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222 lines
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HTML
222 lines
16 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>Integer 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="Integer Division (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="Integer-Functions.html#Integer-Functions" rel="up" title="Integer Functions">
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<link href="Integer-Exponentiation.html#Integer-Exponentiation" rel="next" title="Integer Exponentiation">
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<link href="Integer-Arithmetic.html#Integer-Arithmetic" rel="prev" title="Integer Arithmetic">
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</head>
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<body lang="en">
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<a name="Integer-Division"></a>
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<div class="header">
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<p>
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Next: <a href="Integer-Exponentiation.html#Integer-Exponentiation" accesskey="n" rel="next">Integer Exponentiation</a>, Previous: <a href="Integer-Arithmetic.html#Integer-Arithmetic" accesskey="p" rel="prev">Integer Arithmetic</a>, Up: <a href="Integer-Functions.html#Integer-Functions" accesskey="u" rel="up">Integer Functions</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="Division-Functions"></a>
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<h3 class="section">5.6 Division Functions</h3>
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<a name="index-Integer-division-functions"></a>
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<a name="index-Division-functions"></a>
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<p>Division is undefined if the divisor is zero. Passing a zero divisor to the
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division or modulo functions (including the modular powering functions
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<code>mpz_powm</code> and <code>mpz_powm_ui</code>), will cause an intentional division by
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zero. This lets a program handle arithmetic exceptions in these functions the
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same way as for normal C <code>int</code> arithmetic.
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</p>
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<dl>
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<dt><a name="index-mpz_005fcdiv_005fq"></a>Function: <em>void</em> <strong>mpz_cdiv_q</strong> <em>(mpz_t <var>q</var>, const mpz_t <var>n</var>, const mpz_t <var>d</var>)</em></dt>
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<dt><a name="index-mpz_005fcdiv_005fr"></a>Function: <em>void</em> <strong>mpz_cdiv_r</strong> <em>(mpz_t <var>r</var>, const mpz_t <var>n</var>, const mpz_t <var>d</var>)</em></dt>
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<dt><a name="index-mpz_005fcdiv_005fqr"></a>Function: <em>void</em> <strong>mpz_cdiv_qr</strong> <em>(mpz_t <var>q</var>, mpz_t <var>r</var>, const mpz_t <var>n</var>, const mpz_t <var>d</var>)</em></dt>
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<dt><a name="index-mpz_005fcdiv_005fq_005fui"></a>Function: <em>unsigned long int</em> <strong>mpz_cdiv_q_ui</strong> <em>(mpz_t <var>q</var>, const mpz_t <var>n</var>, unsigned long int <var>d</var><!-- /@w -->)</em></dt>
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<dt><a name="index-mpz_005fcdiv_005fr_005fui"></a>Function: <em>unsigned long int</em> <strong>mpz_cdiv_r_ui</strong> <em>(mpz_t <var>r</var>, const mpz_t <var>n</var>, unsigned long int <var>d</var><!-- /@w -->)</em></dt>
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<dt><a name="index-mpz_005fcdiv_005fqr_005fui"></a>Function: <em>unsigned long int</em> <strong>mpz_cdiv_qr_ui</strong> <em>(mpz_t <var>q</var>, mpz_t <var>r</var>, const <span class="nolinebreak">mpz_t</span> <var>n</var><!-- /@w -->, unsigned long int <var>d</var><!-- /@w -->)</em></dt>
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<dt><a name="index-mpz_005fcdiv_005fui"></a>Function: <em>unsigned long int</em> <strong>mpz_cdiv_ui</strong> <em>(const mpz_t <var>n</var>, unsigned long int <var>d</var><!-- /@w -->)</em></dt>
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<dt><a name="index-mpz_005fcdiv_005fq_005f2exp"></a>Function: <em>void</em> <strong>mpz_cdiv_q_2exp</strong> <em>(mpz_t <var>q</var>, const mpz_t <var>n</var>, <span class="nolinebreak">mp_bitcnt_t</span> <var>b</var><!-- /@w -->)</em></dt>
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<dt><a name="index-mpz_005fcdiv_005fr_005f2exp"></a>Function: <em>void</em> <strong>mpz_cdiv_r_2exp</strong> <em>(mpz_t <var>r</var>, const mpz_t <var>n</var>, <span class="nolinebreak">mp_bitcnt_t</span> <var>b</var><!-- /@w -->)</em></dt>
