2019/05/14 by Martin Lotz, Vanni Noferini, Lotz, Martin +1
Business, Management and Accounting · Computer Science · Mathematics · #15A15 #15A18 #15B52 #60H99 #65F15 #65F35 #Advanced Queuing Theory Analysis #FOS: Mathematics #Mathematical functions and polynomials #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1905.05466
openalex publication_date 2019/05/14 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
We propose a new approach to the theory of conditioning for numerical\nanalysis problems for which both classical and stochastic perturbation theory\nfail to predict the observed accuracy of computed solutions. To motivate our\nideas, we present examples of problems that are discontinuous at a given input\nand have infinite classical and stochastic condition number, but where the\nsolution is still computed to machine precision without relying on structured\nalgorithms. Stimulated by the failure of classical and stochastic perturbation\ntheory in capturing such phenomena, we define and analyse a weak worst-case and\na weak stochastic condition number. This new theory is a more powerful\npredictor of the accuracy of computations than existing tools, especially when\nthe worst-case and the expected sensitivity of a problem to perturbations of\nthe input is not finite. We apply our analysis to the computation of simple\neigenvalues of matrix polynomials, including the more difficult case of\nsingular matrix polynomials. In addition, we show how the weak condition\nnumbers can be estimated in practice.\n