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136 lines
6.6 KiB
HTML
136 lines
6.6 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>Fibonacci 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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<meta name="keywords" content="Fibonacci Numbers Algorithm (GNU MP 6.1.0)">
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<meta name="resource-type" content="document">
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<meta name="distribution" content="global">
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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="Other-Algorithms.html#Other-Algorithms" rel="up" title="Other Algorithms">
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<link href="Lucas-Numbers-Algorithm.html#Lucas-Numbers-Algorithm" rel="next" title="Lucas Numbers Algorithm">
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<link href="Binomial-Coefficients-Algorithm.html#Binomial-Coefficients-Algorithm" rel="prev" title="Binomial Coefficients Algorithm">
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-->
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</style>
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</head>
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<body lang="en">
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<a name="Fibonacci-Numbers-Algorithm"></a>
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<div class="header">
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<p>
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Next: <a href="Lucas-Numbers-Algorithm.html#Lucas-Numbers-Algorithm" accesskey="n" rel="next">Lucas Numbers Algorithm</a>, Previous: <a href="Binomial-Coefficients-Algorithm.html#Binomial-Coefficients-Algorithm" accesskey="p" rel="prev">Binomial Coefficients 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="Fibonacci-Numbers"></a>
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<h4 class="subsection">15.7.4 Fibonacci Numbers</h4>
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<a name="index-Fibonacci-number-algorithm"></a>
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<p>The Fibonacci functions <code>mpz_fib_ui</code> and <code>mpz_fib2_ui</code> are designed
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for calculating isolated <em>F[n]</em> or <em>F[n]</em>,<em>F[n-1]</em>
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values efficiently.
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</p>
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<p>For small <em>n</em>, a table of single limb values in <code>__gmp_fib_table</code> is
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used. On a 32-bit limb this goes up to <em>F[47]</em>, or on a 64-bit limb
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up to <em>F[93]</em>. For convenience the table starts at <em>F[-1]</em>.
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</p>
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<p>Beyond the table, values are generated with a binary powering algorithm,
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calculating a pair <em>F[n]</em> and <em>F[n-1]</em> working from high to
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low across the bits of <em>n</em>. The formulas used are
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</p>
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<div class="example">
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<pre class="example">F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k
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F[2k-1] = F[k]^2 + F[k-1]^2
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F[2k] = F[2k+1] - F[2k-1]
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</pre></div>
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<p>At each step, <em>k</em> is the high <em>b</em> bits of <em>n</em>. If the next bit
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of <em>n</em> is 0 then <em>F[2k]</em>,<em>F[2k-1]</em> is used, or if
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it’s a 1 then <em>F[2k+1]</em>,<em>F[2k]</em> is used, and the process
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repeated until all bits of <em>n</em> are incorporated. Notice these formulas
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require just two squares per bit of <em>n</em>.
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</p>
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<p>It’d be possible to handle the first few <em>n</em> above the single limb table
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with simple additions, using the defining Fibonacci recurrence <em>F[k+1]=F[k]+F[k-1]</em>, but this is not done since it usually
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turns out to be faster for only about 10 or 20 values of <em>n</em>, and
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including a block of code for just those doesn’t seem worthwhile. If they
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really mattered it’d be better to extend the data table.
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</p>
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<p>Using a table avoids lots of calculations on small numbers, and makes small
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<em>n</em> go fast. A bigger table would make more small <em>n</em> go fast, it’s
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just a question of balancing size against desired speed. For GMP the code is
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kept compact, with the emphasis primarily on a good powering algorithm.
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</p>
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<p><code>mpz_fib2_ui</code> returns both <em>F[n]</em> and <em>F[n-1]</em>, but
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<code>mpz_fib_ui</code> is only interested in <em>F[n]</em>. In this case the last
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step of the algorithm can become one multiply instead of two squares. One of
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the following two formulas is used, according as <em>n</em> is odd or even.
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</p>
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<div class="example">
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<pre class="example">F[2k] = F[k]*(F[k]+2F[k-1])
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F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k
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</pre></div>
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<p><em>F[2k+1]</em> here is the same as above, just rearranged to be a
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multiply. For interest, the <em>2*(-1)^k</em> term both here and above
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can be applied just to the low limb of the calculation, without a carry or
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borrow into further limbs, which saves some code size. See comments with
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<code>mpz_fib_ui</code> and the internal <code>mpn_fib2_ui</code> for how this is done.
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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="Lucas-Numbers-Algorithm.html#Lucas-Numbers-Algorithm" accesskey="n" rel="next">Lucas Numbers Algorithm</a>, Previous: <a href="Binomial-Coefficients-Algorithm.html#Binomial-Coefficients-Algorithm" accesskey="p" rel="prev">Binomial Coefficients 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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