Two numerical algorithms are proposed for computing an interval matrix containing the matrix gamma function. In 2014, the author presented algorithms for enclosing all the eigenvalues and basis of invariant subspaces of $A \in \mathbb{C}^{n \times n}$. As byproducts of these algorithms, we can obtain interval matrices containing small matrices whose spectrums are included in that of $A$. In this paper, we interpret the interval matrices containing the basis and small matrices as a result of verified block diagonalization (VBD), and establish a new framework for enclosing matrix functions using the VBD. To achieve enclosure for the gamma function of the small matrices, we derive computable perturbation bounds. We can apply these bounds if input matrices satisfy conditions. We incorporate matrix argument reductions (ARs) to force the input matrices to satisfy the conditions, and develop theories for accelerating the ARs. The first algorithm uses the VBD based on a numerical spectral decomposition, and involves only cubic complexity under an assumption. The second algorithm adopts the VBD based on a numerical Jordan decomposition, and is applicable even for defective matrices. Numerical results show efficiency and robustness of the algorithms.

中文翻译：

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矩阵伽马函数的验证计算

提出了两种数值算法来计算包含矩阵伽马函数的区间矩阵。2014 年，作者提出了封闭 $A \in \mathbb{C}^{n \times n}$ 不变子空间的所有特征值和基的算法。作为这些算法的副产品，我们可以获得包含小矩阵的区间矩阵，这些小矩阵的频谱包含在 $A$ 的频谱中。在本文中，我们解释了包含基矩阵和小矩阵的区间矩阵作为验证块对角化 (VBD) 的结果，并使用 VBD 建立了一个用于封闭矩阵函数的新框架。为了实现小矩阵伽马函数的封闭，我们推导出可计算的扰动边界。如果输入矩阵满足条件，我们可以应用这些边界。我们结合矩阵参数约简 (ARs) 来强制输入矩阵满足条件，并开发加速 ARs 的理论。第一种算法使用基于数值谱分解的 VBD，并且在假设下仅涉及三次复杂度。第二种算法采用基于数值 Jordan 分解的 VBD，甚至适用于有缺陷的矩阵。数值结果显示了算法的效率和鲁棒性。