Structured condition numbers for linear systems with parameterized quasiseparable coefficient matrices

Abstract

Low-rank structured matrices have attracted much attention in the last decades, since they arise in many applications and all share the fundamental property that can be represented by O(n) parameters, where n × n is the size of the matrix. This property has allowed the development of fast algorithms for solving numerically many problems involving low-rank structured matrices by performing operations on the parameters describing the matrices, instead of directly on the matrix entries. Among these problems, the solution of linear systems of equations is probably the most basic and relevant one. Therefore, it is important to measure, via structured computable condition numbers, the relative sensitivity of the solutions of linear systems with low-rank structured coefficient matrices with respect to relative perturbations of the parameters representing such matrices, since this sensitivity determines the maximum accuracy attainable by fast algorithms and allows us to decide which set of parameters is the most convenient from the point of view of accuracy. To develop and analyze such condition numbers is the main goal of this paper. To this purpose, a general expression is obtained for the condition number of the solution of a linear system of equations whose coefficient matrix is any differentiable function of a vector of parameters with respect to perturbations of such parameters. Since there are many different classes of low-rank structured matrices and many different types of parameters Partially supported by Ministerio de Economía y Competitividad of Spain through grants describing them, it is not possible to cover all of them in a single work. Therefore, the general expression of the condition number is particularized to the important case of {1, 1}-quasiseparable matrices and to the quasiseparable and the Givens-vector representations, in order to obtain explicit expressions of the corresponding two condition numbers that can be estimated in O(n) operations. In addition, detailed theoretical and numerical comparisons of these two condition numbers between themselves, and with respect to unstructured condition numbers, are provided, which show that there are situations in which the unstructured condition number is much larger than the structured ones, but that the opposite never happens. The approach presented in this manuscript can be generalized to other classes of low-rank structured matrices and parameterizations.

Structured condition number for multiple right-hand side linear systems with parameterized quasiseparable coefficient matrix

“…In the following theorem, we will study the relationship between K |Ω GV |,|Ω B | (A(Ω GV ), B(Ω B )) and K |Ω QS |,|Ω B | (A(Ω QS ), B(Ω B )). The similar conclusions had been obtained for eigenvalue, generalized eigenvalue computations and linear system solving for {1;1}-quasiseparable matrices in [11,16,17], respectively. Before that, we need to review Lemma 5.1, which describes the perturbation magnitude relationship between the Givens-vector representation via tangents given in Definition 2.3 and the quasiseparable representation given in Definition 2.1.…”

“…Moreover, the maximum rank of the lower and upper submatrix equal to 1. In this paper we adopt the representation of an n-by-n {1;1}-quasiseparable matrices with O(n) parameters instead of its n 2 entries; see more details in [16,17].…”

“…For structured linear systems, it is reasonable to investigate structured perturbations on the input data, because structure-preserving algorithms that preserve the underlying matrix structure can enhance the accuracy and efficiency of solving linear systems. The structured condition numbers for structured linear systems can be found in [16,27,38,39].…”

“…Recently, structured componentwise condition numbers for low-rank structured matrices have been introduced for eigenvalue problems [17], linear systems [16], and generalized eigenvalue problems [11] with parameterized quasiseparable matrices [43]. In this paper, we consider the structured componentwise condition numbers of the multiple right-hand side linear system (1.2) when the coefficient matrix A is a quasiseparable matrix.…”

“…The basic results of {1;1}-quasiseparable matrices are reviewed in Section 2. Different types of condition numbers for multiple right-hand sides linear systems with general parameterized coefficient matrices are investigated in Section 3, which can be used to derive explicit expressions for the condition number of multiple righthand side linear systems with {1;1}-quasiseparable coefficient matrix with respect to the quasiseparable representation [19] and the Givens-vector representation via tangent [16,17,43] in Section 4. We study relationships between different structured and unstructured condition numbers for multiple right-hand side linear systems with parameterized coefficient matrix in Section 5.…”