You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
138 lines
7.0 KiB
HTML
138 lines
7.0 KiB
HTML
4 years ago
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
|
||
|
|
<html>
|
||
|
|
<!-- This manual describes how to install and use the GNU multiple precision
|
||
|
|
arithmetic library, version 6.1.0.
|
||
|
|
|
||
|
|
Copyright 1991, 1993-2015 Free Software Foundation, Inc.
|
||
|
|
|
||
|
|
Permission is granted to copy, distribute and/or modify this document under
|
||
|
|
the terms of the GNU Free Documentation License, Version 1.3 or any later
|
||
|
|
version published by the Free Software Foundation; with no Invariant Sections,
|
||
|
|
with the Front-Cover Texts being "A GNU Manual", and with the Back-Cover
|
||
|
|
Texts being "You have freedom to copy and modify this GNU Manual, like GNU
|
||
|
|
software". A copy of the license is included in
|
||
|
|
GNU Free Documentation License. -->
|
||
|
|
<!-- Created by GNU Texinfo 6.4, http://www.gnu.org/software/texinfo/ -->
|
||
|
|
<head>
|
||
|
|
<title>Exact Division (GNU MP 6.1.0)</title>
|
||
|
|
|
||
|
|
<meta name="description" content="How to install and use the GNU multiple precision arithmetic library, version 6.1.0.">
|
||
|
|
<meta name="keywords" content="Exact Division (GNU MP 6.1.0)">
|
||
|
|
<meta name="resource-type" content="document">
|
||
|
|
<meta name="distribution" content="global">
|
||
|
|
<meta name="Generator" content="makeinfo">
|
||
|
|
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1">
|
||
|
|
<link href="index.html#Top" rel="start" title="Top">
|
||
|
|
<link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
|
||
|
|
<link href="Division-Algorithms.html#Division-Algorithms" rel="up" title="Division Algorithms">
|
||
|
|
<link href="Exact-Remainder.html#Exact-Remainder" rel="next" title="Exact Remainder">
|
||
|
|
<link href="Block_002dWise-Barrett-Division.html#Block_002dWise-Barrett-Division" rel="prev" title="Block-Wise Barrett Division">
|
||
|
|
<style type="text/css">
|
||
|
|
<!--
|
||
|
|
a.summary-letter {text-decoration: none}
|
||
|
|
blockquote.indentedblock {margin-right: 0em}
|
||
|
|
blockquote.smallindentedblock {margin-right: 0em; font-size: smaller}
|
||
|
|
blockquote.smallquotation {font-size: smaller}
|
||
|
|
div.display {margin-left: 3.2em}
|
||
|
|
div.example {margin-left: 3.2em}
|
||
|
|
div.lisp {margin-left: 3.2em}
|
||
|
|
div.smalldisplay {margin-left: 3.2em}
|
||
|
|
div.smallexample {margin-left: 3.2em}
|
||
|
|
div.smalllisp {margin-left: 3.2em}
|
||
|
|
kbd {font-style: oblique}
|
||
|
|
pre.display {font-family: inherit}
|
||
|
|
pre.format {font-family: inherit}
|
||
|
|
pre.menu-comment {font-family: serif}
|
||
|
|
pre.menu-preformatted {font-family: serif}
|
||
|
|
pre.smalldisplay {font-family: inherit; font-size: smaller}
|
||
|
|
pre.smallexample {font-size: smaller}
|
||
|
|
pre.smallformat {font-family: inherit; font-size: smaller}
|
||
|
|
pre.smalllisp {font-size: smaller}
|
||
|
|
span.nolinebreak {white-space: nowrap}
|
||
|
|
span.roman {font-family: initial; font-weight: normal}
|
||
|
|
span.sansserif {font-family: sans-serif; font-weight: normal}
|
||
|
|
ul.no-bullet {list-style: none}
|
||
|
|
-->
|
||
|
|
</style>
|
||
|
|
|
||
|
|
|
||
|
|
</head>
|
||
|
|
|
||
|
|
<body lang="en">
|
||
|
|
<a name="Exact-Division"></a>
|
||
|
|
<div class="header">
|
||
|
|
<p>
|
||
|
|
Next: <a href="Exact-Remainder.html#Exact-Remainder" accesskey="n" rel="next">Exact Remainder</a>, Previous: <a href="Block_002dWise-Barrett-Division.html#Block_002dWise-Barrett-Division" accesskey="p" rel="prev">Block-Wise Barrett 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>
|
||
|
|
</div>
|
||
|
|
<hr>
|
||
|
|
<a name="Exact-Division-1"></a>
|
||
|
|
<h4 class="subsection">15.2.5 Exact Division</h4>
|
||
|
|
|
||
|
|
|
||
|
|
<p>A so-called exact division is when the dividend is known to be an exact
|
||
|
|
multiple of the divisor. Jebelean’s exact division algorithm uses this
|
||
|
|
knowledge to make some significant optimizations (see <a href="References.html#References">References</a>).
