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.

翻译：许多输入为$x$、输出为$y$的数值问题可表述为方程组$F(x, y) = 0$，其目标是对$y$进行求解。条件数衡量了$x$微小扰动导致$y$的变化程度。从这一数值问题出发，通过省略$F$中的任意若干约束，可推导出一个（通常为欠定）子问题。我们提出了一种针对欠定系统的条件数定义，该定义将数值问题的条件数与其子问题的条件数相关联。通过计算两个在经典意义上不具备有限条件数的问题——任意双因子矩阵分解与Tucker分解——的条件性，我们展示了该方法的实际应用价值。