Higham, Nicholas J. and Liu, Xiaobo (2020) A Multiprecision DerivativeFree SchurParlett Algorithm for Computing Matrix Functions. [MIMS Preprint]
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Abstract
The SchurParlett algorithm, implemented in MATLAB as \texttt{funm}, 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 $f$ has a Taylor series expansion with a suitably large radius of convergence. \new{We develop a version of the SchurParlett algorithm that requires only function values and not derivatives. The algorithm requires access to arithmetic of a matrixdependent 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 SchurParlett algorithm is applicable to arbitrary analytic functions $f$ and, like the original SchurParlett algorithm, it generally behaves in a numerically stable fashion. \new{The algorithm is especially useful when the derivatives of $f$ are not readily available or accurately computable.} We apply our algorithm to the matrix MittagLeffler 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.
Item Type:  MIMS Preprint 

Additional Information:  To appear in SIAM Journal on Matrix Analysis and Applications 
Uncontrolled Keywords:  multiprecision algorithm, multiprecision arithmetic, matrix function, Schur decomposition, SchurParlett algorithm, Parlett recurrence, randomized approximate diagonalization, matrix MittagLeffler function 
Subjects:  MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis 
Depositing User:  Nick Higham 
Date Deposited:  18 Jun 2021 16:54 
Last Modified:  18 Jun 2021 16:54 
URI:  https://eprints.maths.manchester.ac.uk/id/eprint/2824 
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A Multiprecision DerivativeFree SchurParlett Algorithm for Computing Matrix Functions. (deposited 07 Sep 2020 13:50)

A Multiprecision DerivativeFree SchurParlett Algorithm for Computing Matrix Functions. (deposited 23 Mar 2021 15:06)
 A Multiprecision DerivativeFree SchurParlett Algorithm for Computing Matrix Functions. (deposited 18 Jun 2021 16:54) [Currently Displayed]

A Multiprecision DerivativeFree SchurParlett Algorithm for Computing Matrix Functions. (deposited 23 Mar 2021 15:06)
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