A note on condition numbers for generalized inverse C^‡_A and their statistical estimation
Mahvish Samar, Abdual Shahkoor
TL;DR
This work analyzes the sensitivity of the generalized inverse $C^{\ddagger}_{A}$ arising from ILSEP by deriving explicit normwise, mixed, and componentwise condition numbers and providing Kronecker-free forms and computable upper bounds. It demonstrates how the Fréchet derivative of $C^{\ddagger}_{A}$ yields conditioning information for ILSEP solutions and residuals, and then develops two families of probabilistic estimators—the probabilistic spectral-norm estimator and small-sample statistical condition estimation (SSCE)—with three tailored algorithms to estimate these condition numbers efficiently. Numerical experiments confirm the reliability of the upper bounds and the effectiveness of the estimation algorithms across varied conditioning of $A$ and $C$, highlighting practical tools for error analysis and stability in ILSEP computations. The results advance perturbation analysis for ILSEP by enabling accurate, scalable conditioning assessments of its generalized inverse.
Abstract
In this paper, we consider the condition number for the generalized inverse C^‡_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^‡_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