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