Unstructured Condition Number Research Articles

Structured conditioning theory for the total least squares problem with linear equality constraint and their estimation

<abstract><p>This article is devoted to the structured and unstructured condition numbers for the total least squares with linear equality constraint (TLSE) problem. By making use of the dual techniques, we investigate three distinct kinds of unstructured condition numbers for a linear function of the TLSE solution and three structured condition numbers for this problem, i.e., normwise, mixed, and componentwise ones, and present their explicit expressions under both unstructured and structured componentwise perturbations. In addition, the relations between structured and unstructured normwise, componentwise, and mixed condition numbers for the TLSE problem are investigated. Furthermore, using the small-sample statistical condition estimation method, we also consider the statistical estimation of both unstructured and structured condition numbers and propose three algorithms. Theoretical and experimental results show that structured condition numbers are always smaller than the corresponding unstructured condition numbers.</p></abstract>

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

In this paper, we consider the structured perturbation analysis for multiple right-hand side linear systems with parameterized coefficient matrix. Especially, we present the explicit expressions for structured condition numbers for multiple right-hand sides linear systems with {1;1}-quasiseparable coefficient matrix in the quasiseparable and the Givens-vector representations. In addition, the comparisons of these two condition numbers between themselves, and with respect to unstructured condition number are investigated.Moreover, the effective structured condition number for multiple right-hand sides linear systems with {1;1}-quasiseparable coefficient matrix is proposed. The relationships between the effective structured condition number and structured condition numbers with respect to the quasiseparable and the Givens-vector representations are also studied.Numerical experiments show that there are situations in which the effective structured condition number can be much smaller than the unstructured ones.

Structured generalized eigenvalue condition numbers for parameterized quasiseparable matrices

In this paper, when A and B are {1;1}-quasiseparable matrices, we consider the structured generalized relative eigenvalue condition numbers of the pair $$(A, \, B)$$ with respect to relative perturbations of the parameters defining A and B in the quasiseparable and the Givens-vector representations of these matrices. A general expression is derived for the condition number of the generalized eigenvalue problems of the pair $$(A,\, B)$$ , where A and B are any differentiable function of a vector of parameters with respect to perturbations of such parameters. Moreover, the explicit expressions of the corresponding structured condition numbers with respect to the quasiseparable and Givens-vector representation via tangents for $$\{1; 1\}$$ -quasiseparable matrices are derived. Our proposed condition numbers can be computed efficiently by utilizing the recursive structure of quasiseparable matrices. We investigate relationships between various condition numbers of structured generalized eigenvalue problem when A and B are {1;1}-quasiseparable matrices. Numerical results show that there are situations in which the unstructured condition number can be much larger than the structured ones.

The Structured Condition Number of a Differentiable Map between Matrix Manifolds, with Applications

We study the structured condition number of differentiable maps between smooth matrix manifolds, developing a theoretical framework that extends previous results for vector subspaces to any smooth manifold. We present algorithms to compute the structured condition number. As special cases of smooth manifolds, we analyze automorphism groups, and Lie and Jordan algebras associated with a scalar product. For such manifolds, we derive a lower bound on the structured condition number that is cheaper to compute than the structured condition number. We provide numerical comparisons between the structured and unstructured condition numbers for the principal matrix logarithm and principal matrix square root of matrices in automorphism groups as well as for the map between matrices in automorphism groups and their polar decomposition. We show that our lower bound can be used as a good estimate for the structured condition number when the matrix argument is well conditioned. We show that the structured and unstructured condition numbers can differ by many orders of magnitude, thus motivating the development of algorithms preserving structure.

Structured Condition Numbers for the Tikhonov Regularization of Discrete Ill-Posed Problems

The possibly most popular regularization method for solving the least squares problem min x parallel to Ax - b parallel to(2) with a highly ill-conditioned or rank deficient coefficient matrix A is the Tikhonov regularization method. In this paper we present the explicit expressions of the normwise, mixed and component wise condition numbers for the Tikhonov regularization when A has linear structures. The structured condition numbers in the special cases of nonlinear structure i.e. Vandermonde and Cauchy matrices are also considered. Some comparisons between structured condition numbers and unstructured condition numbers are made by numerical experiments. In addition, we also derive the normwise, mixed and componentwise condition numbers for the Tikhonov regularization when the coefficient matrix, regularization matrix and right-hand side vector are all perturbed, which generalize the results obtained by Chu et al.

Structured condition numbers for linear systems with parameterized quasiseparable coefficient matrices

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)$\mathcal {O}(n)$ 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 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.

