On using the complex step method for the approximation of Fréchet derivatives of matrix functions in automorphism groups

Abstract

We show, that the Complex Step approximation Im(f (A + ihE))/h to the Fréchet derivative of matrix functions f : R m,n → R m,n 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 Newtons method, we extend the research to the family of Padé iterations, that allows us to introduce iterative schemes for finding function and derivative values while approximately preserving automorphism group structure.

Integral representations for higher-order Fréchet derivatives of matrix functions: Quadrature algorithms and new results on the level-2 condition number

“…Based on Theorem 11, Higham and Relton [8,Algorithm 3.6] propose to compute f (X k ), with X k defined in (21), and then extract its upper-right n × n block in order to obtain…”

“…Recently, Al-Mohy and Arslan [20] have investigated the use of the complex-step approximation for the higher-order Fréchet derivative; see also [21] for related work by Werner. In particular, they show that if f is an analytic function that takes real values at real arguments,…”

“…where ℑ(M ) denotes the (entrywise) imaginary part of a matrix M . Thus, defining X k−1 (h) according to (21), starting with…”

“…Proof. Let spec(A) ⊆ Ω and let X k be the matrix defined in (21). The matrix X k is block upper triangular with all diagonal blocks equal to A, so that spec(X k ) ⊆ Ω and therefore, by the assumption (12), we have…”