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</dl>
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<dl>
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<dt><a name="index-mpz_005ffdiv_005fq"></a>Function: <em>void</em> <strong>mpz_fdiv_q</strong> <em>(mpz_t <var>q</var>, const mpz_t <var>n</var>, const mpz_t <var>d</var>)</em></dt>
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<dt><a name="index-mpz_005ffdiv_005fr"></a>Function: <em>void</em> <strong>mpz_fdiv_r</strong> <em>(mpz_t <var>r</var>, const mpz_t <var>n</var>, const mpz_t <var>d</var>)</em></dt>
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<dt><a name="index-mpz_005ffdiv_005fqr"></a>Function: <em>void</em> <strong>mpz_fdiv_qr</strong> <em>(mpz_t <var>q</var>, mpz_t <var>r</var>, const mpz_t <var>n</var>, const mpz_t <var>d</var>)</em></dt>
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<dt><a name="index-mpz_005ffdiv_005fq_005fui"></a>Function: <em>unsigned long int</em> <strong>mpz_fdiv_q_ui</strong> <em>(mpz_t <var>q</var>, const mpz_t <var>n</var>, unsigned long int <var>d</var><!-- /@w -->)</em></dt>
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<dt><a name="index-mpz_005ffdiv_005fr_005fui"></a>Function: <em>unsigned long int</em> <strong>mpz_fdiv_r_ui</strong> <em>(mpz_t <var>r</var>, const mpz_t <var>n</var>, unsigned long int <var>d</var><!-- /@w -->)</em></dt>
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<dt><a name="index-mpz_005ffdiv_005fqr_005fui"></a>Function: <em>unsigned long int</em> <strong>mpz_fdiv_qr_ui</strong> <em>(mpz_t <var>q</var>, mpz_t <var>r</var>, const <span class="nolinebreak">mpz_t</span> <var>n</var><!-- /@w -->, unsigned long int <var>d</var><!-- /@w -->)</em></dt>
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<dt><a name="index-mpz_005ffdiv_005fui"></a>Function: <em>unsigned long int</em> <strong>mpz_fdiv_ui</strong> <em>(const mpz_t <var>n</var>, unsigned long int <var>d</var><!-- /@w -->)</em></dt>
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<dt><a name="index-mpz_005ffdiv_005fq_005f2exp"></a>Function: <em>void</em> <strong>mpz_fdiv_q_2exp</strong> <em>(mpz_t <var>q</var>, const mpz_t <var>n</var>, <span class="nolinebreak">mp_bitcnt_t</span> <var>b</var><!-- /@w -->)</em></dt>
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<dt><a name="index-mpz_005ffdiv_005fr_005f2exp"></a>Function: <em>void</em> <strong>mpz_fdiv_r_2exp</strong> <em>(mpz_t <var>r</var>, const mpz_t <var>n</var>, <span class="nolinebreak">mp_bitcnt_t</span> <var>b</var><!-- /@w -->)</em></dt>
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</dl>
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<dl>
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<dt><a name="index-mpz_005ftdiv_005fq"></a>Function: <em>void</em> <strong>mpz_tdiv_q</strong> <em>(mpz_t <var>q</var>, const mpz_t <var>n</var>, const mpz_t <var>d</var>)</em></dt>
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<dt><a name="index-mpz_005ftdiv_005fr"></a>Function: <em>void</em> <strong>mpz_tdiv_r</strong> <em>(mpz_t <var>r</var>, const mpz_t <var>n</var>, const mpz_t <var>d</var>)</em></dt>
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<dt><a name="index-mpz_005ftdiv_005fqr"></a>Function: <em>void</em> <strong>mpz_tdiv_qr</strong> <em>(mpz_t <var>q</var>, mpz_t <var>r</var>, const mpz_t <var>n</var>, const mpz_t <var>d</var>)</em></dt>
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<dt><a name="index-mpz_005ftdiv_005fq_005fui"></a>Function: <em>unsigned long int</em> <strong>mpz_tdiv_q_ui</strong> <em>(mpz_t <var>q</var>, const mpz_t <var>n</var>, unsigned long int <var>d</var><!-- /@w -->)</em></dt>
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<dt><a name="index-mpz_005ftdiv_005fr_005fui"></a>Function: <em>unsigned long int</em> <strong>mpz_tdiv_r_ui</strong> <em>(mpz_t <var>r</var>, const mpz_t <var>n</var>, unsigned long int <var>d</var><!-- /@w -->)</em></dt>
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<dt><a name="index-mpz_005ftdiv_005fqr_005fui"></a>Function: <em>unsigned long int</em> <strong>mpz_tdiv_qr_ui</strong> <em>(mpz_t <var>q</var>, mpz_t <var>r</var>, const <span class="nolinebreak">mpz_t</span> <var>n</var><!-- /@w -->, unsigned long int <var>d</var><!-- /@w -->)</em></dt>
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<dt><a name="index-mpz_005ftdiv_005fui"></a>Function: <em>unsigned long int</em> <strong>mpz_tdiv_ui</strong> <em>(const mpz_t <var>n</var>, unsigned long int <var>d</var><!-- /@w -->)</em></dt>
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<dt><a name="index-mpz_005ftdiv_005fq_005f2exp"></a>Function: <em>void</em> <strong>mpz_tdiv_q_2exp</strong> <em>(mpz_t <var>q</var>, const mpz_t <var>n</var>, <span class="nolinebreak">mp_bitcnt_t</span> <var>b</var><!-- /@w -->)</em></dt>
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<dt><a name="index-mpz_005ftdiv_005fr_005f2exp"></a>Function: <em>void</em> <strong>mpz_tdiv_r_2exp</strong> <em>(mpz_t <var>r</var>, const mpz_t <var>n</var>, <span class="nolinebreak">mp_bitcnt_t</span> <var>b</var><!-- /@w -->)</em></dt>
