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95 lines
4.1 KiB
HTML
95 lines
4.1 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>Nth Root 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="Nth Root Algorithm (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="Root-Extraction-Algorithms.html#Root-Extraction-Algorithms" rel="up" title="Root Extraction Algorithms">
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<link href="Perfect-Square-Algorithm.html#Perfect-Square-Algorithm" rel="next" title="Perfect Square Algorithm">
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<link href="Square-Root-Algorithm.html#Square-Root-Algorithm" rel="prev" title="Square Root 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="Nth-Root-Algorithm"></a>
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<div class="header">
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<p>
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Next: <a href="Perfect-Square-Algorithm.html#Perfect-Square-Algorithm" accesskey="n" rel="next">Perfect Square Algorithm</a>, Previous: <a href="Square-Root-Algorithm.html#Square-Root-Algorithm" accesskey="p" rel="prev">Square Root Algorithm</a>, Up: <a href="Root-Extraction-Algorithms.html#Root-Extraction-Algorithms" accesskey="u" rel="up">Root Extraction 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="Nth-Root"></a>
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<h4 class="subsection">15.5.2 Nth Root</h4>
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<a name="index-Root-extraction-algorithm"></a>
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<a name="index-Nth-root-algorithm"></a>
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<p>Integer Nth roots are taken using Newton’s method with the following
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iteration, where <em>A</em> is the input and <em>n</em> is the root to be taken.
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</p>
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<div class="example">
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<pre class="example"> 1 A
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a[i+1] = - * ( --------- + (n-1)*a[i] )
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n a[i]^(n-1)
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</pre></div>
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<p>The initial approximation <em>a[1]</em> is generated bitwise by successively
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powering a trial root with or without new 1 bits, aiming to be just above the
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true root. The iteration converges quadratically when started from a good
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approximation. When <em>n</em> is large more initial bits are needed to get
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good convergence. The current implementation is not particularly well
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optimized.
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</p>
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</body>
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</html>
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