Structured Matrices and Polynomials Unified Superfast Algorithms 1st Edition by Victor Y Pan ISBN 0817642404 9780817642402 by Victor Y. Pan 9780817642402, 9783764342401, 0817642404, 3764342404 instant download after payment.
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ISBN 10: 0817642404
ISBN 13: 9780817642402
Author: Victor Y Pan
Structured matrices serve as a natural bridge between the areas of algebraic computations with polynomials and numerical matrix computations, allowing cross-fertilization of both fields. This book covers most fundamental numerical and algebraic computations with Toeplitz, Hankel, Vandermonde, Cauchy, and other popular structured matrices. Throughout the computations, the matrices are represented by their compressed images, called displacements, enabling both a unified treatment of various matrix structures and dramatic saving of computer time and memory. The resulting superfast algorithms allow further dramatic parallel acceleration using FFT and fast sine and cosine transforms. Included are specific applications to other fields, in particular, superfast solutions to: various fundamental problems of computer algebra; the tangential Nevanlinna--Pick and matrix Nehari problems The primary intended readership for this work includes researchers, algorithm designers, and advanced graduate students in the fields of computations with structured matrices, computer algebra, and numerical rational interpolation. The book goes beyond research frontiers and, apart from very recent research articles, includes yet unpublished results. To serve a wider audience, the presentation unfolds systematically and is written in a user-friendly engaging style. Only some preliminary knowledge of the fundamentals of linear algebra is required. This 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. Examples, tables, figures, exercises, extensive bibliography, and index lend this text to classroom use or self-study.
Structured Matrices and Polynomials Unified Superfast Algorithms 1st Table of contents:
Part I: Foundations and Properties of Structured Matrices
- Chapter 1: Displacement Structures and Generators
- Definition of Displacement Operators (Sylvester, Lyapunov, Pick, etc.)
- Low-Rank Displacement Generators
- Representations of Structured Matrices via Generators
- Properties of Displacement Rank
- Chapter 2: Specific Classes of Structured Matrices
- Toeplitz Matrices: Properties, Inverse, Applications
- Hankel Matrices: Properties, Inverse, Applications
- Cauchy Matrices: Properties, Inverse, Applications
- Vandermonde Matrices: Properties, Inverse, Applications
- Block Structured Matrices: Block Toeplitz, Block Hankel
- Generalized Forms and Mixed Structures
Part II: Polynomials, Rational Functions, and Series
- Chapter 3: Polynomial Algebra Review
- Polynomial Arithmetic (Addition, Multiplication, Division)
- Euclidean Algorithm for Polynomials
- Greatest Common Divisors (GCD) of Polynomials
- Resultants and Discriminants
- Chapter 4: Laurent Series and Rational Functions
- Definition and Properties of Laurent Series
- Connection to Rational Functions
- Pade Approximation and Continued Fractions
- Matrix Polynomials and Operator Polynomials
Part III: Unified Superfast Algorithms
- Chapter 5: Superfast Algorithms for Toeplitz and Hankel Systems
- Levinson and Schur-type Algorithms (brief review of fast algorithms)
- Generalized Schur Algorithm
- The Doubling Algorithm for Toeplitz Systems
- Connection to Euclidean Algorithm for Polynomials
- Chapter 6: Superfast Algorithms for Cauchy and Vandermonde Systems
- Divide-and-Conquer Strategies for Cauchy Matrices
- Fast Multipole Methods (FMM) principles applied to structured matrices
- Algorithms based on Polynomial Interpolation
- Chapter 7: Duality and Transformations between Structured Matrices
- Connections between Toeplitz, Hankel, Cauchy, and Vandermonde Structures
- Transformations preserving or changing displacement rank
- Exploiting Duality for Algorithm Design
- Chapter 8: Superfast Algorithms for General Structured Matrices
- Unified Framework via Displacement Rank
- Recursive Algorithms based on Matrix Partitioning
- Complexity Analysis of Superfast Algorithms (logarithmic factors)
- Numerical Stability Considerations
Part IV: Applications and Extensions
-
Chapter 9: Applications in Signal Processing
- Linear Prediction and System Identification
- Spectral Estimation
- Inverse Scattering
-
Chapter 10: Applications in Control Theory
- Lyapunov and Riccati Equations
- System Realization
-
Chapter 11: Applications in Numerical Analysis
- Preconditioning Techniques for Structured Systems
- Fast Algorithms for Matrix Polynomial Equations
-
Chapter 12: Extensions and Open Problems
- Structured Eigenvalue Problems
- Nonlinear Matrix Equations with Structure
- Parallel and Distributed Superfast Algorithms
- Future Research Directions
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Tags: Victor Y Pan, Matrices, Polynomials
