SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 3, Page 1401-1422, January 2021.
The Schur--Parlett algorithm, implemented in MATLAB as \textttfunm, evaluates an analytic function $f$ at an $n\times n$ matrix argument by using the Schur decomposition and a block recurrence of Parlett. The algorithm requires the ability to compute $f$ and its derivatives, and it requires that $f$ have a Taylor series expansion with a suitably large radius of convergence. We develop a version of the Schur--Parlett algorithm that requires only function values and not derivatives. The algorithm requires access to arithmetic of a matrix-dependent precision at least double the working precision, which is used to evaluate $f$ on the diagonal blocks of order greater than 2 (if there are any) of the reordered and blocked Schur form. The key idea is to compute by diagonalization the function of a small random diagonal perturbation of each diagonal block, where the perturbation ensures that diagonalization will succeed. Our algorithm is inspired by Davies's randomized approximate diagonalization method, but we explain why that is not a reliable numerical method for computing matrix functions. This multiprecision Schur--Parlett algorithm is applicable to arbitrary analytic functions $f$ and, like the original Schur--Parlett algorithm, it generally behaves in a numerically stable fashion. The algorithm is especially useful when the derivatives of $f$ are not readily available or accurately computable. We apply our algorithm to the matrix Mittag--Leffler function and show that it yields results of accuracy similar to, and in some cases much greater than, the state-of-the-art algorithm for this function.

中文翻译：

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用于计算矩阵函数的多精度无导数 Schur--Parlett 算法

SIAM 矩阵分析与应用杂志，第 42 卷，第 3 期，第 1401-1422 页，2021 年 1 月。
Schur-Parlett 算法在 MATLAB 中作为 \textttfunm 实现，通过使用 Schur 分解和 Parlett 的块递归，在 $n\times n$ 矩阵参数处评估解析函数 $f$。该算法需要计算 $f$ 及其导数的能力，并且它要求 $f$ 具有具有适当大收敛半径的泰勒级数展开。我们开发了 Schur-Parlett 算法的一个版本，它只需要函数值而不需要导数。该算法需要访问至少两倍于工作精度的矩阵相关精度的算术，该算法用于在重新排序和阻塞的 Schur 形式的大于 2（如果有）的阶数的对角块上评估 $f$。关键思想是通过对角化计算每个对角块的小随机对角扰动的函数，其中扰动确保对角化成功。我们的算法受到 Davies 的随机近似对角化方法的启发，但我们解释了为什么这不是计算矩阵函数的可靠数值方法。这种多精度 Schur-Parlett 算法适用于任意解析函数 $f$，并且与原始 Schur-Parlett 算法一样，它通常以数值稳定的方式运行。当 $f$ 的导数不容易获得或无法准确计算时，该算法特别有用。我们将我们的算法应用于矩阵 Mittag--Leffler 函数，并表明它产生的精度结果类似于，在某些情况下远大于，