Multiple-precision zero-finding methods and the complexity of elementary function evaluation

Details Multiple-precision zero-finding methods and the complexity of elementary function evaluation Multiple-precision zero-finding methods and the complexity of elementary function evaluation

2010-04-20

arxiv.org/abs/1004.3412v2

Analytic Computational Complexity (edited by J. F. Traub), Academic Press, New York, 1975, 151-176

Computer Science

Numerical Analysis

An old (1975) paper with a postscript describing more recent developments. See also http://wwwmaths.anu.edu.au/~brent/pub/pub0

Scientific paper

We consider methods for finding high-precision approximations to simple zeros of smooth functions. As an application, we give fast methods for evaluating the elementary functions log(x), exp(x), sin(x) etc. to high precision. For example, if x is a positive floating-point number with an n-bit fraction, then (under rather weak assumptions) an n-bit approximation to log(x) or exp(x) may be computed in time asymptotically equal to 13M(n)lg(n), where M(n) is the time required to multiply floating-point numbers with n-bit fractions. Similar results are given for the other elementary functions. Some analogies with operations on formal power series (over a field of characteristic zero) are discussed. In particular, it is possible to compute the first n terms in log(1 + a_1.x + ...) or exp(a_1.x + ...) in time O(M(n)), where M(n) is the time required to multiply two polynomials of degree n - 1. It follows that the first n terms in a q-th power (1 + a_1.x + ...)^q can be computed in time O(M(n)), independent of q. One of the results of this paper is the "Gauss-Legendre" or "Brent-Salamin" algorithm for computing pi. This is the first quadratically convergent algorithm for pi. It was also published in Brent [J. ACM 23 (1976), 242-251], and independently by Salamin [Math. Comp. 30 (1976), 565-570].

Affiliated with

Also associated with

No associations

Rating

Multiple-precision zero-finding methods and the complexity of elementary function evaluation does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Multiple-precision zero-finding methods and the complexity of elementary function evaluation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Multiple-precision zero-finding methods and the complexity of elementary function evaluation will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-472540