Given a matrix-valued function $\mathcal{F}(\lambda)=\sum_{i=1}^d f_i(\lambda) A_i$, with complex matrices $A_i$ and $f_i(\lambda)$ entire functions for $i=1,\ldots,d$, we discuss a method for the numerical approximation of the distance to singularity of $\mathcal{F}(\lambda)$. The closest singular matrix-valued function $\widetilde{\mathcal{F}}(\lambda)$ with respect to the Frobenius norm is approximated using an iterative method. The property of singularity on the matrix-valued function is translated into a numerical constraint for a suitable minimization problem. Unlike the case of matrix polynomials, in the general setting of matrix-valued functions the main issue is that the function $\det ( \widetilde{\mathcal{F}}(\lambda) )$ may have an infinite number of roots. An important feature of the numerical method consists in the possibility of addressing different structures, such as sparsity patterns induced by the matrix coefficients, in which case the search of the closest singular function is restricted to the class of functions preserving the structure of the matrices.

翻译：给定一个矩阵值函数 $\mathcal{F}(\lambda)=\sum_{i=1}^d f_i(\lambda) A_i$，其中 $A_i$ 为复矩阵，$f_i(\lambda)$ 为整函数（$i=1,\ldots,d$），我们讨论了一种数值逼近 $\mathcal{F}(\lambda)$ 奇异性距离的方法。通过迭代方法，我们近似求解了在 Frobenius 范数下距离最近的奇异矩阵值函数 $\widetilde{\mathcal{F}}(\lambda)$。矩阵值函数的奇异性被转化为一个适当极小化问题的数值约束条件。与矩阵多项式情形不同，在矩阵值函数的一般设定中，主要困难在于函数 $\det ( \widetilde{\mathcal{F}}(\lambda) )$ 可能具有无限多个根。该方法的一个重要特性在于能够处理不同的结构约束，例如由矩阵系数诱导的稀疏模式；在此类情形下，对最近奇异函数的搜索被限制在保持矩阵结构的函数类中。