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:1004.3608v1

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The postscript (1999) is now out of date.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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