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138 lines
6.3 KiB
HTML
138 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>Square 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="Square Root 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 name="Generator" content="makeinfo">
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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="Nth-Root-Algorithm.html#Nth-Root-Algorithm" rel="next" title="Nth Root Algorithm">
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<link href="Root-Extraction-Algorithms.html#Root-Extraction-Algorithms" rel="prev" title="Root Extraction Algorithms">
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<style type="text/css">
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a.summary-letter {text-decoration: none}
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div.display {margin-left: 3.2em}
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span.sansserif {font-family: sans-serif; font-weight: normal}
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ul.no-bullet {list-style: none}
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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="Square-Root-Algorithm"></a>
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<div class="header">
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<p>
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Next: <a href="Nth-Root-Algorithm.html#Nth-Root-Algorithm" accesskey="n" rel="next">Nth Root Algorithm</a>, Previous: <a href="Root-Extraction-Algorithms.html#Root-Extraction-Algorithms" accesskey="p" rel="prev">Root Extraction Algorithms</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="Square-Root"></a>
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<h4 class="subsection">15.5.1 Square Root</h4>
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<a name="index-Square-root-algorithm"></a>
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<a name="index-Karatsuba-square-root-algorithm"></a>
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<p>Square roots are taken using the “Karatsuba Square Root” algorithm by Paul
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Zimmermann (see <a href="References.html#References">References</a>).
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</p>
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<p>An input <em>n</em> is split into four parts of <em>k</em> bits each, so with
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<em>b=2^k</em> we have <em>n = a3*b^3 + a2*b^2
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+ a1*b + a0</em>. Part a3 must be “normalized” so that either the high or
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second highest bit is set. In GMP, <em>k</em> is kept on a limb boundary and
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the input is left shifted (by an even number of bits) to normalize.
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</p>
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<p>The square root of the high two parts is taken, by recursive application of
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the algorithm (bottoming out in a one-limb Newton’s method),
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</p>
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<div class="example">
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<pre class="example">s1,r1 = sqrtrem (a3*b + a2)
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</pre></div>
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<p>This is an approximation to the desired root and is extended by a division to
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give <em>s</em>,<em>r</em>,
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</p>
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<div class="example">
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<pre class="example">q,u = divrem (r1*b + a1, 2*s1)
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s = s1*b + q
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r = u*b + a0 - q^2
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</pre></div>
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<p>The normalization requirement on a3 means at this point <em>s</em> is
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either correct or 1 too big. <em>r</em> is negative in the latter case, so
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</p>
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<div class="example">
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<pre class="example">if r < 0 then
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r = r + 2*s - 1
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s = s - 1
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</pre></div>
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<p>The algorithm is expressed in a divide and conquer form, but as noted in the
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paper it can also be viewed as a discrete variant of Newton’s method, or as a
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variation on the schoolboy method (no longer taught) for square roots two
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digits at a time.
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</p>
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<p>If the remainder <em>r</em> is not required then usually only a few high limbs
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of <em>r</em> and <em>u</em> need to be calculated to determine whether an
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adjustment to <em>s</em> is required. This optimization is not currently
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implemented.
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</p>
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<p>In the Karatsuba multiplication range this algorithm is <em>O(1.5*M(N/2))</em>, where <em>M(n)</em> is the time to multiply two numbers
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of <em>n</em> limbs. In the FFT multiplication range this grows to a bound of
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<em>O(6*M(N/2))</em>. In practice a factor of about 1.5 to 1.8 is
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found in the Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range.
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</p>
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<p>The algorithm does all its calculations in integers and the resulting
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<code>mpn_sqrtrem</code> is used for both <code>mpz_sqrt</code> and <code>mpf_sqrt</code>.
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The extended precision given by <code>mpf_sqrt_ui</code> is obtained by
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padding with zero limbs.
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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="Nth-Root-Algorithm.html#Nth-Root-Algorithm" accesskey="n" rel="next">Nth Root Algorithm</a>, Previous: <a href="Root-Extraction-Algorithms.html#Root-Extraction-Algorithms" accesskey="p" rel="prev">Root Extraction Algorithms</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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</body>
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</html>
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