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102 lines
4.1 KiB
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102 lines
4.1 KiB
HTML
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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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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<title>Lucas Numbers 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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<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="Random-Number-Algorithms.html#Random-Number-Algorithms" rel="next" title="Random Number Algorithms">
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<link href="Fibonacci-Numbers-Algorithm.html#Fibonacci-Numbers-Algorithm" rel="prev" title="Fibonacci Numbers Algorithm">
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</head>
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<body lang="en">
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<a name="Lucas-Numbers-Algorithm"></a>
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<div class="header">
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<p>
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Next: <a href="Random-Number-Algorithms.html#Random-Number-Algorithms" accesskey="n" rel="next">Random Number Algorithms</a>, Previous: <a href="Fibonacci-Numbers-Algorithm.html#Fibonacci-Numbers-Algorithm" accesskey="p" rel="prev">Fibonacci Numbers 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="Lucas-Numbers"></a>
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<h4 class="subsection">15.7.5 Lucas Numbers</h4>
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<a name="index-Lucas-number-algorithm"></a>
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<p><code>mpz_lucnum2_ui</code> derives a pair of Lucas numbers from a pair of Fibonacci
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numbers with the following simple formulas.
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</p>
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<div class="example">
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<pre class="example">L[k] = F[k] + 2*F[k-1]
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L[k-1] = 2*F[k] - F[k-1]
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</pre></div>
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<p><code>mpz_lucnum_ui</code> is only interested in <em>L[n]</em>, and some work can be
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saved. Trailing zero bits on <em>n</em> can be handled with a single square
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each.
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</p>
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<div class="example">
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<pre class="example">L[2k] = L[k]^2 - 2*(-1)^k
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</pre></div>
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<p>And the lowest 1 bit can be handled with one multiply of a pair of Fibonacci
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numbers, similar to what <code>mpz_fib_ui</code> does.
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</p>
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<div class="example">
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<pre class="example">L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k
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</pre></div>
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</body>
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</html>
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