矩阵Γ函数的验证计算

摘要：

提出了计算包含矩阵Γ函数的区间矩阵的两种数值算法。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，我们可以得到包含小矩阵的区间矩阵，这些小矩阵的谱包含在$A$的谱中。本文将包含基矩阵和小矩阵的区间矩阵解释为验证块对角化（VBD）的结果，并利用VBD建立了一个新的封闭矩阵函数的框架.为了实现小矩阵的Γ函数的封闭，我们推导出可计算的扰动界。如果输入矩阵满足条件，我们可以应用这些界限。我们将矩阵的论点减少（AR），以迫使输入矩阵，以满足条件，并发展理论加速AR。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立方复杂度。第二种算法采用了基于数值若尔当分解的VBD算法，该算法甚至适用于缺陷矩阵。数值结果表明了算法的有效性和鲁棒性。

英文摘要：

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.

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