On mixed and componentwise condition numbers for Moore–Penrose inverse and linear least squares problems

Abstract

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. In this paper, 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.

Perturbation analysis and condition numbers of scaled total least squares problems

“…In the following part, we show that these mixed and componentwise condition numbers for the STLS problem respectively approach some existing mixed and componentwise condition numbers for the LS problem, i.e., μ M ST LS → μ LS and ν M ST LS → ν LS , when λ → 0. Under the condition that A is of full column rank, Cucker et al [2] derived the following mixed and componentwise condition numbers for the LS problem (2.4):…”

Using dual techniques to derive componentwise and mixed condition numbers for a linear function of a linear least squares solution

“…Note that the norm on the solution space G has not been chosen yet. Following the terminology given in [13] and also used in [9], K(y) is referred to as componentwise (resp. mixed) condition number when · G is componentwise (resp.…”

“…When L is the identity matrix, the expression given in (3.3) is the same as the one established in [9] (using the norm · = · ∞ x ∞ on the solution space). Note that bounds of K ∞ (I, A, b) are also given in [7, p. 34] and in [21, p. 384].…”

“…Componentwise error analysis that provide us with exact expressions or bounds for componentwise condition numbers can be found for example in [5,8,19,26,[28][29][30] for linear systems and in [4,6,9,19,23] for linear least squares. In particular, componentwise backward errors are commonly used as stopping criteria in iterative refinement for solving linear systems (see e.g.…”

“…The conditioning of a nonlinear function of an LLSP solution can also be obtained by replacing in the condition number expression L T by the Jacobian matrix at the solution. When L is the identity matrix and when perturbations on the solution are measured using the infinity norm or a componentwise norm, we obtain the exact formula given in [9]. By considering the special case where we have a residual equal to zero, we obtain componentwise and mixed condition numbers for L T x where x is the solution of a consistent linear system.…”