Pivoting for structured matrices and rational tangential interpolation

Abstract

Gaussian elimination is a standard tool for computing triangular factorizations for general matrices, and thereby solving associated linear systems of equations. As is well-known, when this classical method is implemented in finite-precision-arithmetic, it often fails to compute the solution accurately because of the accumulation of small roundoffs accompanying each elementary floating point operation. This problem motivated a number of interesting and important studies in modern numerical linear algebra; for our purposes in this paper we only mention that starting with the breakthrough work of Wilkinson, several pivoting techniques have been proposed to stabilize the numerical behavior of Gaussian elimination. Interestingly, matrix interpretations of many known and new algorithms for various applied problems can be seen as a way of computing triangular fac-torizations for the associated structured matrices, where different patterns of structure arise in the context of different physical problems. The special structure of such matrices [e.g., Toeplitz, Hankel, Cauchy, Vandermonde, etc.] often allows one to speed-up the computation of its triangular factorization, i.e., to efficiently obtain fast implementations of the Gaussian elimination procedure. There is a vast literature about such methods which are known under different names, e.g., fast Cholesky, fast Gaussian elimination, generalized Schur, or Schur-type algorithms. However, without further improvements they are efficient fast implementations of a numerically inaccurate [for indefinite matrices] method. In this paper we survey recent results on the fast implementation of various pivoting techniques which allowed us to improve numerical accuracy for a variety of fast algorithms. This approach led us to formulate new more accurate numerical methods for factorization of general and of J-unitary rational matrix functions, for solving various tangential interpolation problems, new Toeplitz-like and Toeplitz-plus-Hankel-like solvers, and new divided differences schemes. We beleive that similar methods can be used to design accurate fast algorithm for the other applied problems and a recent work of our colleagues supports this anticipation.

Tensor properties of multilevel Toeplitz and related matrices

“…If A = T 1 ⊗ · · · ⊗ T r is the tensor product of nonsingular Toeplitz matrices, then A −1 = T −1 1 ⊗ · · · ⊗ T −1 r is the tensor product of matrices with low displacement rank (the displacement rank of M is defined to be the rank of M − ZMZ , where M = [δ i,j +1 ] [14,12,17]). We may conjecture that the displacement ranks of the factors pertaining to A −1 remain low also in the case when A is a sum of tensor products of Toeplitz matrices.…”

Approximate factorization of multivariate polynomials using singular value decomposition

“…One approach, already mentioned in section 3.6, is to use sparse numerical interplolation (Giesbrecht et al, 2006) of the bi-or tri-variate factor images. Another is the use of fast structured solvers analogous to the theory of Toeplitz-like matrices (Pan, 2001;Olshevsky, 2003). For the univariate approximate GCD problem, results are reported in (Zhi, 2003;Li et al, 2005).…”

A note on the O(n)-storage implementation of the GKO algorithm and its adaptation to Trummer-like matrices

“…However, both computational practice and theoretical analysis suggest that the GKO algorithm is in practice a reliable algorithm [17]. Several strategies, such as the one proposed by Gu [10], exist in order to avoid generator growth, and they can be applied to both the original GKO algorithm and its space-efficient versions.…”