Doctor of Philosophy
Robert M. Corless
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's 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'' 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.
Summary for Lay Audience
Matrices as arrays of numbers and polynomials as the simplest type of mathematical functions naturally arise in a majority of computational problems. Such useful objects are studied well in the literature due to their vast applications in science and engineering. Solving linear systems of equations and finding roots of polynomials are probably the most well understood problems. However, efforts to introduce more efficient and/or reliable algorithms in general or for special cases are going on.
In this thesis, we focus on a few problems related to matrices and polynomials over complex numbers with a hybrid symbolic-numeric approach. We present algorithms for approximate Greatest Common Divisor (GCD) of polynomials in Bernstein and Lagrange bases. Another problem we have addressed is solving linear systems with parameters as coefficients. Relevant problems to linearization of matrix polynomials in multiple bases are discussed as well.
Rafiee Sevyeri, Leili, "Hybrid Symbolic-Numeric Computing in Linear and Polynomial Algebra" (2020). Electronic Thesis and Dissertation Repository. 7187.