Many numerical problems with input $x$ and output $y$ can be formulated as a 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) relaxation by omitting any number of equations from $F$. We propose a condition number for underdetermined systems that relates the condition number of a numerical problem to those of its relaxations, thereby detecting the ill-conditioned constraints. 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: two-factor matrix decompositions and Tucker decompositions.

翻译：许多具有输入$x$与输出$y$的数值问题可表述为方程组$F(x, y) = 0$，其目标为求解$y$。条件数用于度量$x$的微小扰动导致$y$的变化程度。通过从$F$中省略任意数量的方程，可由此数值问题导出一个（通常为欠定的）松弛系统。本文提出一种针对欠定系统的条件数定义，将原数值问题的条件数与其松弛系统的条件数相关联，从而识别导致病态性的约束条件。我们通过计算两个在经典意义上不具备有限条件数的问题——双因子矩阵分解与Tucker分解——的条件数，展示了该技术的应用。