One of the best known fast computational algorithms is the fast Fourier transform method. Its efficiency is based mainly on the special structure of the discrete Fourier transform matrix. Recently, many other algorithms of this type were discovered, and the theory of structured matrices emerged.

This volume contains 22 survey and research papers devoted to a variety of theoretical and practical aspects of the design of fast algorithms for structured matrices and related issues. Included are several papers containing various affirmative and negative results in this direction. The theory of rational interpolation is one of the excellent sources providing intuition and methods to design fast algorithms. The volume contains several computational and theoretical papers on the topic. There are several papers on new applications of structured matrices, e.g., to the design of fast decoding algorithms, computing state-space realizations, relations to Lie algebras, unconstrained optimization, solving matrix equations, etc.

The book is suitable for mathematicians, engineers, and numerical analysts who design, study, and use fast computational algorithms based on the theory of structured matrices.

This volume is a joint publication of the American Mathematical Society and the Society for Industrial and Applied Mathematics.

Readership

Graduate students and research mathematicians interested in the theory and applications of fast numerical algorithms.

Table of Contents

• V. Olshevsky -- Pivoting for structured matrices and rational tangential interpolation
• G. Heinig -- Inversion of Toeplitz-plus-Hankel matrices with arbitrary rank profile
• D. Fasino and L. Gemignani -- A Lanczos-type algorithm for the QR factorization of Cauchy-like matrices
• D. Fasino, N. Mastronardi, and M. Van Barel -- Fast and stable algorithms for reducing diagonal plus semiseparable matrices to tridiagonal and bidiagonal form
• A. Olshevsky, V. Olshevsky, and J. Wang -- A comrade-matrix-based derivation of the eight versions of fast cosine and sine transforms
• D. A. Bini, L. Gemignani, and B. Meini -- Solving certain matrix equations by means of Toeplitz computations: algorithms and applications
• F. T. Luk and S. Qiao -- A fast singular value algorithm for Hankel matrices
• D. Calvetti, L. Reichel, and F. Sgallari -- A modified companion matrix method based on Newton polynomials
• J. Hendrickx, R. Vandebril, and M. Van Barel -- A fast direct method for solving the two-dimensional Helmholtz equation, with Robbins boundary conditions
• C. Di Fiore -- Structured matrices in unconstrained minimization methods
• N. Ito, W. Schmale, and H. K. Wimmer -- Computation of minimal state space realizations in Jacobson normal form
• A. Mayo -- High order accurate particular solutions of the biharmonic equation on general regions
• T. Wen, A. Edelman, and D. Gorsich -- A fast projected conjugate gradient algorithm for training support vector machines
• V. Olshevsky and M. A. Shokrollahi -- A displacement approach to decoding algebraic codes
• M. Bollhöfer and V. Mehrmann -- Some convergence estimates for algebraic multilevel preconditioners
• D. Noutsos, S. S. Capizzano, and P. Vassalos -- Spectral equivalence and matrix algebra preconditioners for multilevel Toeplitz systems: a negative result
• W. F. Trench -- Spectral distribution of Hermitian Toeplitz matrices formally generated by rational functions
• D. Fasino and S. S. Capizzano -- From Toeplitz matrix sequences to zero distribution of orthogonal polynomials
• K. R. Driessel -- On Lie algebras, submanifolds and structured matrices
• H. Dym -- Riccati equations and bitangential interpolation problems with singular Pick matrices
• V. Bolotnikov, A. Kheifets, and L. Rodman -- Functions with Pick matrices having bounded number of negative eigenvalues
• Yu. M. Arlinskiĭ, S. Hassi, H. S. V. de Snoo, and E. R. Tsekanovskiĭ -- One-dimensional perturbations of selfadjoint operators with finite or discrete spectrum