First order structured perturbation theory for multiple zero eigenvalues of skew-adjoint matrices

Given a matrix belonging to some class of structured matrices, we consider the question of comparing the sensitivity of its eigenvalues under two different kinds of perturbations, either unstructured (i.e., arbitrary) or structured (i.e., those belonging to the same class of matrices as the unperturbed one). In a previous paper (Kressner et al., 2009 [13]), the authors compared the structured and unstructured condition numbers of (possibly multiple) eigenvalues for several different matrix and pencil structures. Only one case was left out of the analysis, namely the one where the asymptotic order of perturbed eigenvalues under structured perturbations is different from the asymptotic order under unstructured ones. This is precisely the case we consider in the present paper: given a matrix which is skew-adjoint with respect to a symmetric scalar product and has a zero eigenvalue with a certain Jordan structure, first order expansions are obtained for the perturbed eigenvalues under structured perturbation, as well as bounds on the structured condition number. Similar results are obtained for structured perturbations of symmetric/skew-symmetric and palindromic matrix pencils.

Structured Condition Numbers for Invariant Subspaces

Invariant subspaces of structured matrices are sometimes better conditioned with respect to structured perturbations than with respect to general perturbations. Sometimes they are not. This paper proposes an appropriate condition number $c_{\S}$, for invariant subspaces subject to structured perturbations. Several examples compare $c_{\S}$ with the unstructured condition number. The examples include block cyclic, Hamiltonian, and orthogonal matrices. This approach extends naturally to structured generalized eigenvalue problems such as palindromic matrix pencils.

Structured condition numbers and backward errors in scalar product spaces

The effect of structure-preserving perturbations on the solution to a linear system, matrix inversion, and distance to singularity is investigated. Particular attention is paid tolinear and nonlinear structures that form Lie algebras, Jordan algebras and automorphism groups of ascalar product. These include complex symmetric, pseudo-symmetric, persymmetric, skewsymmetric, Hamiltonian, unitary, complex orthogonal and symplectic matrices. Under reasonable assumptions on the scalar product, it is shown that there is little or no difference between structured and unstructured condition numbers and distance to singularity for matrices in Lie and Jordan algebras. Hence, for these classes of matrices, the usual unstructured perturbation analysis is sufficient. It is shown that this is not true in general for structures in automorphism groups. Bounds and computable expressions for the structured condition numbers for a linear system and matrix inversion are derived for these nonlinear structures.Structured backward errors for the approximate solution of linear systems are also considered. Conditions are given for the structured backward error to be finite. For Lie and Jordan algebras it is proved that, whenever the structured backward error is finite, it is within a small factor of or equal to the unstructured one. The same conclusion holds for orthogonal and unitary structures but cannot easily be extended to other matrix groups. This work extends and unifies earlier analyses.

Eigenvalues, pseudospectrum and structured perturbations

We investigate the behavior of eigenvalues under structured perturbations. We show that for many common structures such as (complex) symmetric, Toeplitz, symmetric Toeplitz, circulant and others the structured condition number is equal to the unstructured condition number for normwise perturbations, and prove similar results for real perturbations. An exception are complex skewsymmetric matrices. We also investigate componentwise complex and real perturbations. Here Hermitian and skew-Hermitian matrices are exceptional for real perturbations. Furthermore we characterize the structured (complex and real) pseudospectrum for a number of structures and show that often there is little or no significant difference to the usual, unstructured pseudospectrum.

Structured perturbations of group inverse and singular linear system with index one

In this paper, we study the normwise perturbation theory of singular linear structured system with index one. The structures under investigation are Toeplitz, circulant, Hankel matrices. We show that the structured condition number is smaller than unstructured condition number. For specific right-hand side, we present the condition number for structured system.

Structured perturbations of Drazin inverse

In this paper the normwise perturbation theory of the Drazin inverse is investigated. We study the following structure matrices: Toeplitz, Hankel matrices. When we take the structure of the matrix, we get that the structured condition number is small compared with unstructured condition number. For specific right hand side we give the estimate and formula of the condition number for structured system.

Structured Perturbations Part I: Normwise Distances

In this paper we study the condition number of linear systems, the condition number of matrix inversion, and the distance to the nearest singular matrix, all problems with respect to normwise structured perturbations. The structures under investigation are symmetric, persymmetric, skewsymmetric, symmetric Toeplitz, general Toeplitz, circulant, Hankel, and persymmetric Hankel matrices (some results on other structures such as tridiagonal and tridiagonal Toeplitz matrices, both symmetric and general, are presented as well). We show that for a given matrix the worst case structured condition number for all right-hand sides is equal to the unstructured condition number. For a specific right-hand side we give various explicit formulas and estimations for the condition numbers for linear systems, especially for the ratio of the condition numbers with respect to structured and unstructured perturbations. Moreover, the condition number of matrix inversion is shown to be the same for structured and unstructured perturbations, and the same is proved for the distance to the nearest singular matrix. It follows a generalization of the classical Eckart--Young theorem, namely, that the reciprocal of the condition number is equal to the distance to the nearest singular matrix for all structured perturbations mentioned above.