Logarithmic depth circuits for algebraic functions

Abstract

This paper describes circuits for computation of various algebraic functions on polynomials, power series, integers, and reals for which it has been a long standing open problem to compute in depth less then (log n)2. Let R[x] be the polynomials and power series over a commutative ring which supports a fast Fourier transform and let L[x] be the polynomials and power series over the rationals L. For polynomials of degree n-1, we give circuits of depth O(log n) for computing - the m-th power of a polynomial and the product of m polynomials in R[x], where m=nO(1) - the symmetric functions on R[x] - the remainder and quotient of division of polynomials in L[x] - interpolation of a polynomial in L[x]. For power series with n given low order terms, we give circuits of depth O(log n) for computing the first n low order terms of - the m-th power of a power series in R[x] and the product of m power series in R[x] where m=nO(1) - the composition of power series in R[x] - the reciprocal of a power series and the division of two power series in L[x] -the reversion of a power series in L[x] - various elementary functions applied to power series in L[x] such as (fixed) powers, roots, exponentation, logarithm, sin, cos, arctangent, and hyperbolic cosine. For integers represented by n bit binary numbers, we give boolean circuits (whose gates compute the boolean operations ∧, ∨, and ¬) of depth O(log n (loglog n) 2) for computing: - the m-th power of an integer and the product of m = nO(1) integers, - the remainder and quotient of the division of two integers. There are many immediate consequences of this result. For reals on a finite interval [a,b] represented as floating point numbers within relative accuracy o(2-n), we have boolean circuits of depth O (log n(loglog n) 2) for computing within relative accuracy o(2-n): - the m-th power of a real and the product of m = nO(1) reals - the reciprocal of a real and division of reals - various elementary functions on reals. Also, as a consequence of the above, for polynomials and power series in L[x] we have uniform boolean circuits of depth O(log n(loglog n)2) for all the above listed problems for polynomials and power series, and also: - evaluation of a polynomial or power series in L[x] at n points, within relative accuracy o(2-n). All our circuits may be uniformly constructed by a deterministic Turning machine with space O(log n) and have constant indegree.