A note on condition numbers for generalized inverse C^{\ddag}_A and their statistical estimation

TL;DR

This paper derives explicit formulas for condition numbers of the generalized inverse C^{ ext{dagger}}_A, applies them to indefinite least squares problems, and proposes reliable statistical estimation algorithms validated by numerical experiments.

Contribution

It provides explicit expressions for various condition numbers of C^{ ext{dagger}}_A and develops high-reliability estimation algorithms for these condition numbers.

Findings

Explicit formulas for normwise, mixed, and componentwise condition numbers.

Algorithms for probabilistic spectral norm estimation and statistical condition estimation.

Numerical experiments demonstrating the effectiveness of the proposed methods.

Abstract

In this paper, we consider the condition number for the generalized inverse C^{\ddag}_A. We first present the explicit expression of normwise mixed and componentwise condition numbers. Then, we derive the explicit expression of normwise condition number without Kronecker product using the classical method for condition numbers. With the intermediate result, i.e., the derivative of C^{\ddag}_A, we can recover the explicit expressions of condition numbers for solution of Indefinite least squares problem with equality constraint. To estimate these condition numbers with high reliability, we choose the probabilistic spectral norm estimator and the small-sample statistical condition estimation method and devise three algorithms. Numerical experiments are provided to illustrate the obtained results