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118 lines
5.2 KiB
HTML
118 lines
5.2 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>Basecase 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="Basecase Multiplication (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="Multiplication-Algorithms.html#Multiplication-Algorithms" rel="up" title="Multiplication Algorithms">
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<link href="Karatsuba-Multiplication.html#Karatsuba-Multiplication" rel="next" title="Karatsuba Multiplication">
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<link href="Multiplication-Algorithms.html#Multiplication-Algorithms" rel="prev" title="Multiplication Algorithms">
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</style>
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</head>
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<body lang="en">
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<a name="Basecase-Multiplication"></a>
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<div class="header">
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<p>
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Next: <a href="Karatsuba-Multiplication.html#Karatsuba-Multiplication" accesskey="n" rel="next">Karatsuba Multiplication</a>, Previous: <a href="Multiplication-Algorithms.html#Multiplication-Algorithms" accesskey="p" rel="prev">Multiplication Algorithms</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="Basecase-Multiplication-1"></a>
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<h4 class="subsection">15.1.1 Basecase Multiplication</h4>
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<p>Basecase NxM multiplication is a straightforward rectangular set of
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cross-products, the same as long multiplication done by hand and for that
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reason sometimes known as the schoolbook or grammar school method. This is an
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<em>O(N*M)</em> algorithm. See Knuth section 4.3.1 algorithm M
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(see <a href="References.html#References">References</a>), and the <samp>mpn/generic/mul_basecase.c</samp> code.
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</p>
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<p>Assembly implementations of <code>mpn_mul_basecase</code> are essentially the same
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as the generic C code, but have all the usual assembly tricks and
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obscurities introduced for speed.
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</p>
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<p>A square can be done in roughly half the time of a multiply, by using the fact
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that the cross products above and below the diagonal are the same. A triangle
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of products below the diagonal is formed, doubled (left shift by one bit), and
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then the products on the diagonal added. This can be seen in
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<samp>mpn/generic/sqr_basecase.c</samp>. Again the assembly implementations take
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essentially the same approach.
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</p>
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<div class="example">
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<pre class="example"> u0 u1 u2 u3 u4
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+---+---+---+---+---+
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u0 | d | | | | |
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+---+---+---+---+---+
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u1 | | d | | | |
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+---+---+---+---+---+
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u2 | | | d | | |
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+---+---+---+---+---+
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u3 | | | | d | |
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+---+---+---+---+---+
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u4 | | | | | d |
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+---+---+---+---+---+
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</pre></div>
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<p>In practice squaring isn’t a full 2x faster than multiplying, it’s
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usually around 1.5x. Less than 1.5x probably indicates
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<code>mpn_sqr_basecase</code> wants improving on that CPU.
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</p>
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<p>On some CPUs <code>mpn_mul_basecase</code> can be faster than the generic C
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<code>mpn_sqr_basecase</code> on some small sizes. <code>SQR_BASECASE_THRESHOLD</code> is
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the size at which to use <code>mpn_sqr_basecase</code>, this will be zero if that
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routine should be used always.
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</p>
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</body>
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</html>
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