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Fast On 2 implementation of Gaussian elimination with partial pivoting is designed for matrices possessing Cauchy-like displacement structure. We show how Toeplitzz like, Toeplitz-plus-Hankelllike and Vandermonde-like matrices can be transformed into Cauchyylike matrices by using Discrete Fourier, Cosine or Sine Transform matrices. In particular this allows… (More)

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)

- I Gohberg, V Olshevsky, Comp
- 1994

Fast algorithms for computing the product by a vector are presented for a number of classes of matrices whose properties relate to the properties of Toeplitz, Vandermonde or Cauchy matrices (these matrices are deened using the concept of displacement of a matrix) and also for their inverses. All the actions which are not dependent upon the coordinates of… (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)

- Vadim Olshevsky, M. Amin Shokrollahi
- 2003

Algebraic coding theory is one of the areas that routinely gives rise to computational problems involving various structured matrices, such as Hankel, Vandermonde, Cauchy matrices, and certain generalizations thereof. Their structure has often been used to derive efficient algorithms; however, the use of the structure was pattern-specific, without applying… (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)

- Numerische Mathematik, I Gohberg, V Olshevsky
- 1994

This paper contains two fast algorithms for inversion of Chebyshev{ Vandermonde matrices of the rst and second kind. They are based on special representations of the Bezoutians of Chebyshev polynomials of both kinds. The paper also contains the results of numerical experiments which show that the algorithms proposed here are not only much faster, but also… (More)

In an earlier paper GKO95 w e exploited the displacement structure of Cauchy-like matrices to derive for them a fast On 2 implementation of Gaussian elimination with partial pivoting. One application is to the rapid and numerically accurate solution of linear systems with Toeplitz-like coeecient matrices, based on the fact that the latter can be transformed… (More)

In this paper we carry over the Björck-Pereyra algorithm for solving Vandermonde linear systems to what we suggest to call Szegö-Vandermonde systems V Φ (x), i.e., polynomial-Vandermonde systems where the corresponding polynomial system Φ is the Szegö polyno-mials. The properties of the corresponding unitary Hessenberg matrix allow us to derive a fast O(n… (More)

" Life as we know it would considerably different if, from 1965 Cooley-Turkey paper onwards the FFT community has made systematic and heavy use of matrix-vector notation. Indeed, couching results and algorithms in matrix-vector notation the FFT literature can be unified and made more understandable to the outsider. " Charles Van Loan, " Computational… (More)