On parallel complexity of analytic functions

Abstract In this paper, we study the parallel complexity of analytic functions. We investigate the complexity of computing the derivatives, integrals, and zeros of N C or logarithmic-space computable analytic functions, where N C denotes the complexity class of sets acceptable by polynomial-size, polylogarithmic-depth, uniform Boolean circuits. It is shown that the derivatives and integrals of N C (or logarithmic-space) computable analytic functions remain N C (or, respectively, logarithmic-space) computable. We also study the problem of finding all zeros of an NC computable analytic function inside an N C computable Jordan curve, and show that, under a uniformity condition on the function values on the Jordan curve, all zeros can be found in NC .

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