Which variables of a numerical problem cause ill-conditioning?

Abstract

We study a broad class of numerical problems that can be defined as the solution of a system of (nonlinear) equations for a subset of the dependent variables. Given a system of the form $F(x,y,z) = c$ with multivariate input $x$ and dependent variables $y$ and $z$, we define and give concrete expressions for the condition number of solving for a value of $y$ such that $F(x,y,z) = c$ for some unspecified $z$. This condition number can be used to determine which of the dependent variables of a numerical problem are the most ill-conditioned. We show how this can be used to explain the condition number of the problem of solving for all dependent variables, even if the solution is not unique. The concepts are illustrated with Tucker decomposition of tensors as an example problem.