameterized arithmetic circuit for the
evaluation of algebraic families of multivariate polynomials. Based on this
notion, we present a computation model, adapted to Scientific Computing, which
captures all known branching parsimonious symbolic algorithms in effective
Algebraic Geometry. We justify this model by arguments from Software
Engineering. Finally we exhibit a class of simple elimination problems of
effective Algebraic Geometry which require exponential time to be solved by
branching parsimonious algorithms of our computation model.Comment: 70 pages. arXiv admin note: substantial text overlap with
arXiv:1201.434
Development of symbolic algorithms for certain algebraic processes
This study investigates the problem of computing the exact greatest common divisor of two polynomials relative to an orthogonal basis, defined over the rational number field. The main objective of the study is to design and implement an effective and efficient symbolic algorithm for the general class of dense polynomials, given the rational number defining terms of their basis. From a general algorithm using the comrade matrix approach, the nonmodular and modular techniques are prescribed. If the coefficients of the generalized polynomials are multiprecision integers, multiprecision arithmetic will be required in the construction of the comrade matrix and the corresponding systems coefficient matrix. In addition, the application of the nonmodular elimination technique on this coefficient matrix extensively applies multiprecision rational number operations. The modular technique is employed to minimize the complexity involved in such computations. A divisor test algorithm that enables the detection of an unlucky reduction is a crucial device for an effective implementation of the modular technique. With the bound of the true solution not known a priori, the test is devised and carefully incorporated into the modular algorithm. The results illustrate that the modular algorithm illustrate its best performance for the class of relatively prime polynomials. The empirical computing time results show that the modular algorithm is markedly superior to the nonmodular algorithms in the case of sufficiently dense Legendre basis polynomials with a small GCD solution. In the case of dense Legendre basis polynomials with a big GCD solution, the modular algorithm is significantly superior to the nonmodular algorithms in higher degree polynomials. For more definitive conclusions, the computing time functions of the algorithms that are presented in this report have been worked out. Further investigations have also been suggested
RealCertify: a Maple package for certifying non-negativity
Let (resp. ) be the field of rational (resp. real)
numbers and be variables. Deciding the non-negativity
of polynomials in over or over semi-algebraic
domains defined by polynomial constraints in is a classical
algorithmic problem for symbolic computation.
The Maple package \textsc{RealCertify} tackles this decision problem by
computing sum of squares certificates of non-negativity for inputs where such
certificates hold over the rational numbers. It can be applied to numerous
problems coming from engineering sciences, program verification and
cyber-physical systems. It is based on hybrid symbolic-numeric algorithms based
on semi-definite programming.Comment: 4 pages, 2 table
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