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The use of normalized binomial coefficients as weights to accelerate convergence of oscillatory first‐order iterations
Author(s) -
Hakala Reino W.
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.560010633
Subject(s) - iterated function , mathematics , sequence (biology) , convergence (economics) , binomial coefficient , order (exchange) , binomial (polynomial) , omega , algebraic number , process (computing) , mathematical analysis , statistics , combinatorics , computer science , physics , genetics , finance , quantum mechanics , economics , biology , economic growth , operating system
Normalized binomial coefficients can be used as weights in averaging the members of a sequence of iterates resulting from an oscillatory process of first order, to obtain an improved estimate of the fixed point of the process. This procedure results in convergence whether the original oscillatory first‐order process is convergent or divergent, but not too rapidly divergent. By way of example, this method causes the omega technique to converge much more rapidly and results in convergence even when the sequence obtained by the unmodified omega technique diverges. Normalized‐binomial‐coefficient‐weighted averaging of three successive iterates, though appreciably simpler, is a close approximation to Aitken's δ 2 process in cases of oscillatory iterations, and is profitably used in place of the latter process, especially in algebraic developments.