## The complexity of multiple-precision arithmetic

32. R. P. Brent,
The complexity of multiple-precision arithmetic, in
* The Complexity of Computational Problem Solving*
(edited by R. S. Anderssen and R. P. Brent),
University of Queensland Press, Brisbane, 1976, 126-165.
Retyped and postscript added 1999.
arXiv:1004.3608v2
Abstract:
pdf (74K).

Paper:
pdf (268K).

Original paper (scanned):
pdf (700K).

## Abstract

In studying the complexity of iterative processes it is usually assumed
that the arithmetic operations of addition, multiplication, and division
can be performed in certain constant times. This assumption is invalid
if the precision required increases as the computation proceeds. We give
upper and lower bounds on the number of single-precision operations
required to perform various multiple-precision operations, and deduce
some interesting consequences concerning the relative efficiencies of
methods for solving nonlinear equations using variable-length
multiple-precision arithmetic.
## Comments

Related papers (written later) are Brent
[28,
34].
The postscript (1999) is now out of date. For an excellent survey of more
recent results and improvements, see
Daniel J. Bernstein, *Fast multiplication and its applications*,
Algorithmic Number Theory, MSRI Publications, Volume 44, 2008.
Available here.

## Errata

In the original version:
page 141, line 3: replace "Proof" by "Remark".

page 157, last line: replace "<" by ">".

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