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112 lines
5.1 KiB
HTML
112 lines
5.1 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>Toom 4-Way Multiplication (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="Toom 4-Way Multiplication (GNU MP 6.1.0)">
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<meta name="resource-type" content="document">
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<meta name="distribution" content="global">
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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="Multiplication-Algorithms.html#Multiplication-Algorithms" rel="up" title="Multiplication Algorithms">
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<link href="Higher-degree-Toom_0027n_0027half.html#Higher-degree-Toom_0027n_0027half" rel="next" title="Higher degree Toom'n'half">
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<link href="Toom-3_002dWay-Multiplication.html#Toom-3_002dWay-Multiplication" rel="prev" title="Toom 3-Way Multiplication">
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<style type="text/css">
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</style>
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</head>
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<body lang="en">
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<a name="Toom-4_002dWay-Multiplication"></a>
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<div class="header">
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<p>
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Next: <a href="Higher-degree-Toom_0027n_0027half.html#Higher-degree-Toom_0027n_0027half" accesskey="n" rel="next">Higher degree Toom'n'half</a>, Previous: <a href="Toom-3_002dWay-Multiplication.html#Toom-3_002dWay-Multiplication" accesskey="p" rel="prev">Toom 3-Way Multiplication</a>, Up: <a href="Multiplication-Algorithms.html#Multiplication-Algorithms" accesskey="u" rel="up">Multiplication 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="Toom-4_002dWay-Multiplication-1"></a>
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<h4 class="subsection">15.1.4 Toom 4-Way Multiplication</h4>
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<a name="index-Toom-multiplication-1"></a>
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<p>Karatsuba and Toom-3 split the operands into 2 and 3 coefficients,
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respectively. Toom-4 analogously splits the operands into 4 coefficients.
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Using the notation from the section on Toom-3 multiplication, we form two
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polynomials:
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</p>
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<div class="display">
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<pre class="display"><em>X(t) = x3*t^3 + x2*t^2 + x1*t + x0</em>
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<em>Y(t) = y3*t^3 + y2*t^2 + y1*t + y0</em>
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</pre></div>
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<p><em>X(t)</em> and <em>Y(t)</em> are evaluated and multiplied at 7 points, giving
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values of <em>W(t)</em> at those points. In GMP the following points are used,
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</p>
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<blockquote>
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<table>
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<tr><td>Point</td><td>Value</td></tr>
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<tr><td><em>t=0</em></td><td><em>x0 * y0</em>, which gives w0 immediately</td></tr>
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<tr><td><em>t=1/2</em></td><td><em>(x3+2*x2+4*x1+8*x0) * (y3+2*y2+4*y1+8*y0)</em></td></tr>
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<tr><td><em>t=-1/2</em></td><td><em>(-x3+2*x2-4*x1+8*x0) * (-y3+2*y2-4*y1+8*y0)</em></td></tr>
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<tr><td><em>t=1</em></td><td><em>(x3+x2+x1+x0) * (y3+y2+y1+y0)</em></td></tr>
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<tr><td><em>t=-1</em></td><td><em>(-x3+x2-x1+x0) * (-y3+y2-y1+y0)</em></td></tr>
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<tr><td><em>t=2</em></td><td><em>(8*x3+4*x2+2*x1+x0) * (8*y3+4*y2+2*y1+y0)</em></td></tr>
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<tr><td><em>t=inf</em></td><td><em>x3 * y3</em>, which gives w6 immediately</td></tr>
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</table>
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</blockquote>
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<p>The number of additions and subtractions for Toom-4 is much larger than for Toom-3.
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But several subexpressions occur multiple times, for example <em>x2+x0</em>, occurs
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for both <em>t=1</em> and <em>t=-1</em>.
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</p>
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<p>Toom-4 is asymptotically <em>O(N^1.404<!-- /@w -->)</em>, the exponent being
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<em>log(7)/log(4)</em>, representing 7 recursive multiplies of 1/4 the
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original size each.
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</p>
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</body>
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</html>
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