On some complexity issues of NC analytic functions

Authors:
Fuxiang Yu
Affiliations:
Department of Computer Science, State University of New York at Stony Brook, Stony Brook, NY
Venue:
TAMC'06 Proceedings of the Third international conference on Theory and Applications of Models of Computation
Year:
2006

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Abstract

This paper studies the complexity of derivatives and integration of NC real functions (not necessarily analytic) and NC analytic functions. We show that for NC real functions, derivatives and integration are infeasible, but analyticity helps to reduce the complexity. For example, the integration of a log-space computable real function f is as hard as #P, but if f is an analytic function, then the integration is log-space computable. As an application, we study the problem of finding all zeros of an NC analytic function inside a Jordan curve and show that, under a uniformity condition on the function values of the Jordan curve, the zeros are all NC computable.