Vadim Olshevsky

Fast Gaussian elimination with partial pivoting for matrices with displacement structure

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, 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 O(n 3 ) operations.

Complexity of multiplication with vectors for structured matrices

Abstract Fast algorithms for computing the product with 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 defined using the concept of displacement of a matrix) and also for their inverses. All the actions which are not dependent upon the coordinates of the input vector are singled out in a separate preprocessing stage. The proposed algorithms are based on new representations of these matrices, involving factor circulants.

Displacement structure approach to discrete-trigonometric-transform based preconditioners of G.Strang type and of T.Chan type

In this paper adisplacement structure technique is used to design a class of newpreconditioners for theconjugate 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 preconditioners belonging to any of 8 classes of matrices diagonalized by the correspondingdiscrete cosine or sine transforms. Under the standard Wiener class assumption theclustering property is established for all of these preconditioners, guaranteeing a rapid convergence of the preconditioned conjugate gradient method. The formulas for the G.Strang-type preconditioners have another important application: they suggest a wide variety of newO(m logm) algorithms for multiplication of a Toeplitz matrix by a vector, based on any of the 8 DCT’s and DST’s. Recentlytransformations of Toeplitz matrices to Vandermonde-like or Cauchy-like matrices have been found to be useful in developing accuratedirect methods for Toeplitz linear equations. Here it is suggested to further extend the range of the transformation approach by exploring it foriterative methods; this technique allowed us to reduce the complexity of each iteration of the preconditioned conjugate gradient method to 4 discrete transforms per iteration.

Fast Algorithms with Preprocessing for Matrix-Vector Multiplication Problems

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 of at most O(n log2n) flops and in a number of cases they 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 multiplication by an arbitrary vector. Effective algorithms for computing the Vandermonde determinant and the determination of a Cauchy matrix are given.

Circulants, displacements and decompositions of matrices

In this paper are suggested new formulas for representation of matrices and their inverses in the form of sums of products of factor circulants, which are based on the analysis of the factor cyclic displacement of matrices. The results in applications to Toeplitz matrices generalized the Gohberg-Semencul, Ben-Artzi-Shalom and Heinig-Rost formulas and are useful for complexity analysis.

Displacement-structure approach to polynomial Vandermonde and related matrices

Abstract We introduce a new class of what we call polynomial Vandermonde-like matrices. This class generalizes the polynomial Vandermonde matrices studied earlier by various authors, who derived explicit inversion formulas and fast algorithms for inversion and for solving the associated linear systems. A displacement-structure approach allows us to carry over all these results to the wider class of polynomial Vandermonde-like matrices.

Fast state space algorithms for matrix Nehari and Nehari-Takagi interpolation problems

Numerical algorithms with complexityO(n2) operations are proposed for solving matrix Nehari and Nehari-Takagi problems withn interpolation points. The algorithms are based on explicit formulas for the solutions and on theorems about cascade decomposition of rational matrix function given in a state space form. The method suggests also fast algorithms for LDU factorizations of structured matrices. The numerical behavior of the designed algorithms is studied for a wide set of examples.

The Fast Generalized Parker-Traub Algorithm for Inversion of Vandermonde and Related Matrices

In this paper we compare the numerical properties of the well-knownfastO(n2) Traub and Bjorck?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 onlyfastO(n2) inversion of a Vandermonde matrix, but it also gives moreaccuracy. We also show that the Parker?Traub algorithm is connected to the well-known concept ofdisplacement rank,introduced by Kailath, Kung, and Morf about two decades ago, and therefore this algorithm can be generalized to invert the more general class ofVandermonde-likematrices, naturally suggested by the idea of displacement.

A unified superfast algorithm for boundary rational tangential interpolation problems and for inversion and factorization of dense structured matrices

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. It is widely known that the now classical algorithm for solving this problem proposed by Nevanlinna in 1929 can be seen as a way of computing the Cholesky factorization for the corresponding Pick matrix. Moreover; the classical Nevanlinna algorithm takes advantage of the special structure of the Pick matrix to compute this triangular factorization in only O(n/sup 2/) arithmetic operations, where n is the number of interpolation points, or equivalently, the size of the Pick matrix. Since the structure-ignoring standard Cholesky algorithm [though applicable to the wider class of general matrices] has much higher complexity O(n/sup 3/), the Nevanlinna algorithm is an example of what is now called fast algorithms. In this paper we use a divide-and-conquer approach to propose a new superfast O(n log/sup 3/ n) algorithm to construct solutions for the more general boundary tangential Nevanlinna-Pick problem. This dramatic speed-up is achieved via a new divide-and-conquer algorithm for factorization of rational matrix functions; this superfast algorithm seems to have a practical and theoretical significance itself. It can be used to solve similar rational interpolation problems [e.g., the matrix Nehari problem], and a variety, of engineering problems. It can also be used for inversion and triangular factorization of matrices with displacement structure, including Hankel-like, Vandermonde-like, and Cauchy-like matrices.

Displacement structure approach to Chebyshev-Vandermonde and related matrices

In this paper we use the displacement structure concept to introduce a new class of matrices, designated asChebyshev-Vandermonde-like matrices, generalizing ordinary Chebyshev-Vandermonde matrices, studied earlier by different authors. Among other results the displacement structure approach allows us to give a nice explanation for the form of the Gohberg-Olshevsky formulas for the inverses of ordinary Chebyshev-Vandermonde matrices. Furthermore, the fact that the displacement structure is inherited by Schur complements leads to a fastO(n2) implementation of Gaussian elimination withpartial pivoting for Chebyshev-Vandermonde-like matrices.