What part of a numerical problem is ill-conditioned?

05/19/2023

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by   Nick Dewaele, et al.

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Many numerical problems with input x and output y can be formulated as an system of equations F(x, y) = 0 where the goal is to solve for y. The condition number measures the change of y for small perturbations to x. From this numerical problem, one can derive a (typically underdetermined) subproblem by omitting any number of constraints from F. We propose a condition number for underdetermined systems that relates the condition number of a numerical problem to those of its subproblems. We illustrate the use of our technique by computing the condition of two problems that do not have a finite condition number in the classic sense: any two-factor matrix decompositions and Tucker decompositions.