A Multiprecision Derivative-Free Schur-Parlett Algorithm for Computing Matrix Functions

The Schur–Parlett algorithm, implemented in MATLAB as funm, computes a function f(A) of an n × n matrix A 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 has 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 uses higher precision arithmetic 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 triangular block, where the perturbation ensures that diagonalization will succeed. This multiprecision Schur–Parlett algorithm is applicable to arbitrary functions f and, like the original Schur–Parlett algorithm, it generally behaves in a numerically stable fashion. 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. 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. The algorithm will be useful for evaluating any matrix function for which the derivatives of the underlying function are not readily available or accurately computable.

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