Premium
Some applications of nonlinear convergence accelerators
Author(s) -
Weniger E. J.,
Grotendorst J.,
Steinborn E. O.
Publication year - 2009
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560280818
Subject(s) - transformation (genetics) , convergence (economics) , series (stratigraphy) , monotonic function , nonlinear system , mathematics , function (biology) , convergent series , calculus (dental) , computer science , mathematical analysis , physics , power series , quantum mechanics , dentistry , evolutionary biology , biology , economics , gene , economic growth , medicine , paleontology , biochemistry , chemistry
The efficient and reliable evaluation of infinite series is a frequently occurring problem. In many cases one is confronted with series expansions which converge so slowly that an evaluation by a direct summation of their terms would not be feasible, or even with series expansions which diverge. Consequently, alternative methods for their evaluation have to be used. In this article we want to report our experiences with two nonlinear convergence accelerators, the Shanks transformation and the u transformation of Levin. The Shanks transformation was designed to accelerate linear convergence whereas the u transformation is especially powerful for slowly converging monotonic series and for some converging or diverging alternating series. The numerical examples presented are drawn from molecular multicenter integrals and from special function theory.