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

An elementary proof of Barnett's theorem about the greatest common divisor of several univarlate polynomials

AbstractThis articule provides a new proof of Barnett's theorem giving the degree of the greatest common divisor of several univariate polynomials with coefficients in a field in terms of the rank of a well-defined matrix. The new proof is elementary and self-contained (no use of Jordan form or invariant factors), and it is based on some easy to state properties of subresultants. Moreover this proof allows one to generalize Barnett's results to the case when the considered polynomials have their coefficients in an integral domain

Hybrid Symbolic-Numeric Computing in Linear and Polynomial Algebra

In this thesis, we introduce hybrid symbolic-numeric methods for solving problems in linear and polynomial algebra. We mainly address the approximate GCD problem for polynomials, and problems related to parametric and polynomial matrices. For symbolic methods, our main concern is their complexity and for the numerical methods we are more concerned about their stability. The thesis consists of 5 articles which are presented in the following order: Chapter 1, deals with the fundamental notions of conditioning and backward error. Although our results are not novel, this chapter is a novel explication of conditioning and backward error that underpins the rest of the thesis. In Chapter 2, we adapt Victor Y. Pan\u27s root-based algorithm for finding approximate GCD to the case where the polynomials are expressed in Bernstein bases. We use the numerically stable companion pencil of G. F. Jónsson to compute the roots, and the Hopcroft-Karp bipartite matching method to find the degree of the approximate GCD. We offer some refinements to improve the process. In Chapter 3, we give an algorithm with similar idea to Chapter 2, which finds an approximate GCD for a pair of approximate polynomials given in a Lagrange basis. More precisely, we suppose that these polynomials are given by their approximate values at distinct known points. We first find each of their roots by using a Lagrange basis companion matrix for each polynomial. We introduce new clustering algorithms and use them to cluster the roots of each polynomial to identify multiple roots, and then marry the two polynomials using a Maximum Weight Matching (MWM) algorithm, to find their GCD. In Chapter 4, we define ``generalized standard triples\u27\u27 X, zC1 - C0, Y of regular matrix polynomials P(z) in order to use the representation X(zC1 - C0)-1 Y=P-1(z). This representation can be used in constructing algebraic linearizations; for example, for H(z) = z A(z)B(z) + C from linearizations for A(z) and B(z). This can be done even if A(z) and B(z) are expressed in differing polynomial bases. Our main theorem is that X can be expressed using the coefficients of 1 in terms of the relevant polynomial basis. For convenience we tabulate generalized standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases; for the Bernstein basis; for Lagrange interpolational bases; and for Hermite interpolational bases. We account for the possibility of common similarity transformations. We give explicit proofs for the less familiar bases. Chapter 5 is devoted to parametric linear systems (PLS) and related problems, from a symbolic computational point of view. PLS are linear systems of equations in which some symbolic parameters, that is, symbols that are not considered to be candidates for elimination or solution in the course of analyzing the problem, appear in the coefficients of the system. We assume that the symbolic parameters appear polynomially in the coefficients and that the only variables to be solved for are those of the linear system. It is well-known that it is possible to specify a covering set of regimes, each of which is a semi-algebraic condition on the parameters together with a solution description valid under that condition.We provide a method of solution that requires time polynomial in the matrix dimension and the degrees of the polynomials when there are up to three parameters. Our approach exploits the Hermite and Smith normal forms that may be computed when the system coefficient domain is mapped to the univariate polynomial domain over suitably constructed fields. Our approach effectively identifies intrinsic singularities and ramification points where the algebraic and geometric structure of the matrix changes. Specially parametric eigenvalue problems can be addressed as well. Although we do not directly address the problem of computing the Jordan form, our approach allows the construction of the algebraic and geometric eigenvalue multiplicities revealed by the Frobenius form, which is a key step in the construction of the Jordan form of a matrix

Association of Christians in the Mathematical Sciences Proceedings 2019

The conference proceedings of the Association of Christians in the Mathematical Sciences biannual conference, May 29-June 1, 2019 at Indiana Wesleyan University

Computing the greatest common divisor of polynomials using the comrade matrix

The comrade matrix of a polynomial is an analogue of the companion matrix when the matrix is expressed in terms of a general basis such that the basis is a set of orthogonal polynomials satisfying the three-term recurrence relation. We present the algorithms for computing the comrade matrix, and the coefficient matrix of the corresponding linear systems derived from the recurrence relation. The computing times of these algorithms are analyzed. The computing time bounds, which dominate these times, are obtained as functions of the degree and length of the integers that represent the rational number coefficients of the input polynomials. The ultimate aim is to apply these computing time bounds in the analysis of the performance of the generalized polynomial greatest common divisor algorithms

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

The Craft of Economic Modeling

This book is a practical guide to building economic models, both macroeconomic and multisectoral. It uses free software available from the Internet together with regularly updated databanks including the quarterly national accounts of the United States. It does NOT deal with some recent fads in economic model building, such as Real Business Cycles, Computable General Equilibriium models, nor DSGE models, all of which are, in the author's opinion, "alternative reality" models at best