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.
170 lines
7.8 KiB
HTML
170 lines
7.8 KiB
HTML
<!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>Karatsuba Multiplication (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="Karatsuba Multiplication (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="Multiplication-Algorithms.html#Multiplication-Algorithms" rel="up" title="Multiplication Algorithms">
|
|
<link href="Toom-3_002dWay-Multiplication.html#Toom-3_002dWay-Multiplication" rel="next" title="Toom 3-Way Multiplication">
|
|
<link href="Basecase-Multiplication.html#Basecase-Multiplication" rel="prev" title="Basecase Multiplication">
|
|
<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="Karatsuba-Multiplication"></a>
|
|
<div class="header">
|
|
<p>
|
|
Next: <a href="Toom-3_002dWay-Multiplication.html#Toom-3_002dWay-Multiplication" accesskey="n" rel="next">Toom 3-Way Multiplication</a>, Previous: <a href="Basecase-Multiplication.html#Basecase-Multiplication" accesskey="p" rel="prev">Basecase Multiplication</a>, Up: <a href="Multiplication-Algorithms.html#Multiplication-Algorithms" accesskey="u" rel="up">Multiplication Algorithms</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
|
|
</div>
|
|
<hr>
|
|
<a name="Karatsuba-Multiplication-1"></a>
|
|
<h4 class="subsection">15.1.2 Karatsuba Multiplication</h4>
|
|
<a name="index-Karatsuba-multiplication"></a>
|
|
|
|
<p>The Karatsuba multiplication algorithm is described in Knuth section 4.3.3
|
|
part A, and various other textbooks. A brief description is given here.
|
|
</p>
|
|
<p>The inputs <em>x</em> and <em>y</em> are treated as each split into two parts of
|
|
equal length (or the most significant part one limb shorter if N is odd).
|
|
</p>
|
|
<div class="example">
|
|
<pre class="example"> high low
|
|
+----------+----------+
|
|
| x1 | x0 |
|
|
+----------+----------+
|
|
|
|
+----------+----------+
|
|
| y1 | y0 |
|
|
+----------+----------+
|
|
</pre></div>
|
|
|
|
<p>Let <em>b</em> be the power of 2 where the split occurs, i.e. if x0 is
|
|
<em>k</em> limbs (y0 the same) then
|
|
<em>b=2^(k*mp_bits_per_limb)</em>.
|
|
With that <em>x=x1*b+x0</em> and <em>y=y1*b+y0</em>, and the
|
|
following holds,
|
|
</p>
|
|
<div class="display">
|
|
<pre class="display"><em>x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0</em>
|
|
</pre></div>
|
|
|
|
<p>This formula means doing only three multiplies of (N/2)x(N/2) limbs,
|
|
whereas a basecase multiply of NxN limbs is equivalent to four
|
|
multiplies of (N/2)x(N/2). The factors <em>(b^2+b)</em> etc represent
|
|
the positions where the three products must be added.
|
|
</p>
|
|
<div class="example">
|
|
<pre class="example"> high low
|
|
+--------+--------+ +--------+--------+
|
|
| x1*y1 | | x0*y0 |
|
|
+--------+--------+ +--------+--------+
|
|
+--------+--------+
|
|
add | x1*y1 |
|
|
+--------+--------+
|
|
+--------+--------+
|
|
add | x0*y0 |
|
|
+--------+--------+
|
|
+--------+--------+
|
|
sub | (x1-x0)*(y1-y0) |
|
|
+--------+--------+
|
|
</pre></div>
|
|
|
|
<p>The term <em>(x1-x0)*(y1-y0)</em> is best calculated as an
|
|
absolute value, and the sign used to choose to add or subtract. Notice the
|
|
sum <em>high(x0*y0)+low(x1*y1)</em> occurs twice, so it’s possible to do <em>5*k</em> limb
|
|
additions, rather than <em>6*k</em>, but in GMP extra function call overheads
|
|
outweigh the saving.
|
|
</p>
|
|
<p>Squaring is similar to multiplying, but with <em>x=y</em> the formula reduces to
|
|
an equivalent with three squares,
|
|
</p>
|
|
<div class="display">
|
|
<pre class="display"><em>x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2</em>
|
|
</pre></div>
|
|
|
|
<p>The final result is accumulated from those three squares the same way as for
|
|
the three multiplies above. The middle term <em>(x1-x0)^2</em> is now
|
|
always positive.
|
|
</p>
|
|
<p>A similar formula for both multiplying and squaring can be constructed with a
|
|
middle term <em>(x1+x0)*(y1+y0)</em>. But those sums can exceed
|
|
<em>k</em> limbs, leading to more carry handling and additions than the form
|
|
above.
|
|
</p>
|
|
<p>Karatsuba multiplication is asymptotically an <em>O(N^1.585<!-- /@w -->)</em> algorithm,
|
|
the exponent being <em>log(3)/log(2)</em>, representing 3 multiplies
|
|
each <em>1/2</em> the size of the inputs. This is a big improvement over the
|
|
basecase multiply at <em>O(N^2)</em> and the advantage soon overcomes the extra
|
|
additions Karatsuba performs. <code>MUL_TOOM22_THRESHOLD</code> can be as little
|
|
as 10 limbs. The <code>SQR</code> threshold is usually about twice the <code>MUL</code>.
|
|
</p>
|
|
<p>The basecase algorithm will take a time of the form <em>M(N) = a*N^2 + b*N + c</em> and the Karatsuba algorithm <em>K(N) = 3*M(N/2) + d*N + e</em>, which expands to <em>K(N) = 3/4*a*N^2 + 3/2*b*N + 3*c + d*N + e</em>. The
|
|
factor <em>3/4</em> for <em>a</em> means per-crossproduct speedups in the
|
|
basecase code will increase the threshold since they benefit <em>M(N)</em> more
|
|
than <em>K(N)</em>. And conversely the <em>3/2</em> for <em>b</em> means
|
|
linear style speedups of <em>b</em> will increase the threshold since they
|
|
benefit <em>K(N)</em> more than <em>M(N)</em>. The latter can be seen for
|
|
instance when adding an optimized <code>mpn_sqr_diagonal</code> to
|
|
<code>mpn_sqr_basecase</code>. Of course all speedups reduce total time, and in
|
|
that sense the algorithm thresholds are merely of academic interest.
|
|
</p>
|
|
|
|
<hr>
|
|
<div class="header">
|
|
<p>
|
|
Next: <a href="Toom-3_002dWay-Multiplication.html#Toom-3_002dWay-Multiplication" accesskey="n" rel="next">Toom 3-Way Multiplication</a>, Previous: <a href="Basecase-Multiplication.html#Basecase-Multiplication" accesskey="p" rel="prev">Basecase Multiplication</a>, Up: <a href="Multiplication-Algorithms.html#Multiplication-Algorithms" accesskey="u" rel="up">Multiplication Algorithms</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
|
|
</div>
|
|
|
|
|
|
|
|
</body>
|
|
</html>
|