# Structured Matrices in Mathematics, Computer Science, and Engineering I

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*Vadim Olshevsky*

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.

#### Readership

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

## Structured Matrices in Mathematics, Computer Science, and Engineering I

- Contents vii8 free
- Foreword xi12 free
- Part I. Interpolation and Approximation 116 free
- Structured matrices, reproducing kernels and interpolation 318
- A superfast algorithm for confluent rational tangential interpolation problem via matrix-vector multiplication for confluent Cauchy-like matrices 3146
- The maximal-volume concept in approximation by low-rank matrices 4762
- A matrix interpretation of the extended Euclidean algorithm 5368
- The essential polynomial approach to convergence of matrix Padé approximants 7186

- Part II. System Theory, Signal and Image Processing 89104
- Systems of low Hankel rank: A survey 91106
- Tensor approximation and signal processing applications 103118
- Exploiting Toeplitz-like structure in adaptive filtering algorithms using signal flow graphs 135150
- The structured total least squares problem 157172
- Exploiting Toeplitz structure in atmospheric image restoration 177192

- Part III. Control Theory 191206
- A survey of model reduction methods for large-scale systems 193208
- Theory and computations of some inverse eigenvalue problems for the quadratic pencil 221236
- Partial eigenvalue assignment for large linear control systems 241256
- A hybrid method for the numerical solution of discrete-time algebraic Riccati equations 255270

- Part IV. Spectral Properties. Conditioning 271286