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.
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978-0-8218-2092-6 (9780821820926)
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Fast algorithms: The Schur algorithm for matrices with Hessenberg displacement structure by G. Heinig and V. Olshevsky Fast inversion algorithms for a class of block structured matrices by Y. Eidelman and I. Gohberg A fast and stable solver for recursively semi-separable systems of linear equations by S. Chandrasekaran and M. Gu Numerical issues: Stability properties of several variants of the unitary Hessenberg $QR$ algorithm by M. Stewart Comparison of algorithms for Toeplitz least squares and symmetric positive definite linear systems by M. Kim, H. Park, and L. Elden Stability of Toeplitz matrix inversion formulas by G. Heinig Necessary and sufficient conditions for accurate and efficient rational function evaluation and factorizations of rational matrices by J. Demmel and P. Koev Updating and downdating of orthonormal polynomial vectors and some applications by M. Van Barel and A. Bultheel Rank-revealing decompositions of symmetric Toeplitz matrices by P. C. Hansen and P. Yalamov Iterative methods. Preconditioners: A survey of preconditioners for ill-conditioned Toeplitz systems by R. H. Chan, M. K. Ng, and A. M. Yip Preconditioning of Hermitian block-Toeplitz-Toeplitz-block matrices by level-1 preconditioners by D. Potts and G. Steidl Linear algebra and various applications: Approximate displacement rank and applications by D. A. Bini and B. Meini Properties of some generalizations of Kac-Murdock-Szego matrices by W. F. Trench Efficient inversion formulas for Toeplitz-plus-Hankel matrices using trigonometric transformations by G. Heinig and K. Rost On a generalization of Poincare's theorem for matrix difference equations arising from root-finding problems by L. Gemignani Completions of triangular matrices: A survey of results and open problems by L. Rodman Positive representation formulas for finite difference discretizations of (elliptic) second order PDEs by S. S. Capizzano and C. T. Possio On some problems involving invariant norms and Hadamard products by P. Tilli A generalization of the Perron-Frobenius theorem for non-linear perturbations of Stiltjes matrices by Y. S. Choi, I. Koltracht, and P. J. McKenna The Rhombus matrix: Definition and properties by M. J. C. Gover and A. M. Byrne.
