On using the complex step method for the approximation of Fr\'echet derivatives of matrix functions in automorphism groups

TL;DR

This paper explores the use of the complex step method to approximate Fréchet derivatives of matrix functions like the sign, square root, and polar mappings, extending previous work to iterative schemes that preserve automorphism group structures.

Contribution

It introduces iterative schemes based on Padé iterations for approximating derivatives of matrix functions while maintaining automorphism group properties.

Findings

Complex step method is applicable to various matrix functions.

Padé iterations can approximate derivatives while preserving structure.

Extended methods improve computational approaches for matrix functions.

Abstract

We show, that the Complex Step approximation to the Fr\'echet derivative of real matrix functions is applicable to the matrix sign, square root and polar mapping using iterative schemes. While this property was already discovered for the matrix sign using Newton's method, we extend the research to the family of Pad\'e iterations, that allows us to introduce iterative schemes for finding function and derivative values while approximately preserving automorphism group structure.