Structured eigenvalue condition numbers for parameterized quasiseparable matrices

Abstract

The development of fast algorithms for performing computations with n × n low-rank structured matrices has been a very active area of research during the last two decades, as a consequence of the numerous applications where these matrices arise. The key ideas behind these fast algorithms are that low-rank structured matrices can be described in terms of O(n) parameters and that these algorithms operate on the parameters instead on the matrix entries. Therefore, the sensitivity of any computed quantity should be measured with respect to the possible variations that the parameters dening these matrices may suer, since this determines the maximum accuracy of a given fast computation. In other words, it is necessary to develop condition numbers with respect to parameters for dierent magnitudes and classes of low-rank structured matrices, but, as far as we know, this has not yet been accomplished in any case. In this paper, we derive structured relative eigenvalue condition numbers for the important class of low-rank structured matrices known as {1; 1}-quasiseparable matrices with respect to relative perturbations of the parameters in the quasiseparable and in the Givens-vector representations of these matrices, and we provide fast algorithms for computing them. Comparisons among the new structured condition numbers and the unstructured one are also presented, as well as numerical experiments showing that the structured condition numbers can be small in situations where the unstructured one is huge. In addition, the approach presented in this paper is general and may be extended to other problems and classes of low-rank structured matrices.

Structured condition numbers for linear systems with parameterized quasiseparable coefficient matrices

“…This useful property is stated in the following proposition. Since many interesting classes of matrices can be represented by sets of parameters different from their entries (see Theorem 4.1, for example), we generalize the definitions above to these representations and, following the ideas in [5], we will focus on componentwise relative condition numbers for representations.…”

“…Such application is to determine which representation is better to use, from the point of view of accuracy, for developing a fast algorithm for solving a linear system, since the most sensible choice is the one with smallest condition number. In this work, we will prove that, as it happens in the case of the eigenvalues [5], the condition number with respect to any quasiseparable representation is the same, and cannot be too much larger than the condition number with respect to the Givens-vector representation, which is always the smallest. Moreover, we will show how these two condition numbers can be reliably estimated in O(n) flops.…”

“…This situation makes necessary the development of structured condition numbers with respect to the parameters on which fast algorithms operate and of reliable methods to estimate a posteriori the backward errors on such parameters from the residuals of the computed outputs of the algorithms, because in this way the forward errors committed by fast algorithms for low-rank structured matrices may be reliably and optimally estimated a posteriori as the product of the structured condition numbers times the backward errors on the parameters. This is a challenging research plan that has been initiated recently in [5], where for the first time in the literature some structured condition numbers (for eigenvalues in that case) with respect to some parameterizations of a certain class of low-rank structured matrices were introduced and analyzed. Among many other results, it was proved in [5] that simple eigenvalues of {1, 1}-quasiseparable matrices [6,16] may be much less sensitive to perturbations of the parameters describing the matrices than to perturbations of the entries of the matrix.…”

“…This is a challenging research plan that has been initiated recently in [5], where for the first time in the literature some structured condition numbers (for eigenvalues in that case) with respect to some parameterizations of a certain class of low-rank structured matrices were introduced and analyzed. Among many other results, it was proved in [5] that simple eigenvalues of {1, 1}-quasiseparable matrices [6,16] may be much less sensitive to perturbations of the parameters describing the matrices than to perturbations of the entries of the matrix. In this paper, we extend the development of structured condition numbers with respect to perturbations of the parameters to the fundamental case of the solutions of linear systems of equations with low-rank structured coefficient matrices, and we will prove that, also in this case, the solutions may be much better conditioned with respect to perturbations of the parameters than with respect to perturbations of the entries of the matrix, but that the opposite can not happen.…”

“…In this paper, we extend the development of structured condition numbers with respect to perturbations of the parameters to the fundamental case of the solutions of linear systems of equations with low-rank structured coefficient matrices, and we will prove that, also in this case, the solutions may be much better conditioned with respect to perturbations of the parameters than with respect to perturbations of the entries of the matrix, but that the opposite can not happen. This work is influenced by the recent references [5,11], which deal with the sensitivity of eigenvalues of some low-rank structured matrices, but also by the classical reference [13], in which the use of differential calculus for developing condition numbers was introduced.…”