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98 lines
4.5 KiB
HTML
98 lines
4.5 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>Rational Internals (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="Rational Internals (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="Internals.html#Internals" rel="up" title="Internals">
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<link href="Float-Internals.html#Float-Internals" rel="next" title="Float Internals">
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<link href="Integer-Internals.html#Integer-Internals" rel="prev" title="Integer Internals">
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</style>
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</head>
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<body lang="en">
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<a name="Rational-Internals"></a>
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<div class="header">
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<p>
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Next: <a href="Float-Internals.html#Float-Internals" accesskey="n" rel="next">Float Internals</a>, Previous: <a href="Integer-Internals.html#Integer-Internals" accesskey="p" rel="prev">Integer Internals</a>, Up: <a href="Internals.html#Internals" accesskey="u" rel="up">Internals</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="Rational-Internals-1"></a>
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<h3 class="section">16.2 Rational Internals</h3>
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<a name="index-Rational-internals"></a>
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<p><code>mpq_t</code> variables represent rationals using an <code>mpz_t</code> numerator and
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denominator (see <a href="Integer-Internals.html#Integer-Internals">Integer Internals</a>).
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</p>
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<p>The canonical form adopted is denominator positive (and non-zero), no common
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factors between numerator and denominator, and zero uniquely represented as
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0/1.
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</p>
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<p>It’s believed that casting out common factors at each stage of a calculation
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is best in general. A GCD is an <em>O(N^2)</em> operation so it’s better to do
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a few small ones immediately than to delay and have to do a big one later.
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Knowing the numerator and denominator have no common factors can be used for
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example in <code>mpq_mul</code> to make only two cross GCDs necessary, not four.
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</p>
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<p>This general approach to common factors is badly sub-optimal in the presence
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of simple factorizations or little prospect for cancellation, but GMP has no
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way to know when this will occur. As per <a href="Efficiency.html#Efficiency">Efficiency</a>, that’s left to
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applications. The <code>mpq_t</code> framework might still suit, with
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<code>mpq_numref</code> and <code>mpq_denref</code> for direct access to the numerator and
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denominator, or of course <code>mpz_t</code> variables can be used directly.
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</p>
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</body>
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</html>
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