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119 lines
5.9 KiB
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119 lines
5.9 KiB
HTML
4 years ago
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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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<html>
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<!-- This manual describes how to install and use the GNU multiple precision
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arithmetic library, version 6.1.0.
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Copyright 1991, 1993-2015 Free Software Foundation, Inc.
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Permission is granted to copy, distribute and/or modify this document under
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the terms of the GNU Free Documentation License, Version 1.3 or any later
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version published by the Free Software Foundation; with no Invariant Sections,
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with the Front-Cover Texts being "A GNU Manual", and with the Back-Cover
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Texts being "You have freedom to copy and modify this GNU Manual, like GNU
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software". A copy of the license is included in
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GNU Free Documentation License. -->
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<!-- Created by GNU Texinfo 6.4, http://www.gnu.org/software/texinfo/ -->
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<head>
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<title>Lehmer's Algorithm (GNU MP 6.1.0)</title>
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<meta name="description" content="How to install and use the GNU multiple precision arithmetic library, version 6.1.0.">
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<meta name="keywords" content="Lehmer's Algorithm (GNU MP 6.1.0)">
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<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1">
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<link href="index.html#Top" rel="start" title="Top">
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<link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
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<link href="Greatest-Common-Divisor-Algorithms.html#Greatest-Common-Divisor-Algorithms" rel="up" title="Greatest Common Divisor Algorithms">
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<link href="Subquadratic-GCD.html#Subquadratic-GCD" rel="next" title="Subquadratic GCD">
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<link href="Binary-GCD.html#Binary-GCD" rel="prev" title="Binary GCD">
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</style>
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</head>
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<body lang="en">
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<a name="Lehmer_0027s-Algorithm"></a>
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<div class="header">
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<p>
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Next: <a href="Subquadratic-GCD.html#Subquadratic-GCD" accesskey="n" rel="next">Subquadratic GCD</a>, Previous: <a href="Binary-GCD.html#Binary-GCD" accesskey="p" rel="prev">Binary GCD</a>, Up: <a href="Greatest-Common-Divisor-Algorithms.html#Greatest-Common-Divisor-Algorithms" accesskey="u" rel="up">Greatest Common Divisor Algorithms</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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</div>
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<hr>
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<a name="Lehmer_0027s-algorithm"></a>
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<h4 class="subsection">15.3.2 Lehmer’s algorithm</h4>
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<p>Lehmer’s improvement of the Euclidean algorithms is based on the observation
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that the initial part of the quotient sequence depends only on the most
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significant parts of the inputs. The variant of Lehmer’s algorithm used in GMP
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splits off the most significant two limbs, as suggested, e.g., in “A
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Double-Digit Lehmer-Euclid Algorithm” by Jebelean (see <a href="References.html#References">References</a>). The
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quotients of two double-limb inputs are collected as a 2 by 2 matrix with
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single-limb elements. This is done by the function <code>mpn_hgcd2</code>. The
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resulting matrix is applied to the inputs using <code>mpn_mul_1</code> and
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<code>mpn_submul_1</code>. Each iteration usually reduces the inputs by almost one
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limb. In the rare case of a large quotient, no progress can be made by
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examining just the most significant two limbs, and the quotient is computed
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using plain division.
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</p>
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<p>The resulting algorithm is asymptotically <em>O(N^2)</em>, just as the Euclidean
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algorithm and the binary algorithm. The quadratic part of the work are
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the calls to <code>mpn_mul_1</code> and <code>mpn_submul_1</code>. For small sizes, the
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linear work is also significant. There are roughly <em>N</em> calls to the
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<code>mpn_hgcd2</code> function. This function uses a couple of important
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optimizations:
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</p>
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<ul>
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<li> It uses the same relaxed notion of correctness as <code>mpn_hgcd</code> (see next
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section). This means that when called with the most significant two limbs of
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two large numbers, the returned matrix does not always correspond exactly to
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the initial quotient sequence for the two large numbers; the final quotient
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may sometimes be one off.
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</li><li> It takes advantage of the fact the quotients are usually small. The division
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operator is not used, since the corresponding assembler instruction is very
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slow on most architectures. (This code could probably be improved further, it
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uses many branches that are unfriendly to prediction).
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</li><li> It switches from double-limb calculations to single-limb calculations half-way
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through, when the input numbers have been reduced in size from two limbs to
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one and a half.
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</li></ul>
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<hr>
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<div class="header">
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<p>
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Next: <a href="Subquadratic-GCD.html#Subquadratic-GCD" accesskey="n" rel="next">Subquadratic GCD</a>, Previous: <a href="Binary-GCD.html#Binary-GCD" accesskey="p" rel="prev">Binary GCD</a>, Up: <a href="Greatest-Common-Divisor-Algorithms.html#Greatest-Common-Divisor-Algorithms" accesskey="u" rel="up">Greatest Common Divisor Algorithms</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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</div>
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</body>
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</html>
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