Mixed, Componentwise, and Structured Condition Numbers

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

“…They were Gohberg and Koltracht [6] who named Skeel's condition number mixed to distinguish it from componentwise condition numbers such as that in [13]. They also gave explicit expressions for both mixed and componentwise condition numbers, always for square systems of linear equations.…”

“…To be precise, two different kinds of condition number were studied: first, those measuring the errors in the output using norms and the input perturbations componentwise, and second, those measuring both the error in the output and the perturbation in the input componentwise. The resulting condition numbers are called mixed and componentwise, respectively, by Gohberg and Koltracht [6]. We will use this terminology throughout this paper.…”

Condition Numbers and Local Perturba¬tion Bounds for the Matrix Equation Xs ± AHXtA = Q

“…Hence, (14) holds. As function F (X, S) is Lipschitzian at S, the heuristic rule [ 13 ] may be applied to estimate the actual error in the computed solution for computing environment with rounding unit macheps: when macheps k(S) < 1, then about − log 10 (macheps k(S)) are the true decimal digits in the result.…”

“…This example is taken from [ 2 ]. On Table 2 are given the results obtained for the relative norm-wise condition numbers k A and k Q (13), the overall relative norm-wise condition number k 1 (14), the mixed condition numberk(σ) (17), the component-wise condition numberk(σ) (18) and the ratio est(δ) δ x of the local estimate est(δ) (16), based on the norm-wise condition numbers to the estimated value-the relative perturbations δ X in the solution X. The numerical example demonstrates the effectiveness of the bounds proposed.…”

Applications of statistical condition estimation to the solution of linear systems

“…Structured condition analysis has been involved in a lot of recent work, e.g. [3][4][5]. As an example, Higham and Higham [3] define the structured componentwise backward error and apply it to symmetric matrices and Toeplitz matrices as examples.…”