Stochastic conditionin g of matrix functions

Serge Gratton (1, 2) , David Titley-Peloquin (3)

1 IRIT-APO - Algorithmes Parallèles et Optimisation
2 Toulouse INP - Institut National Polytechnique (Toulouse)
3 CERFACS - Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique

We investigate the sensitivity of matrix functions to random noise in their input. We propose the notion of a stochastic condition number, which determines, to first order, the sensitivity of a matrix function to random noise. We derive an upper bound on the stochastic condition number that can be estimated efficiently by using "small-sample" estimation techniques. The bound can be used to estimate the median, or any other quantile, of the error in a function's output when its input is subjected to random noise. We give numerical experiments illustrating the effectiveness of our stochastic error estimate.

Mots clés: Sensitivity analysis; Matrix functions; Uncertainty propagation; Conditioning; Perturbation analysis

Domaines: Arithmétique des ordinateurs; Analyse numérique [cs.NA]

Serge Gratton, David Titley-Peloquin. Stochastic conditionin g of matrix functions. SIAM/ASA Journal on Uncertainty Quantification, 2014, 2 (1), pp.763-783. ⟨10.1137/140973827⟩. ⟨hal-02147970⟩