Titlebook: Structured Matrices and Polynomials; Unified Superfast Al Victor Y. Pan Textbook 2001 Birkhäuser Boston 2001 Interpolation.Matrix.algebra.a

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, 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 displa

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Structured Matrices and Displacement Operators,ower of the approach based on the displacement transformations of two kinds that extend successful algorithms from one class of structured matrices to various other classes. We also briefly comment on parallel implementation of computations with structured matrices.

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Computations with Structured Matrices. Introduction,covered in some technical detail in Sections 1.2-1.5. The rest of the chapter is a non-technical but extensive introduction to unified superfast algorithms for computation with structured matrices. Further elaboration and technical details are given in Chapters 4–7.

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Matrix Structures of Vandermonde and Cauchy Types and Polynomial and Rational Computations,he related computations with matrices of Vandermonde and Cauchy types. We also show applications to loss-resilient encoding/decoding and to diagonalization of matrix algebras, which leads us to the discrete sine and cosine transforms. Figures 3.3– 3.6 summarize correlation among computational problems in these areas.

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Conclusion,e matrices . Subsequently, various effective algorithms were developed along this line for several other classes of structured matrices as well. This included superfast algorithms, running in nearly linear time, up to (poly)logarithmic factors and using small (linear) memory space.

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Toeplitz/Hankel Matrix Structure and Polynomial Computations,n and the estimates for the arithmetic cost of multiplying Toeplitz, Hankel, Vandermonde, and Cauchy matrices by vectors, the results of these two chapters are little used in Chapters 4–7. Some sample pseudocodes for the algorithms of Sections 2.4 and 2.5 are collected in Section 2.15.

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Newton-Structured Numerical Iteration,e reader to some open problems in the ongoing research. More than in other chapters, the presentation omits proofs, derivations, and other technical details, which we replace by references to bibliography.