Huaian Diao

On Drazin inverse of singular Toeplitz matrix

In this paper we show that the Drazin inverse of singular Toeplitz matrix can be represented as a sum of products of lower and upper triangular Toeplitz matrices and give the explicit expression, which generalizes the well-known result by Gohberg and Semencul regarding the inverse of Toeplitz matrix.

Mixed, componentwise condition numbers and small sample statistical condition estimation of Sylvester equations

SUMMARY We present a componentwise perturbation analysis for the continuous-time Sylvester equations. Componentwise, mixed condition numbers and new perturbation bounds are derived for the matrix equations. The small sample statistical method can also be applied for the condition estimation. These condition numbers and perturbation bounds are tested on numerical examples and compared with the normwise condition number. The numerical examples illustrate that the mixed condition number gives sharper bounds than the normwise one. Copyright

On condition numbers for Moore-Penrose inverse and linear least squares problem involving Kronecker products

SUMMARY In this paper, we investigate the normwise, mixed, and componentwise condition numbers and their upper bounds for the Moore–Penrose inverse of the Kronecker product and more general matrix function compositions involving Kronecker products. We also present the condition numbers and their upper bounds for the associated Kronecker product linear least squares solution with full column rank. In practice, the derived upper bounds for the mixed and componentwise condition numbers for Kronecker product linear least squares solution can be efficiently estimated using the Hager–Higham Algorithm. Copyright

Structured condition numbers of structured Tikhonov regularization problem and their estimations

Both structured componentwise and structured normwise perturbation analysis of the Tikhonov regularization are presented. The structured matrices under consideration include: Toeplitz, Hankel, Vandermonde, and Cauchy matrices. Structured normwise, mixed and componentwise condition numbers for the Tikhonov regularization are introduced and their explicit expressions are derived. For the general linear structure, based on the derived expressions, we prove structured condition numbers are smaller than their corresponding unstructured counterparts. By means of the power method and small sample statistical condition estimation (SCE), fast condition estimation algorithms are proposed. Our estimation methods can be integrated into Tikhonov regularization algorithms that use the generalized singular value decomposition (GSVD). For large scale linear structured Tikhonov regularization problems, we show how to incorporate the SCE into the preconditioned conjugate gradient (PCG) method to get the posterior error estimations. The structured condition numbers and perturbation bounds are tested on some numerical examples and compared with their unstructured counterparts. Our numerical examples demonstrate that the structured mixed condition numbers give sharper perturbation bounds than existing ones, and the proposed condition estimation algorithms are reliable. Also, an image restoration example is tested to show the effectiveness of the SCE for large scale linear structured Tikhonov regularization problems.

Condition numbers for the outer inverse and constrained singular linear system

In this paper, we investigate the condition number of the outer inverse AT,S(2) and outer inverse AT,S(2) solution of constrained linear system Ax = b, x ∈ T, where A is a real m × n matrix, b and x are real vectors, T is a subspace. Let α and β be two positive real numbers, when we consider the weighted Frobenius norm ∥[αA,βb]∥QP(F) on the data we get the formula of condition number for the outer inverse AT,S(2) solution of constrained linear system. For the normwise condition number, the sensitivity of the relative condition number itself is studied, and the componentwise perturbation is also investigated.

A condition analysis of the weighted linear least squares problem using dual norms

Abstract In this paper, based on the theory of adjoint operators and dual norms, we define condition numbers for a linear solution function of the weighted linear least squares problem. The explicit expressions of the normwise and componentwise condition numbers derived in this paper can be computed at low cost when the dimension of the linear function is low due to dual operator theory. Moreover, we use the augmented system to perform a componentwise perturbation analysis of the solution and residual of the weighted linear least squares problems. We also propose two efficient condition number estimators. Our numerical experiments demonstrate that our condition numbers give accurate perturbation bounds and can reveal the conditioning of individual components of the solution. Our condition number estimators are accurate as well as efficient.

On componentwise condition numbers for eigenvalue problems with structured matrices

We give explicit expressions for the componentwise condition number for eigenvalue problems with structured matrices. We will consider only linear structures and show a general result from which expressions for the condition numbers follow. We obtain explicit expressions for the following structures: Toeplitz and Hankel. Details for other linear structures should follow in a straightforward manner from our general result. Copyright

On structured componentwise condition numbers for Hamiltonian eigenvalue problems

Abstract In this paper, we introduce the structured componentwise perturbation analysis to Hamiltonian eigenvalue problems. The explicit expressions for the relative structured componentwise condition number for Hamiltonian eigenvalue problems are derived. Also, we will compare the relative structured componentwise condition number with the relative unstructured componentwise condition number and the relative structured normwise condition number, respectively. We show the superiority of the relative structured componentwise condition number through numerical examples.

Condition numbers for a linear function of the solution of the linear least squares problem with equality constraints

Abstract In this paper, we consider the normwise, mixed and componentwise condition numbers for a linear function L x of the solution x to the linear least squares problem with equality constraints (LSE). The explicit expressions of the normwise, mixed and componentwise condition numbers are derived. Also, we revisit some previous results on the condition numbers of linear least squares problem (LS) and LSE. It is shown that some previous explicit condition number expressions on LS and LSE can be recovered from our new derived condition numbers’ formulas. The sharp upper bounds for the derived normwise, mixed and componentwise condition numbers are obtained, which can be estimated efficiently by means of the classical Hager–Higham algorithm for estimating matrix one-norm. Moreover, the proposed condition estimation methods can be incorporated into the generalized QR factorization method for solving LSE. The numerical examples show that when the coefficient matrices of LSE are sparse and badly-scaled, the mixed and componentwise condition numbers can give sharp perturbation bounds, on the other hand normwise condition numbers can severely overestimate the exact relative errors because normwise condition numbers ignore the data sparsity and scaling. However, from the numerical experiments for random LSE problems, if the data is not either sparse or badly scaled, it is more suitable to adopt the normwise condition number to measure the conditioning of LSE since the explicit formula of the normwise condition number is more compact.