A MULTIPRECISION DERIVATIVE-FREE SCHUR PARLETT ALGORITHM FOR COMPUTING MATRIX FUNCTIONS.

The Schur--Parlett algorithm, implemented in MATLAB as funm, evaluates an analytic function f at an n X 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 / 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 diago-nalization 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 / 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. [ABSTRACT FROM AUTHOR]

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