Given a matrix-valued function $\mathcal{F}(\lambda)=\sum_{i=1}^d f_i(\lambda) A_i$, with complex matrices $A_i$ and $f_i(\lambda)$ analytic functions for $i=1,\ldots,d$, we discuss a method for the numerical approximation of the distance to singularity for $\mathcal{F}(\lambda)$. The closest matrix-valued function $\widetilde {\mathcal{F}}(\lambda)$ with respect to the Frobenius norm is approximated using an iterative method. The condition 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. The main feature of the numerical method consists in the possibility of extending it to different structures, such as sparsity patterns induced by the matrix coefficients.

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，$f_i(\lambda)$ 为解析函数（$i=1,\ldots,d$），本文讨论了一种对该函数奇异性距离进行数值逼近的方法。利用迭代方法，我们近似得到了在 Frobenius 范数下最接近的矩阵值函数 $\widetilde {\mathcal{F}}(\lambda)$。将矩阵值函数的奇异性条件转化为一个适当的极小化问题的数值约束。与矩阵多项式情形不同，在矩阵值函数的一般设定中，主要问题在于 $\det ( \widetilde{\mathcal{F}}(\lambda) )$ 可能具有无穷多个根。该数值方法的主要特点在于其可扩展至不同结构，例如由矩阵系数诱导的稀疏模式。