The complex step approximation to the higher order Frechet derivatives of a matrix function

Küçük Resim Yok

Tarih

2021

Dergi Başlığı

Dergi ISSN

Cilt Başlığı

Yayıncı

Springer

Erişim Hakkı

info:eu-repo/semantics/closedAccess

Özet

Thekth Frechet derivative of a matrix functionfis a multilinear operator from a cartesian product ofksubsets of the spaceDOUBLE-STRUCK CAPITAL C-nxn into itself. We show that thekth Frechet derivative of a real-valued matrix functionfat a real matrixAin real direction matrices E-1, E-2, horizontal ellipsis, E-k can be computed using the complex step approximation. We exploit the algorithm of Higham and Relton (SIAM J. Matrix Anal. Appl.35(3):1019-1037,2014) with the complex step approximation and mixed derivative of complex step and central finite difference scheme. Comparing with their approach, our cost analysis and numerical experiment reveal thathalfandseven-eighthsof the computational cost can be saved for the complex step and mixed derivative, respectively. Whenfhas an algorithm that computes its action on a vector, the computational cost drops down significantly as the dimension of the problem andkincrease.

Açıklama

Anahtar Kelimeler

Matrix function, Frechet derivative, Higher order Frechet derivative, Complex step approximation, Action of matrix functions

Kaynak

Numerical Algorithms

WoS Q Değeri

Q1

Scopus Q Değeri

Q2

Cilt

87

Sayı

3

Künye