An integral approximation error is estimated for an operator exponential function, and the unimprovability of the estimate with respect to order is investigated. The smoothness of the initial vector in terms of the order of accuracy of the Cayley transform method is substantiated for approximations of operator exponential and cosine functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. P. Gavrilyuk and V. L. Makarov, Strongly Positive Operators and Numerical Algorithms without Saturation of Accuracy [in Russian], Inst. Mat. NAN Ukr., Kyiv (2004).
V. L. Makarov and V. L. Ryabichev, “Unimprovable accuracy estimates of the Cayley transform method for finding an operator cosine function,” Dop. NAN Ukr., No. 12, 21–25 (2002).
V. L. Makarov, V. B. Vasilik, and V. L. Ryabichev, “Nonimprovable order-of-magnitude estimates of the rate of convergence of the Caley transform method for approximation of an operator exponent,” Cybern. Syst. Analysis, 38, No. 4, 632–636 (2002).
V. L. Makarov, N. V. Maiko, and V. L. Ryabichev, “Approximation accuracy of an operator exponential function,” Visnyk Kyiv. Univ., Ser. Fiz.-Mat. Nauky, Issue 4, 192–197 (2002).
S. M. Torba, “Direct and inverse theorems of approximate methods for the solution of an abstract Cauchy problem,” Ukr. Math. J., 59, No. 6, 919–937 (2007).
F. Riesz and B. Sz.-Nagy, Functional Analysis, Dover Books on Advanced Mathematics, New York, Dover Publications Inc. (1990).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 145–152, September–October 2009.
Rights and permissions
About this article
Cite this article
Maiko, N.V., Ryabichev, V.L. Approximation theorems for operator exponential and cosine functions. Cybern Syst Anal 45, 800–807 (2009). https://doi.org/10.1007/s10559-009-9145-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10559-009-9145-x