Abstract.Fast 0(n2) implementation of Gaussian elimination with partial pivoting is designed for matrices possessing Cauchy-like displacement structure. We show how Toeplitz-like, Toeplitz-plus-Hankel-like and Vandermondelike matrices can be transformed into Cauchy-like matrices by using Discrete Fourier, Cosine or Sine Transform matrices.In particular this allows us to propose a new fast 0{n2) Toeplitz solver GKO, which incorporates partial pivoting. A large set of numerical examples showed that GKO demonstrated stable numerical behavior and can be recommended for solving linear systems, especially with nonsymmetric, indefinite and ill-conditioned positive definite Toeplitz matrices. It is also useful for block Toeplitz and mosaic Toeplitz ( Toeplitz-block ) matrices.The algorithms proposed in this paper suggest an attractive alternative to look-ahead approaches, where one has to jump over ill-conditioned leading submatrices, which in the worst case requires 0(n3) operations.0. Introduction 0.1. Displacement structure. Let matrices F, A e Qnxn be given. Let R £ Cnx" be a matrix satisfying a Sylvester-type equation (0.1) V{F,A}(R) = F-R-R-A = G-B, with some rectangular matrices G £ C"Xa, B £ Cax" , where the number a is small in comparison with « . The pair of matrices G, B in (0.1 ) is referred to as a {F, A}-generator of R and the smallest possible inner size a among all {F, ^-generators is called the {F, A}-displacement rank of R . The concept of displacement structure was first introduced in [21] using the Stein-type displacement operator V{f,a}(')' Cnxn -► Qnxn given byReceived by the editor August 4, 1994 and, in revised form, November 17, 1994 1991 Mathematics Subject Classification. Primary 15A06, 15A09, 15A23, 15A57, 65F05, 65H10.Key words and phrases. Gaussian elimination, partial pivoting, displacement structure, Toeplitzlike, Cauchy-like, Vandermonde-like, Toeplitz-plus-Hankel matrix, Schur algorithm, Levinson algorithm.The research of the third author was supported in part by the Army Research Office under Grant DAAH04-93-G-0029. This manuscript is submitted for publication with the understanding that the U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon. ©1995 American Mathematical Society 1557License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1558 I. gohberg, t. kailath, and v. olshevsky The variant (0.1 ) of displacement equation appeared later in [ 19]. The left-hand sides of (0.2) and (0.1) are often referred to as Toeplitz-like and Hankel-like displacement operators, respectively. The most general form of displacement structure, which clearly includes (0.1) and (0.2), was introduced in [24] using the operator (0.3) V{n,A,F,A}(R) = n-F-A*-F.R-A*.A standard Gaussian elimination scheme applied for triangular factorization of R would require 0(«3) operations. At the same time, displacement structure allows us to speed-up the triangular factorization of ...

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Fast O(n 2) implementation of Gaussian elimination with partial pivoting is designed for matrices possessing Cauchy-like displacement structure. We show how Toeplitz{ like, Toeplitz-plus-Hankel{like and Vandermonde-like matrices can be transformed into Cauchy{like matrices by using Discrete Fourier, Cosine or Sine Transform matrices. In particular this allows us to propose a new fast O(n 2) Toeplitz solver GKO, which incorporates partial pivoting. A large set of numerical examples showed that GKO demonstrated stable numerical behavior and can be recommended for solving linear systems, especially with nonsymmetric, inde nite and ill-conditioned positive de nite Toeplitz matrices. It is also useful for block Toeplitz and mosaic Toeplitz (Toeplitz{block) matrices. The algorithms proposed in this paper suggest an attractive alternative to lookahead approaches, where one has to jump over ill-conditioned leading submatrices, which in the worse case requires O(n 3) operations.

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 known estimates. In these algorithms, in a separate preprocessing phase, are singled out all the actions on the preparation of a given matrix, which aimed at the reduction of the complexity of the second stage of computations directly connected with the multiplication by an arbitrary vector. Incidentally, the e ective algorithms for computing the Vandermonde determinant and the determinant of a Cauchy matrix, are given.

The QR iteration method for tridiagonal matrices is in the heart of one classical method to solve the general eigenvalue problem. In this paper we consider the more general class of quasiseparable matrices that includes not only tridiagonal but also companion, comrade, unitary Hessenberg and semiseparble matrices. A fast QR iteration method exploiting the Hermitian quasiseparable structure (and thus generalizing the classical tridiagonal scheme) is presented. The algorithm is based on an earlier work [Y. Eidelman and I. Gohberg, A modification of the Dewilde-van der Veen method for inversion of finite structured matrices, Linear Algebra Appl. 343-344 (2002) 419-450], and it applies to the general case of Hermitian quasiseparable matrices of an arbitrary order.The algorithm operates on generators (i.e., a linear set of parameters defining the quasiseparable matrix), and the storage and the cost of one iteration are only linear. The results of some numerical experiments are presented. 306 Y. Eidelman et al. / Linear Algebra and its Applications 404 (2005) 305-324An application of this method to solve the general eigenvalue problem via quasiseparable matrices will be analyzed separately elsewhere.