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113 lines
6.0 KiB
HTML
113 lines
6.0 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>Divide and Conquer Division (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="Divide and Conquer Division (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="Division-Algorithms.html#Division-Algorithms" rel="up" title="Division Algorithms">
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<link href="Block_002dWise-Barrett-Division.html#Block_002dWise-Barrett-Division" rel="next" title="Block-Wise Barrett Division">
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<link href="Basecase-Division.html#Basecase-Division" rel="prev" title="Basecase Division">
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-->
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</style>
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</head>
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<body lang="en">
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<a name="Divide-and-Conquer-Division"></a>
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<div class="header">
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<p>
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Next: <a href="Block_002dWise-Barrett-Division.html#Block_002dWise-Barrett-Division" accesskey="n" rel="next">Block-Wise Barrett Division</a>, Previous: <a href="Basecase-Division.html#Basecase-Division" accesskey="p" rel="prev">Basecase Division</a>, Up: <a href="Division-Algorithms.html#Division-Algorithms" accesskey="u" rel="up">Division 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="Divide-and-Conquer-Division-1"></a>
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<h4 class="subsection">15.2.3 Divide and Conquer Division</h4>
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<p>For divisors larger than <code>DC_DIV_QR_THRESHOLD</code>, division is done by dividing.
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Or to be precise by a recursive divide and conquer algorithm based on work by
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Moenck and Borodin, Jebelean, and Burnikel and Ziegler (see <a href="References.html#References">References</a>).
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</p>
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<p>The algorithm consists essentially of recognising that a 2NxN division
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can be done with the basecase division algorithm (see <a href="Basecase-Division.html#Basecase-Division">Basecase Division</a>),
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but using N/2 limbs as a base, not just a single limb. This way the
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multiplications that arise are (N/2)x(N/2) and can take advantage of
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Karatsuba and higher multiplication algorithms (see <a href="Multiplication-Algorithms.html#Multiplication-Algorithms">Multiplication Algorithms</a>). The two “digits” of the quotient are formed by recursive
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Nx(N/2) divisions.
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</p>
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<p>If the (N/2)x(N/2) multiplies are done with a basecase multiplication
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then the work is about the same as a basecase division, but with more function
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call overheads and with some subtractions separated from the multiplies.
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These overheads mean that it’s only when N/2 is above
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<code>MUL_TOOM22_THRESHOLD</code> that divide and conquer is of use.
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</p>
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<p><code>DC_DIV_QR_THRESHOLD</code> is based on the divisor size N, so it will be somewhere
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above twice <code>MUL_TOOM22_THRESHOLD</code>, but how much above depends on the
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CPU. An optimized <code>mpn_mul_basecase</code> can lower <code>DC_DIV_QR_THRESHOLD</code> a
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little by offering a ready-made advantage over repeated <code>mpn_submul_1</code>
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calls.
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</p>
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<p>Divide and conquer is asymptotically <em>O(M(N)*log(N))</em> where
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<em>M(N)</em> is the time for an NxN multiplication done with FFTs. The
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actual time is a sum over multiplications of the recursed sizes, as can be
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seen near the end of section 2.2 of Burnikel and Ziegler. For example, within
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the Toom-3 range, divide and conquer is <em>2.63*M(N)</em>. With higher
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algorithms the <em>M(N)</em> term improves and the multiplier tends to <em>log(N)</em>. In practice, at moderate to large sizes, a 2NxN division
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is about 2 to 4 times slower than an NxN multiplication.
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</p>
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<hr>
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<div class="header">
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<p>
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Next: <a href="Block_002dWise-Barrett-Division.html#Block_002dWise-Barrett-Division" accesskey="n" rel="next">Block-Wise Barrett Division</a>, Previous: <a href="Basecase-Division.html#Basecase-Division" accesskey="p" rel="prev">Basecase Division</a>, Up: <a href="Division-Algorithms.html#Division-Algorithms" accesskey="u" rel="up">Division 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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