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102 lines
4.6 KiB
HTML
102 lines
4.6 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>Small Quotient 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="Small Quotient Division (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="Division-Algorithms.html#Division-Algorithms" rel="up" title="Division Algorithms">
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<link href="Greatest-Common-Divisor-Algorithms.html#Greatest-Common-Divisor-Algorithms" rel="next" title="Greatest Common Divisor Algorithms">
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<link href="Exact-Remainder.html#Exact-Remainder" rel="prev" title="Exact Remainder">
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<style type="text/css">
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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="Small-Quotient-Division"></a>
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<div class="header">
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<p>
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Previous: <a href="Exact-Remainder.html#Exact-Remainder" accesskey="p" rel="prev">Exact Remainder</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="Small-Quotient-Division-1"></a>
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<h4 class="subsection">15.2.7 Small Quotient Division</h4>
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<p>An NxM division where the number of quotient limbs Q=N-M is
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small can be optimized somewhat.
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</p>
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<p>An ordinary basecase division normalizes the divisor by shifting it to make
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the high bit set, shifting the dividend accordingly, and shifting the
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remainder back down at the end of the calculation. This is wasteful if only a
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few quotient limbs are to be formed. Instead a division of just the top
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<em>2*Q</em> limbs of the dividend by the top Q limbs of the divisor can be
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used to form a trial quotient. This requires only those limbs normalized, not
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the whole of the divisor and dividend.
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</p>
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<p>A multiply and subtract then applies the trial quotient to the M-Q
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unused limbs of the divisor and N-Q dividend limbs (which includes Q
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limbs remaining from the trial quotient division). The starting trial
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quotient can be 1 or 2 too big, but all cases of 2 too big and most cases of 1
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too big are detected by first comparing the most significant limbs that will
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arise from the subtraction. An addback is done if the quotient still turns
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out to be 1 too big.
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</p>
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<p>This whole procedure is essentially the same as one step of the basecase
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algorithm done in a Q limb base, though with the trial quotient test done only
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with the high limbs, not an entire Q limb “digit” product. The correctness
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of this weaker test can be established by following the argument of Knuth
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section 4.3.1 exercise 20 but with the <em>v2*q>b*r+u2</em> condition appropriately relaxed.
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</p>
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</body>
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</html>
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