Fast Numerical Multivariate Multipoint Evaluation

Abstract

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We design nearly-linear time numerical algorithms for the problem of multivariate multipoint evaluation over the fields of rational, real and complex numbers. We consider both exact and approximate versions of the algorithm. The input to the algorithms are (1) coefficients of an m-variate polynomial f with degree d in each variable, and (2) points a 1 , . . . , a N each of whose coordinate has value bounded by one and bit-complexity s. Approximate version: Given additionally an accuracy parameter t, the algorithm computes rational numbers β 1 , . . . , for all i, and has a running time of ((Nm + d m )(s + t)) 1+o( 1) for all m and all sufficiently large d. Exact version (when over rationals): Given additionally a bound c on the bit-complexity of all evaluations, the algorithm computes the rational numbers f (a 1 ), . . . , f (a N ), in time ((Nm + d m )(s + c)) 1+o(1) for all m and all sufficiently large d. .