Arithmetic with real algebraic numbers is in NC

Authors:
B. Mishra;P. Pedersen
Affiliations:
New York University;New York University
Venue:
ISSAC '90 Proceedings of the international symposium on Symbolic and algebraic computation
Year:
1990

Citing 8
Cited 1

On computing the determinant in small parallel time using a small number of processors

Information Processing Letters
Parallel algorithms for algebraic problems

SIAM Journal on Computing
The complexity of elementary algebra and geometry

Journal of Computer and System Sciences
A fast parallel algorithm to compute the rank of a matrix over an arbitrary field

Combinatorica
Thom's lemma, the coding of real algebraic numbers and the computation of the topology of semi-algebraic sets

Journal of Symbolic Computation
Elements of computer algebra with applications

Elements of computer algebra with applications
Complexity of computation on real algebraic numbers

Journal of Symbolic Computation
Fast parallel calculation of the rank of matrices over a field of arbitrary characteristic

FCT '85 Fundamentals of Computation Theory

Computation in real closed infinitesimal and transcendental extensions of the rationals

CADE'13 Proceedings of the 24th international conference on Automated Deduction

Quantified Score

Hi-index	0.00

Visualization

Abstract

We describe NC algorithms for doing exact arithmetic with real algebraic numbers in the sign-coded representation introduced by Coste and Roy [CoR 1988]. We present polynomial sized circuits of depth &Ogr;(log3 N) for the monadic operations -&agr;, 1/&agr;, as well as &agr; + r, &agr; · r, and sgn(&agr; - r), where r is rational and &agr; is real algebraic. We also present polynomial sized circuits of depth &Ogr;(log7 N) for the dyadic operations &agr;+&bgr;, &agr;·&bgr;, and sgn(&agr; - &bgr;), where &agr; and &bgr; are both real algebraic. Our algorithms employ a strengthened form of the NC polynomial-consistency algorithm of Ben-Or, Kozen, and Reif [BKR 1986].