Structured Matrices in Mathematics, Computer Science, and Engineering II
About this Volume
Edited by: Vadim Olshevsky
2001: Volume: 281
ISBNs: 978-0-8218-2092-6 (print); 978-0-8218-7871-2 (online)
Many important problems in applied sciences, mathematics, and engineering can be reduced to matrix problems. Moreover, various applications often introduce a special structure into the corresponding matrices, so that their entries can be described by a certain compact formula. Classic examples include Toeplitz matrices, Hankel matrices, Vandermonde matrices, Cauchy matrices, Pick matrices, Bezoutians, controllability and observability matrices, and others. Exploiting these and the more general structures often allows us to obtain elegant solutions to mathematical problems as well as to design more efficient practical algorithms for a variety of applied engineering problems.
Structured matrices have been under close study for a long time and in quite diverse (and seemingly unrelated) areas, for example, mathematics, computer science, and engineering. Considerable progress has recently been made in all these areas, and especially in studying the relevant numerical and computational issues. In the past few years, a number of practical algorithms blending speed and accuracy have been developed. This significant growth is fully reflected in these volumes, which collect 38 papers devoted to the numerous aspects of the topic.
The collection of the contributions to these volumes offers a flavor of the plethora of different approaches to attack structured matrix problems. The reader will find that the theory of structured matrices is positioned to bridge diverse applications in the sciences and engineering, deep mathematical theories, as well as computational and numerical issues. The presentation fully illustrates the fact that the techniques of engineers, mathematicians, and numerical analysts nicely complement each other, and they all contribute to one unified theory of structured matrices.
The book is published in two volumes. The first contains articles on interpolation, system theory, signal and image processing, control theory, and spectral theory. Articles in the second volume are devoted to fast algorithms, numerical and iterative methods, and various applications.
Graduate students and research mathematicians interested in linear and multilinear algebra, matrix theory, operator theory, numerical analysis, and systems theory and control.
Table of Contents
Part V. Fast Algorithms
- G. Heinig and V. Olshevsky – The Schur algorithm for matrices with Hessenberg displacement structure
- Y. Eidelman and I. Gohberg – Fast inversion algorithms for a class of block structured matrices
- S. Chandrasekaran and Ming Gu – A fast and stable solver for recursively semi-separable systems of linear equations
Part VI. Numerical Issues
- Michael Stewart – Stability properties of several variants of the unitary Hessenberg $QR$ algorithm
- Myungwon Kim, Haesun Park and Lars Eldén – Comparison of algorithms for Toeplitz least squares and symmetric positive definite linear systems
- Georg Heinig – Stability of Toeplitz matrix inversion formulas
- James Demmel and Plamen Koev – Necessary and sufficient conditions for accurate and efficient rational function evaluation and factorizations of rational matrices
- Marc Van Barel and Adhemar Bultheel – Updating and downdating of orthonormal polynomial vectors and some applications
- Per Christian Hansen and Plamen Yalamov – Rank-revealing decompositions of symmetric Toeplitz matrices
Part VII. Iterative Methods. Preconditioners
- Raymond H. Chan, Michael K. Ng and Andy M. Yip – A survey of preconditioners for ill-conditioned Toeplitz systems
- Daniel Potts and Gabriele Steidl – Preconditioning of Hermitian block-Toeplitz-Toeplitz-block matrices by level-1 preconditioners
Part VIII. Linear Algebra and Various Applications
- Dario Andrea Bini and Beatrice Meini – Approximate displacement rank and applications
- William F. Trench – Properties of some generalizations of Kac-Murdock-Szegő matrices
- Georg Heinig and Karla Rost – Efficient inversion formulas for Toeplitz-plus-Hankel matrices using trigonometric transformations
- Luca Gemignani – On a generalization of Poincaré’s theorem for matrix difference equations arising from root-finding problems
- Leiba Rodman – Completions of triangular matrices: a survey of results and open problems
- S. Serra Capizzano and C. Tablino Possio – Positive representation formulas for finite difference discretizations of (elliptic) second order PDEs
- Paolo Tilli – On some problems involving invariant norms and Hadamard products
- Y. S. Choi, I. Koltracht and P. J. McKenna – A generalization of the Perron-Frobenius theorem for non-linear perturbations of Stiltjes matrices
- M. J. C. Gover and A. M. Byrne – The rhombus matrix: definition and properties