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98 lines
4.8 KiB
HTML
98 lines
4.8 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>Higher degree Toom'n'half (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="Higher degree Toom'n'half (GNU MP 6.1.0)">
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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="FFT-Multiplication.html#FFT-Multiplication" rel="next" title="FFT Multiplication">
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<link href="Toom-4_002dWay-Multiplication.html#Toom-4_002dWay-Multiplication" rel="prev" title="Toom 4-Way Multiplication">
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</head>
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<body lang="en">
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<a name="Higher-degree-Toom_0027n_0027half"></a>
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<div class="header">
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<p>
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Next: <a href="FFT-Multiplication.html#FFT-Multiplication" accesskey="n" rel="next">FFT Multiplication</a>, Previous: <a href="Toom-4_002dWay-Multiplication.html#Toom-4_002dWay-Multiplication" accesskey="p" rel="prev">Toom 4-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="Higher-degree-Toom_0027n_0027half-1"></a>
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<h4 class="subsection">15.1.5 Higher degree Toom’n’half</h4>
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<a name="index-Toom-multiplication-2"></a>
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<p>The Toom algorithms described above (see <a href="Toom-3_002dWay-Multiplication.html#Toom-3_002dWay-Multiplication">Toom 3-Way Multiplication</a>,
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see <a href="Toom-4_002dWay-Multiplication.html#Toom-4_002dWay-Multiplication">Toom 4-Way Multiplication</a>) generalizes to split into an arbitrary
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number of pieces. In general a split of two equally long operands into
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<em>r</em> pieces leads to evaluations and pointwise multiplications done at
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<em>2*r-1</em> points. To fully exploit symmetries it would be better to have
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a multiple of 4 points, that’s why for higher degree Toom’n’half is used.
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</p>
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<p>Toom’n’half means that the existence of one more piece is considered for a
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single operand. It can be virtual, i.e. zero, or real, when the two operand
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are not exactly balanced. By choosing an even <em>r</em>,
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Toom-<em>r+1/2</em> requires <em>2r</em> points, a multiple of four.
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</p>
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<p>The four-plets of points include 0, <em>inf</em>, +1, -1 and
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<em>+-2^i</em>, <em>+-2^-i</em> . Each of them giving shortcuts for the
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evaluation phase and for some steps in the interpolation phase. Further tricks
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are used to reduce the memory footprint of the whole multiplication algorithm
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to a memory buffer equanl in size to the result of the product.
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</p>
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<p>Current GMP uses both Toom-6’n’half and Toom-8’n’half.
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</p>
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</body>
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</html>
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