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139 lines
6.3 KiB
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139 lines
6.3 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>Factorial Algorithm (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="Factorial Algorithm (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="Other-Algorithms.html#Other-Algorithms" rel="up" title="Other Algorithms">
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<link href="Binomial-Coefficients-Algorithm.html#Binomial-Coefficients-Algorithm" rel="next" title="Binomial Coefficients Algorithm">
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<link href="Prime-Testing-Algorithm.html#Prime-Testing-Algorithm" rel="prev" title="Prime Testing Algorithm">
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</head>
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<body lang="en">
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<a name="Factorial-Algorithm"></a>
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<div class="header">
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<p>
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Next: <a href="Binomial-Coefficients-Algorithm.html#Binomial-Coefficients-Algorithm" accesskey="n" rel="next">Binomial Coefficients Algorithm</a>, Previous: <a href="Prime-Testing-Algorithm.html#Prime-Testing-Algorithm" accesskey="p" rel="prev">Prime Testing Algorithm</a>, Up: <a href="Other-Algorithms.html#Other-Algorithms" accesskey="u" rel="up">Other 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="Factorial"></a>
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<h4 class="subsection">15.7.2 Factorial</h4>
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<a name="index-Factorial-algorithm"></a>
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<p>Factorials are calculated by a combination of two algorithms. An idea is
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shared among them: to compute the odd part of the factorial; a final step
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takes account of the power of <em>2</em> term, by shifting.
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</p>
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<p>For small <em>n</em>, the odd factor of <em>n!</em> is computed with the simple
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observation that it is equal to the product of all positive odd numbers
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smaller than <em>n</em> times the odd factor of <em>[n/2]!</em>,
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where <em>[x]</em> is the integer part of <em>x</em>, and so on
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recursively. The procedure can be best illustrated with an example,
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</p>
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<blockquote>
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<p><em>23! = (23.21.19.17.15.13.11.9.7.5.3)(11.9.7.5.3)(5.3)2^{19}</em>
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</p></blockquote>
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<p>Current code collects all the factors in a single list, with a loop and no
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recursion, and compute the product, with no special care for repeated chunks.
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</p>
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<p>When <em>n</em> is larger, computation pass trough prime sieving. An helper
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function is used, as suggested by Peter Luschny:
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</p>
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<div class="example">
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<pre class="example"> n
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-----
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n! | | L(p,n)
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msf(n) = -------------- = | | p
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[n/2]!^2.2^k p=3
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</pre></div>
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<p>Where <em>p</em> ranges on odd prime numbers. The exponent <em>k</em> is chosen to
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obtain an odd integer number: <em>k</em> is the number of 1 bits in the binary
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representation of <em>[n/2]</em>. The function L<em>(p,n)</em>
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can be defined as zero when <em>p</em> is composite, and, for any prime
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<em>p</em>, it is computed with:
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</p>
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<div class="example">
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<pre class="example"> ---
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\ n
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L(p,n) = / [---] mod 2 <= log (n) .
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--- p^i p
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i>0
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</pre></div>
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<p>With this helper function, we are able to compute the odd part of <em>n!</em>
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using the recursion implied by <em>n!=[n/2]!^2*msf(n)*2^k</em>. The recursion stops using the
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small-<em>n</em> algorithm on some <em>[n/2^i]</em>.
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</p>
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<p>Both the above algorithms use binary splitting to compute the product of many
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small factors. At first as many products as possible are accumulated in a
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single register, generating a list of factors that fit in a machine word. This
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list is then split into halves, and the product is computed recursively.
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</p>
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<p>Such splitting is more efficient than repeated Nx1 multiplies since it
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forms big multiplies, allowing Karatsuba and higher algorithms to be used.
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And even below the Karatsuba threshold a big block of work can be more
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efficient for the basecase algorithm.
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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="Binomial-Coefficients-Algorithm.html#Binomial-Coefficients-Algorithm" accesskey="n" rel="next">Binomial Coefficients Algorithm</a>, Previous: <a href="Prime-Testing-Algorithm.html#Prime-Testing-Algorithm" accesskey="p" rel="prev">Prime Testing Algorithm</a>, Up: <a href="Other-Algorithms.html#Other-Algorithms" accesskey="u" rel="up">Other 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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