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In this paper the problem of complexity of multiplication of a matrix with a vector is studied for Toeplitz, Hankel, Vandermonde and Cauchy matrices and for matrices connected with them (i.e. for transpose, inverse and transpose to inverse matrices). The proposed algorithms have complexities at most O(n log 2 n) ops and in a number of cases improve the… (More)

Using methods originating in numerical analysis, we will develop a unified framework for derivation of efficient list decoding algorithms for algebraic-geometric codes. We will demonstrate our method by accelerating Sudan's list decoding algorithm for Reed-Solomon codes [22], its generalization to algebraic-geometric codes by Shokrollahi and Wasserman [21],… (More)

In this paper a displacement structure technique is used to design a class of new precondition-ers for the conjugate gradient method applied to the solution of large Toeplitz linear equations. Explicit formulas are suggested for the G.Strang-type and for the T.Chan-type precondition-ers belonging to any of 8 classes of matrices diagonalized by the… (More)

In this paper we survey several recent results that highlight an interplay between a relatively new class of quasiseparable matrices and polynomials. Quasiseparable matrices generalize two classical matrix classes, Jacobi (tridiagonal) matrices and unitary Hessenberg matrices that are known to correspond to real orthogonal polynomials and Szegö polynomials,… (More)

The classical scalar Nevanlinna-Pick interpolation problem has a long and distinguished history, appearing in a variety of applications in mathematics and electrical engineering. There is a vast literature on this problem and on its various far reaching generalizations; for a quick historical survey see [1] and [38]. It is widely known that the now… (More)

Polynomial and rational interpolation and multipoint evaluation are classical subjects, which remain central for the theory and practice of algebraic and numerical computing and have extensive applications to sciences, engineering and signal and image processing. In spite of long and intensive study of these subjects, several major problems remained open.… (More)

In this paper we compare the numerical properties of the well-known fast O(n 2) Traub and Bjj orck-Pereyra algorithms, which both use the special structure of a Vandermonde matrix to rapidly compute the entries of its inverse. The results of numerical experiments suggest that the Parker variant of what we shall call the Parker-Traub algorithm allows one not… (More)

Many important problems in pure and applied mathematics and engineering can be reduced to linear algebra on dense structured matrices. The structure of these dense matrices is understood in the sense that their n z entries can be "compressed" to a smaller number O(n) of parameters. Operating directly on these paxameters allows one to design efficient fast… (More)

In this paper we derive a fast O(n 2) algorithm for solving linear systems where the coefficient matrix is a polynomial-Vandermonde matrix V R (x) = [r j−1 (x i)] with polynomials {r k (x)} related to a Hessenberg quasiseparable matrix. The result generalizes the well-known Björck-Pereyra algorithm for classical Vandermonde systems involving monomials. It… (More)