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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Some basic information on information-based complexity theory

Author: Beresford N. Parlett
Journal: Bull. Amer. Math. Soc. 26 (1992), 3-27
MSC (2000): Primary 68Q30; Secondary 65Y20
MathSciNet review: 1102755
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Abstract: Numerical analysts might be expected to pay close attention to a branch of complexity theory called information-based complexity theory (IBCT), which produces an abundance of impressive results about the quest for approximate solutions to mathematical problems. Why then do most numerical analysts turn a cold shoulder to IBCT? Close analysis of two representative papers reveals a mixture of nice new observations, error bounds repackaged in new language, misdirected examples, and misleading theorems. Some elements in the framework of IBCT, erected to support a rigorous yet flexible theory, make it difficult to judge whether a model is off-target or reasonably realistic. For instance, a sharp distinction is made between information and algorithms restricted to this information. Yet the information itself usually comes from an algorithm, so the distinction clouds the issues and can lead to true but misleading inferences. Another troublesome aspect of IBCT is a free parameter F, the class of admissible problem instances. By overlooking F’s membership fee, the theory sometimes distorts the economics of problem solving in a way reminiscent of agricultural subsidies. The current theory’s surprising results pertain only to unnatural situations, and its genuinely new insights might serve us better if expressed in the conventional modes of error analysis and approximation theory.

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Article copyright: © Copyright 1992 American Mathematical Society