Given two bounded linear operators F , G on a Banach space X such that G 2 F = G F 2 = 0, we derive an explicit expression for the Drazin inverse of F + G. For this purpose, firstly, we obtain a formula for the resolvent of an auxiliary operator matrix in the form M = F I G F G . From the provided representation of (F + G) D several special cases are considered. In particular, we recover the case G F = 0 studied by Hartwig et al. [R.E. Hartwig, G. Wang, Y. Wei, Some additive results on Drazin inverse, Linear Algebra Appl. 322 (2001) 207-217] for matrices and by Djordjević and Wei [D.S. Djordjević, Y. Wei, Additive results for the generalized Drazin inverse, J. Aust. Math. Soc. 73 (1) (2002) 115-126] for operators. Finally, we apply our results to obtain representations for the Drazin inverse of operator matrices in the form M = A B C D which are extensions of some cases given in the literature.

No abstract

Abstract. Least squares problems minx b − Ax 2 where the matrix A ∈ C m×n (m ≥ n) has some particular structure arise frequently in applications. Polynomial data fitting is a well-known instance of problems that yield highly structured matrices, but many other examples exist. Very often, structured matrices have huge condition numbers κ 2 (A) = A 2 A † 2 (A † is the Moore-Penrose pseudoinverse of A) and therefore standard algorithms fail to compute accurate minimum 2-norm solutions of least squares problems. In this work, we introduce a framework that allows us to compute minimum 2-norm solutions of many classes of structured least squares problems accurately, i.e., with errors x 0 − x 0 2 / x 0 2 = O(u), where u is the unit roundoff, independently of the magnitude of κ 2 (A) for most vectors b. The cost of these accurate computations is O(n 2 m) flops, i.e., roughly the same cost as standard algorithms for least squares problems. The approach in this work relies in computing first an accurate rank-revealing decomposition of A, an idea that has been widely used in recent decades to compute, for structured ill-conditioned matrices, singular value decompositions, eigenvalues, and eigenvectors in the Hermitian case and solutions of linear systems with high relative accuracy. In order to prove that accurate solutions are computed, a new multiplicative perturbation theory of the least squares problem is needed. The results presented in this paper are valid for both full rank and rank deficient problems and also in the case of underdetermined linear systems (m < n). Among other types of matrices, the new method applies to rectangular Cauchy, Vandermonde, and graded matrices, and detailed numerical tests for Cauchy matrices are presented. 1. Introduction. Structured matrices arise frequently in theory and applications [44,45]. As a consequence, the design and analysis of special algorithms for performing structured matrix computations is a classical area of numerical linear algebra that attracts the attention of many researchers. Special algorithms for solving structured linear systems of equations or structured eigenvalue problems are included in many standard references [15,24,29,35,51], but special algorithms for solving structured least squares (LS) problems do not appear so often in the literature. The goal of special algorithms is to exploit the structure to increase the speed of computations, and/or to decrease storage requirements, and/or to improve the accuracy of the solutions with respect to standard algorithms. On this latter goal, let us mention that algorithms for solving structured linear systems of equations more accurately than standard methods have been developed from the early days of numerical linear

No abstract

We study properties of the Drazin index of regular elements in a ring with a unity 1. We give expressions for generalized inverses of 1 − ba in terms of generalized inverses of 1 − ab. In our development we prove that the Drazin index of 1 − ba is equal to the Drazin index of 1 − ab.2000 Mathematics subject classification: primary 15A09; secondary 16U99.