On the condition of some problems in linear algebra

Abstract

Classical condition numbers are normwise: they measure the size of both input perturbations and output errors using some norms. To take into account the relative of each data component, and, in particular, a possible data sparseness, componentwise condition numbers have been increasingly considered. These are mostly of two kinds: mixed and componentwise [21]. In Chapter 2, we give explicit expressions, computable from the data, for the mixed and componentwise condition numbers for the computation of the Moore-Penrose inverse as well as for the computation of solutions and residues of linear least squares problems. In both cases the data matrices have full column (row) rank. For several classes of structured rectangular matrices, we provide explicit expressions for both mixed and componentwise structured condition numbers in Chapter 3 and the structures we consider are upper triangular, Toeplitz, Hankel, Vandermonde, and Cauchy matrices. Such expressions for many other classes of matrices can be similarly derived. In Chapter 4, we give explicit expressions for the mixed 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: symmetric, Hermitian, skewsymmetric, skewhermitian, Toeplitz, and Hankel. Details for other linear structures should follow in a straightforward manner from our general result. In Chapter 5 we do a smoothed analysis —in the sense of [15, 45]— of the condition number for the Moore-Penrose inverse. Usual average analysis follows in a trivial manner as follow similar analyses for the condition number of the polar factorization.