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130 lines
6.4 KiB
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130 lines
6.4 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>Subquadratic GCD (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="Subquadratic GCD (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="Extended-GCD.html#Extended-GCD" rel="next" title="Extended GCD">
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<link href="Lehmer_0027s-Algorithm.html#Lehmer_0027s-Algorithm" rel="prev" title="Lehmer's Algorithm">
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</head>
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<body lang="en">
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<a name="Subquadratic-GCD"></a>
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<div class="header">
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<p>
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Next: <a href="Extended-GCD.html#Extended-GCD" accesskey="n" rel="next">Extended GCD</a>, Previous: <a href="Lehmer_0027s-Algorithm.html#Lehmer_0027s-Algorithm" accesskey="p" rel="prev">Lehmer's Algorithm</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="Subquadratic-GCD-1"></a>
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<h4 class="subsection">15.3.3 Subquadratic GCD</h4>
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<p>For inputs larger than <code>GCD_DC_THRESHOLD</code>, GCD is computed via the HGCD
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(Half GCD) function, as a generalization to Lehmer’s algorithm.
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</p>
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<p>Let the inputs <em>a,b</em> be of size <em>N</em> limbs each. Put <em>S = floor(N/2) + 1</em>. Then HGCD(a,b) returns a transformation
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matrix <em>T</em> with non-negative elements, and reduced numbers <em>(c;d) =
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T^{-1} (a;b)</em>. The reduced numbers <em>c,d</em> must be larger than <em>S</em>
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limbs, while their difference <em>abs(c-d)</em> must fit in <em>S</em> limbs. The
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matrix elements will also be of size roughly <em>N/2</em>.
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</p>
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<p>The HGCD base case uses Lehmer’s algorithm, but with the above stop condition
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that returns reduced numbers and the corresponding transformation matrix
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half-way through. For inputs larger than <code>HGCD_THRESHOLD</code>, HGCD is
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computed recursively, using the divide and conquer algorithm in “On
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Schönhage’s algorithm and subquadratic integer GCD computation” by Möller
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(see <a href="References.html#References">References</a>). The recursive algorithm consists of these main
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steps.
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</p>
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<ul>
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<li> Call HGCD recursively, on the most significant <em>N/2</em> limbs. Apply the
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resulting matrix <em>T_1</em> to the full numbers, reducing them to a size just
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above <em>3N/2</em>.
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</li><li> Perform a small number of division or subtraction steps to reduce the numbers
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to size below <em>3N/2</em>. This is essential mainly for the unlikely case of
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large quotients.
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</li><li> Call HGCD recursively, on the most significant <em>N/2</em> limbs of the reduced
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numbers. Apply the resulting matrix <em>T_2</em> to the full numbers, reducing
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them to a size just above <em>N/2</em>.
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</li><li> Compute <em>T = T_1 T_2</em>.
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</li><li> Perform a small number of division and subtraction steps to satisfy the
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requirements, and return.
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</li></ul>
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<p>GCD is then implemented as a loop around HGCD, similarly to Lehmer’s
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algorithm. Where Lehmer repeatedly chops off the top two limbs, calls
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<code>mpn_hgcd2</code>, and applies the resulting matrix to the full numbers, the
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sub-quadratic GCD chops off the most significant third of the limbs (the
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proportion is a tuning parameter, and <em>1/3</em> seems to be more efficient
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than, e.g, <em>1/2</em>), calls <code>mpn_hgcd</code>, and applies the resulting
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matrix. Once the input numbers are reduced to size below
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<code>GCD_DC_THRESHOLD</code>, Lehmer’s algorithm is used for the rest of the work.
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</p>
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<p>The asymptotic running time of both HGCD and GCD is <em>O(M(N)*log(N))</em>,
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where <em>M(N)</em> is the time for multiplying two <em>N</em>-limb numbers.
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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="Extended-GCD.html#Extended-GCD" accesskey="n" rel="next">Extended GCD</a>, Previous: <a href="Lehmer_0027s-Algorithm.html#Lehmer_0027s-Algorithm" accesskey="p" rel="prev">Lehmer's Algorithm</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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