The complex step approximation to the higher order Frechet derivatives of a matrix function
Küçük Resim Yok
Tarih
2020
Yazarlar
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