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<dd><a name="index-Bit-shift-right"></a>
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<br>
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<p>Divide <var>n</var> by <var>d</var>, forming a quotient <var>q</var> and/or remainder
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<var>r</var>. For the <code>2exp</code> functions, <em><var>d</var>=2^<var>b</var></em>.
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The rounding is in three styles, each suiting different applications.
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</p>
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<ul>
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<li> <code>cdiv</code> rounds <var>q</var> up towards <em>+infinity</em>, and <var>r</var> will
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have the opposite sign to <var>d</var>. The <code>c</code> stands for “ceil”.
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</li><li> <code>fdiv</code> rounds <var>q</var> down towards <em>-infinity</em>, and
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<var>r</var> will have the same sign as <var>d</var>. The <code>f</code> stands for
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“floor”.
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</li><li> <code>tdiv</code> rounds <var>q</var> towards zero, and <var>r</var> will have the same sign
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as <var>n</var>. The <code>t</code> stands for “truncate”.
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</li></ul>
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<p>In all cases <var>q</var> and <var>r</var> will satisfy
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<em><var>n</var>=<var>q</var>*<var>d</var>+<var>r</var></em>, and
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<var>r</var> will satisfy <em>0<=abs(<var>r</var>)<abs(<var>d</var>)</em>.
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</p>
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<p>The <code>q</code> functions calculate only the quotient, the <code>r</code> functions
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only the remainder, and the <code>qr</code> functions calculate both. Note that for
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<code>qr</code> the same variable cannot be passed for both <var>q</var> and <var>r</var>, or
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results will be unpredictable.
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</p>
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<p>For the <code>ui</code> variants the return value is the remainder, and in fact
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returning the remainder is all the <code>div_ui</code> functions do. For
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<code>tdiv</code> and <code>cdiv</code> the remainder can be negative, so for those the
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return value is the absolute value of the remainder.
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</p>
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<p>For the <code>2exp</code> variants the divisor is <em>2^<var>b</var></em>. These
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functions are implemented as right shifts and bit masks, but of course they
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round the same as the other functions.
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</p>
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<p>For positive <var>n</var> both <code>mpz_fdiv_q_2exp</code> and <code>mpz_tdiv_q_2exp</code>
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are simple bitwise right shifts. For negative <var>n</var>, <code>mpz_fdiv_q_2exp</code>
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is effectively an arithmetic right shift treating <var>n</var> as twos complement
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the same as the bitwise logical functions do, whereas <code>mpz_tdiv_q_2exp</code>
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effectively treats <var>n</var> as sign and magnitude.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005fmod"></a>Function: <em>void</em> <strong>mpz_mod</strong> <em>(mpz_t <var>r</var>, const mpz_t <var>n</var>, const mpz_t <var>d</var>)</em></dt>
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<dt><a name="index-mpz_005fmod_005fui"></a>Function: <em>unsigned long int</em> <strong>mpz_mod_ui</strong> <em>(mpz_t <var>r</var>, const mpz_t <var>n</var>, unsigned long int <var>d</var><!-- /@w -->)</em></dt>
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<dd><p>Set <var>r</var> to <var>n</var> <code>mod</code> <var>d</var>. The sign of the divisor is
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ignored; the result is always non-negative.