|
||
|
|
</p>
|
||
|
|
<p>The idea can be illustrated in decimal for example with 368154 divided by
|
||
|
|
543. Because the low digit of the dividend is 4, the low digit of the
|
||
|
|
quotient must be 8. This is arrived at from <em>4*7 mod 10</em>, using the fact 7 is the modular inverse of 3 (the low digit of
|
||
|
|
the divisor), since <em>3*7
|
||
|
|
≡ 1 mod 10</em>. So <em>8*543=4344</em> can be
|
||
|
|
subtracted from the dividend leaving 363810. Notice the low digit has become
|
||
|
|
zero.
|
||
|
|
</p>
|
||
|
|
<p>The procedure is repeated at the second digit, with the next quotient digit 7
|
||
|
|
(<em>7 ≡ 1*7 mod 10</em>), subtracting
|
||
|
|
<em>7*543=3801</em>, leaving 325800. And finally at
|
||
|
|
the third digit with quotient digit 6 (<em>8*7
|
||
|
|
mod 10</em>), subtracting <em>6*543=3258</em> leaving 0.
|
||
|
|
So the quotient is 678.
|
||
|
|
</p>
|
||
|
|
<p>Notice however that the multiplies and subtractions don’t need to extend past
|
||
|
|
the low three digits of the dividend, since that’s enough to determine the
|
||
|
|
three quotient digits. For the last quotient digit no subtraction is needed
|
||
|
|
at all. On a 2NxN division like this one, only about half the work of
|
||
|
|
a normal basecase division is necessary.
|
||
|
|
</p>
|
||
|
|
<p>For an NxM exact division producing Q=N-M quotient limbs, the
|
||
|
|
saving over a normal basecase division is in two parts. Firstly, each of the
|
||
|
|
Q quotient limbs needs only one multiply, not a 2x1 divide and
|
||
|
|
multiply. Secondly, the crossproducts are reduced when <em>Q>M</em> to
|
||
|
|
<em>Q*M-M*(M+1)/2</em>, or when <em>Q<=M</em> to <em>Q*(Q-1)/2</em>. Notice the savings are complementary. If Q is big then many
|
||
|
|
divisions are saved, or if Q is small then the crossproducts reduce to a small
|
||
|
|
number.
|
||
|
|
</p>
|
||
|
|
<p>The modular inverse used is calculated efficiently by <code>binvert_limb</code> in
|
||
|
|
<samp>gmp-impl.h</samp>. This does four multiplies for a 32-bit limb, or six for a
|
||
|
|
64-bit limb. <samp>tune/modlinv.c</samp> has some alternate implementations that
|
||
|
|
might suit processors better at bit twiddling than multiplying.
|
||
|
|
</p>
|
||
|
|
<p>The sub-quadratic exact division described by Jebelean in “Exact Division
|
||
|
|
with Karatsuba Complexity” is not currently implemented. It uses a
|
||
|
|
rearrangement similar to the divide and conquer for normal division
|
||
|
|
(see <a href="Divide-and-Conquer-Division.html#Divide-and-Conquer-Division">Divide and Conquer Division</a>), but operating from low to high. A
|
||
|
|
further possibility not currently implemented is “Bidirectional Exact Integer
|
||
|
|
Division” by Krandick and Jebelean which forms quotient limbs from both the
|
||
|
|
high and low ends of the dividend, and can halve once more the number of
|
||
|
|
crossproducts needed in a 2NxN division.
|
||
|
|
</p>
|
||
|
|
<p>A special case exact division by 3 exists in <code>mpn_divexact_by3</code>,
|
||
|
|
supporting Toom-3 multiplication and <code>mpq</code> canonicalizations. It forms
|
||
|
|
quotient digits with a multiply by the modular inverse of 3 (which is
|
||
|
|
<code>0xAA..AAB</code>) and uses two comparisons to determine a borrow for the next
|
||
|
|
limb. The multiplications don’t need to be on the dependent chain, as long as
|
||
|
|
the effect of the borrows is applied, which can help chips with pipelined
|
||
|
|
multipliers.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<hr>
|
||
|
|
<div class="header">
|
||
|
|
<p>
|
||
|
|
Next: <a href="Exact-Remainder.html#Exact-Remainder" accesskey="n" rel="next">Exact Remainder</a>, Previous: <a href="Block_002dWise-Barrett-Division.html#Block_002dWise-Barrett-Division" accesskey="p" rel="prev">Block-Wise Barrett 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>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
</body>
|
||
|
|
</html>
|