Matrix Polynomials
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| Author | : I. Gohberg |
| Publisher | : SIAM |
| Total Pages | : 423 |
| Release | : 2009-07-23 |
| Genre | : Mathematics |
| ISBN | : 0898716810 |
This book is the definitive treatment of the theory of polynomials in a complex variable with matrix coefficients. Basic matrix theory can be viewed as the study of the special case of polynomials of first degree; the theory developed in Matrix Polynomials is a natural extension of this case to polynomials of higher degree. It has applications in many areas, such as differential equations, systems theory, the Wiener-Hopf technique, mechanics and vibrations, and numerical analysis. Although there have been significant advances in some quarters, this work remains the only systematic development of the theory of matrix polynomials. The book is appropriate for students, instructors, and researchers in linear algebra, operator theory, differential equations, systems theory, and numerical analysis. Its contents are accessible to readers who have had undergraduate-level courses in linear algebra and complex analysis.
| Author | : Victor Y. Pan |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 299 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 1461201292 |
This user-friendly, engaging textbook makes the material accessible to graduate students and new researchers who wish to study the rapidly exploding area of computations with structured matrices and polynomials. The book goes beyond research frontiers and, apart from very recent research articles, includes previously unpublished results.
| Author | : Dario Bini |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 433 |
| Release | : 2012-12-06 |
| Genre | : Computers |
| ISBN | : 1461202655 |
Our Subjects and Objectives. This book is about algebraic and symbolic computation and numerical computing (with matrices and polynomials). It greatly extends the study of these topics presented in the celebrated books of the seventies, [AHU] and [BM] (these topics have been under-represented in [CLR], which is a highly successful extension and updating of [AHU] otherwise). Compared to [AHU] and [BM] our volume adds extensive material on parallel com putations with general matrices and polynomials, on the bit-complexity of arithmetic computations (including some recent techniques of data compres sion and the study of numerical approximation properties of polynomial and matrix algorithms), and on computations with Toeplitz matrices and other dense structured matrices. The latter subject should attract people working in numerous areas of application (in particular, coding, signal processing, control, algebraic computing and partial differential equations). The au thors' teaching experience at the Graduate Center of the City University of New York and at the University of Pisa suggests that the book may serve as a text for advanced graduate students in mathematics and computer science who have some knowledge of algorithm design and wish to enter the exciting area of algebraic and numerical computing. The potential readership may also include algorithm and software designers and researchers specializing in the design and analysis of algorithms, computational complexity, alge braic and symbolic computing, and numerical computation.
| Author | : Percy Deift |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 273 |
| Release | : 2000 |
| Genre | : Mathematics |
| ISBN | : 0821826956 |
This volume expands on a set of lectures held at the Courant Institute on Riemann-Hilbert problems, orthogonal polynomials, and random matrix theory. The goal of the course was to prove universality for a variety of statistical quantities arising in the theory of random matrix models. The central question was the following: Why do very general ensembles of random n times n matrices exhibit universal behavior as n > infinity? The main ingredient in the proof is the steepest descent method for oscillatory Riemann-Hilbert problems. Titles in this series are copublished with the Courant Institute of Mathematical Sciences at New York University.
| Author | : Philip Saltenberger |
| Publisher | : Logos Verlag Berlin GmbH |
| Total Pages | : 194 |
| Release | : 2019-05-30 |
| Genre | : Mathematics |
| ISBN | : 3832549145 |
In this thesis, a novel framework for the construction and analysis of strong linearizations for matrix polynomials is presented. Strong linearizations provide the standard means to transform polynomial eigenvalue problems into equivalent generalized eigenvalue problems while preserving the complete finite and infinite eigenstructure of the problem. After the transformation, the QZ algorithm or special methods appropriate for structured linearizations can be applied for finding the eigenvalues efficiently. The block Kronecker ansatz spaces proposed here establish an innovative and flexible approach for the construction of strong linearizations in the class of strong block minimal bases pencils. Moreover, they represent a new vector-space-setting for linearizations of matrix polynomials that additionally provides a common basis for various existing techniques on this task (such as Fiedler-linearizations). New insights on their relations, similarities and differences are revealed. The generalized eigenvalue problems obtained often allow for an efficient numerical solution. This is discussed with special attention to structured polynomial eigenvalue problems whose linearizations are structured as well. Structured generalized eigenvalue problems may also lead to equivalent structured (standard) eigenvalue problems. Thereby, the transformation produces matrices that can often be regarded as selfadjoint or skewadjoint with respect to some indefinite inner product. Based on this observation, normal matrices in indefinite inner product spaces and their spectral properties are studied and analyzed. Multiplicative and additive canonical decompositions respecting the matrix structure induced by the inner product are established.