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</p>
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<p><code>mpz_mod_ui</code> is identical to <code>mpz_fdiv_r_ui</code> above, returning the
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remainder as well as setting <var>r</var>. See <code>mpz_fdiv_ui</code> above if only
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the return value is wanted.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005fdivexact"></a>Function: <em>void</em> <strong>mpz_divexact</strong> <em>(mpz_t <var>q</var>, const mpz_t <var>n</var>, const mpz_t <var>d</var>)</em></dt>
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<dt><a name="index-mpz_005fdivexact_005fui"></a>Function: <em>void</em> <strong>mpz_divexact_ui</strong> <em>(mpz_t <var>q</var>, const mpz_t <var>n</var>, unsigned long <var>d</var>)</em></dt>
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<dd><a name="index-Exact-division-functions"></a>
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<p>Set <var>q</var> to <var>n</var>/<var>d</var>. These functions produce correct results only
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when it is known in advance that <var>d</var> divides <var>n</var>.
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</p>
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<p>These routines are much faster than the other division functions, and are the
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best choice when exact division is known to occur, for example reducing a
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rational to lowest terms.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005fdivisible_005fp"></a>Function: <em>int</em> <strong>mpz_divisible_p</strong> <em>(const mpz_t <var>n</var>, const mpz_t <var>d</var>)</em></dt>
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<dt><a name="index-mpz_005fdivisible_005fui_005fp"></a>Function: <em>int</em> <strong>mpz_divisible_ui_p</strong> <em>(const mpz_t <var>n</var>, unsigned long int <var>d</var>)</em></dt>
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<dt><a name="index-mpz_005fdivisible_005f2exp_005fp"></a>Function: <em>int</em> <strong>mpz_divisible_2exp_p</strong> <em>(const mpz_t <var>n</var>, mp_bitcnt_t <var>b</var>)</em></dt>
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<dd><a name="index-Divisibility-functions"></a>
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<p>Return non-zero if <var>n</var> is exactly divisible by <var>d</var>, or in the case of
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<code>mpz_divisible_2exp_p</code> by <em>2^<var>b</var></em>.
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</p>
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<p><var>n</var> is divisible by <var>d</var> if there exists an integer <var>q</var> satisfying
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<em><var>n</var> = <var>q</var>*<var>d</var></em>. Unlike the other division
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functions, <em><var>d</var>=0</em> is accepted and following the rule it can be seen
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that only 0 is considered divisible by 0.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005fcongruent_005fp"></a>Function: <em>int</em> <strong>mpz_congruent_p</strong> <em>(const mpz_t <var>n</var>, const mpz_t <var>c</var>, const mpz_t <var>d</var>)</em></dt>
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<dt><a name="index-mpz_005fcongruent_005fui_005fp"></a>Function: <em>int</em> <strong>mpz_congruent_ui_p</strong> <em>(const mpz_t <var>n</var>, unsigned long int <var>c</var>, unsigned long int <var>d</var>)</em></dt>
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<dt><a name="index-mpz_005fcongruent_005f2exp_005fp"></a>Function: <em>int</em> <strong>mpz_congruent_2exp_p</strong> <em>(const mpz_t <var>n</var>, const mpz_t <var>c</var>, mp_bitcnt_t <var>b</var>)</em></dt>
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<dd><a name="index-Divisibility-functions-1"></a>
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<a name="index-Congruence-functions"></a>
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<p>Return non-zero if <var>n</var> is congruent to <var>c</var> modulo <var>d</var>, or in the
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case of <code>mpz_congruent_2exp_p</code> modulo <em>2^<var>b</var></em>.
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</p>
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<p><var>n</var> is congruent to <var>c</var> mod <var>d</var> if there exists an integer <var>q</var>
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satisfying <em><var>n</var> = <var>c</var> + <var>q</var>*<var>d</var></em>. Unlike
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the other division functions, <em><var>d</var>=0</em> is accepted and following the
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rule it can be seen that <var>n</var> and <var>c</var> are considered congruent mod 0
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only when exactly equal.
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</p></dd></dl>
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<hr>
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<div class="header">
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<p>
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Next: <a href="Integer-Exponentiation.html#Integer-Exponentiation" accesskey="n" rel="next">Integer Exponentiation</a>, Previous: <a href="Integer-Arithmetic.html#Integer-Arithmetic" accesskey="p" rel="prev">Integer Arithmetic</a>, Up: <a href="Integer-Functions.html#Integer-Functions" accesskey="u" rel="up">Integer Functions</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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