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96 lines
4.4 KiB
HTML
96 lines
4.4 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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<title>Basecase Division (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="Basecase Division (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="Division-Algorithms.html#Division-Algorithms" rel="up" title="Division Algorithms">
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<link href="Divide-and-Conquer-Division.html#Divide-and-Conquer-Division" rel="next" title="Divide and Conquer Division">
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<link href="Single-Limb-Division.html#Single-Limb-Division" rel="prev" title="Single Limb Division">
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</head>
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<a name="Basecase-Division"></a>
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<div class="header">
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<p>
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Next: <a href="Divide-and-Conquer-Division.html#Divide-and-Conquer-Division" accesskey="n" rel="next">Divide and Conquer Division</a>, Previous: <a href="Single-Limb-Division.html#Single-Limb-Division" accesskey="p" rel="prev">Single Limb Division</a>, Up: <a href="Division-Algorithms.html#Division-Algorithms" accesskey="u" rel="up">Division 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="Basecase-Division-1"></a>
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<h4 class="subsection">15.2.2 Basecase Division</h4>
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<p>Basecase NxM division is like long division done by hand, but in base
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<em>2^mp_bits_per_limb</em>. See Knuth
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section 4.3.1 algorithm D, and <samp>mpn/generic/sb_divrem_mn.c</samp>.
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</p>
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<p>Briefly stated, while the dividend remains larger than the divisor, a high
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quotient limb is formed and the Nx1 product <em>q*d</em> subtracted at
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the top end of the dividend. With a normalized divisor (most significant bit
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set), each quotient limb can be formed with a 2x1 division and a
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1x1 multiplication plus some subtractions. The 2x1 division is
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by the high limb of the divisor and is done either with a hardware divide or a
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multiply by inverse (the same as in <a href="Single-Limb-Division.html#Single-Limb-Division">Single Limb Division</a>) whichever is
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faster. Such a quotient is sometimes one too big, requiring an addback of the
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divisor, but that happens rarely.
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</p>
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<p>With Q=N-M being the number of quotient limbs, this is an
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<em>O(Q*M)</em> algorithm and will run at a speed similar to a basecase
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QxM multiplication, differing in fact only in the extra multiply and
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divide for each of the Q quotient limbs.
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</p>
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</body>
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</html>
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