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

The need to compute matrix functions arises in many applications in science and engineering. Specialized methods exist for evaluating particular matrix functions, for example, the well-known scaling and squaring algorithm for the matrix exponential, Newton’s method for matrix sign function, and the inverse scaling and squaring method for the matrix logarithm. For some functions a specialized method is not available, in which case a general purpose algorithm is needed. The Schur-Parlett algorithm computes a general function $f$ of a matrix, with the function dependence restricted to the evaluation of $f$ on the diagonal blocks of the reordered and blocked Schur form. It evaluates $f$ on the nontrivial diagonal blocks via a Taylor series, so it requires the derivatives of $f$ and it also requires the Taylor series to have a sufficiently large radius of convergence. However, the derivatives are not always available or accurately computable.

In our recent preprint, Nick Higham and I present 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. Our algorithm is inspired by Davies’s randomized approximate diagonalization method, but we show that the measure of error that underlies randomized approximate diagonalization makes it not a reliable numerical method for computing matrix functions. The key idea in our algorithm 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. Multiprecision algorithms have already been developed for the matrix exponential and the matrix logarithm, and those algorithms are tightly coupled to the functions in question, whereas here we place no restrictions on the function. And the algorithm we propose is not restricted only to a working precision of double since its framework is precision independent. We test our algorithm in double precision on a variety of test problems, and the results show that the algorithm generally behaves in a numerically stable fashion. 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.

In the following example we employ the algorithm to compute in double precision the matrix Airy function, and compare the result with that produced by a spectral decomposition method (which breaks down if A is defective, and will be inaccurate if A is close to being defective).

We observe that the normwise relative difference between our algorithm and the spectral decomposition method is tiny. This is just one of the many matrix functions that no specialized algorithm is available for, and the MATLAB $\texttt{funm}$ function, which implements the Schur-Parlett algorithm, is not able to compute. Our new algorithm greatly expands the class of readily computable matrix functions, and its MATLAB code is available here.