论文标题
验证的矩阵γ函数的计算
Verified computation of matrix gamma function
论文作者
论文摘要
提出了两种数值算法用于计算包含矩阵函数的间隔矩阵。在2014年，作者提出了用于封闭$ a \ in \ mathbb {c}^{n \ times n} $的所有特征值和不变子空间的基础的算法。作为这些算法的副产品，我们可以获得包含小矩阵的间隔矩阵，其频谱包含在$ a $中。在本文中，我们解释了由于经过验证的块对角（VBD）而导致的，该间隔矩阵包含基础和小矩阵，并建立了使用VBD封闭矩阵函数的新框架。为了实现小矩阵的伽马功能的外壳，我们得出了可计算的扰动边界。如果输入矩阵满足条件，我们可以应用这些边界。我们将矩阵参数减少（AR）纳入迫使输入矩阵以满足条件并发展加速ARS的理论。第一个算法基于数值频谱分解使用VBD，并且在假设下仅涉及立方复杂性。第二种算法基于数值Jordan分解采用VBD，甚至适用于有缺陷的矩阵。数值结果显示算法的效率和鲁棒性。
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.