| Author | : I. Gohberg |
| Publisher | : Birkhäuser |
| Total Pages | : 220 |
| Release | : 2013-11-21 |
| Genre | : Science |
| ISBN | : 3034854722 |
This paper is a largely expository account of the theory of p x p matrix polyno mials associated with Hermitian block Toeplitz matrices and some related problems of interpolation and extension. Perhaps the main novelty is the use of reproducing kernel Pontryagin spaces to develop parts of the theory in what hopefully the reader will regard as a reasonably lucid way. The topics under discussion are presented in a series of short sections, the headings of which give a pretty good idea of the overall contents of the paper. The theory is a rich one and the present paper in spite of its length is far from complete. The author hopes to fill in some of the gaps in future publications. The story begins with a given sequence h_n" ... , hn of p x p matrices with h-i = hj for j = 0, ... , n. We let k = O, ... ,n, (1.1) denote the Hermitian block Toeplitz matrix based on ho, ... , hk and shall denote its 1 inverse H k by (k)] k [ r = .. k = O, ... ,n, (1.2) k II} . '-0 ' I- whenever Hk is invertible.
| Author | : Ákos Pintér |
| Publisher | : MDPI |
| Total Pages | : 154 |
| Release | : 2021-09-03 |
| Genre | : Mathematics |
| ISBN | : 303650818X |
Polynomials play a crucial role in many areas of mathematics including algebra, analysis, number theory, and probability theory. They also appear in physics, chemistry, and economics. Especially extensively studied are certain infinite families of polynomials. Here, we only mention some examples: Bernoulli, Euler, Gegenbauer, trigonometric, and orthogonal polynomials and their generalizations. There are several approaches to these classical mathematical objects. This Special Issue presents nine high quality research papers by leading researchers in this field. I hope the reading of this work will be useful for the new generation of mathematicians and for experienced researchers as well.
| Author | : E.V. Krishnamurthy |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 170 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 1461251184 |
This book is written as an introduction to polynomial matrix computa tions. It is a companion volume to an earlier book on Methods and Applications of Error-Free Computation by R. T. Gregory and myself, published by Springer-Verlag, New York, 1984. This book is intended for seniors and graduate students in computer and system sciences, and mathematics, and for researchers in the fields of computer science, numerical analysis, systems theory, and computer algebra. Chapter I introduces the basic concepts of abstract algebra, including power series and polynomials. This chapter is essentially meant for bridging the gap between the abstract algebra and polynomial matrix computations. Chapter II is concerned with the evaluation and interpolation of polynomials. The use of these techniques for exact inversion of poly nomial matrices is explained in the light of currently available error-free computation methods. In Chapter III, the principles and practice of Fourier evaluation and interpolation are described. In particular, the application of error-free discrete Fourier transforms for polynomial matrix computations is consi dered.
| Author | : Paul Nevai |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 494 |
| Release | : 1989-12-31 |
| Genre | : Mathematics |
| ISBN | : 9780792305699 |
This volume contains the Proceedings of the NATO Advanced Study Institute on "Orthogonal Polynomials and Their Applications" held at The Ohio State University in Columbus, Ohio, U.S.A. between May 22,1989 and June 3,1989. The Advanced Study Institute primarily concentrated on those aspects of the theory and practice of orthogonal polynomials which surfaced in the past decade when the theory of orthogonal polynomials started to experience an unparalleled growth. This progress started with Richard Askey's Regional Confer ence Lectures on "Orthogonal Polynomials and Special Functions" in 1975, and subsequent discoveries led to a substantial revaluation of one's perceptions as to the nature of orthogonal polynomials and their applicability. The recent popularity of orthogonal polynomials is only partially due to Louis de Branges's solution of the Bieberbach conjecture which uses an inequality of Askey and Gasper on Jacobi polynomials. The main reason lies in their wide applicability in areas such as Pade approximations, continued fractions, Tauberian theorems, numerical analysis, probability theory, mathematical statistics, scattering theory, nuclear physics, solid state physics, digital signal processing, electrical engineering, theoretical chemistry and so forth. This was emphasized and convincingly demonstrated during the presentations by both the principal speakers and the invited special lecturers. The main subjects of our Advanced Study Institute included complex orthogonal polynomials, signal processing, the recursion method, combinatorial interpretations of orthogonal polynomials, computational problems, potential theory, Pade approximations, Julia sets, special functions, quantum groups, weighted approximations, orthogonal polynomials associated with root systems, matrix orthogonal polynomials, operator theory and group representations.
| Author | : Tadeusz Kaczorek |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 514 |
| Release | : 2007-01-19 |
| Genre | : Technology & Engineering |
| ISBN | : 1846286050 |
This book reviews new results in the application of polynomial and rational matrices to continuous- and discrete-time systems. It provides the reader with rigorous and in-depth mathematical analysis of the uses of polynomial and rational matrices in the study of dynamical systems. It also throws new light on the problems of positive realization, minimum-energy control, reachability, and asymptotic and robust stability.
