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122 lines
6.4 KiB
HTML
122 lines
6.4 KiB
HTML
<!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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<head>
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<title>Other 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="Other Multiplication (GNU MP 6.1.0)">
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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="Unbalanced-Multiplication.html#Unbalanced-Multiplication" rel="next" title="Unbalanced Multiplication">
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<link href="FFT-Multiplication.html#FFT-Multiplication" rel="prev" title="FFT Multiplication">
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</head>
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<body lang="en">
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<a name="Other-Multiplication"></a>
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<div class="header">
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<p>
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Next: <a href="Unbalanced-Multiplication.html#Unbalanced-Multiplication" accesskey="n" rel="next">Unbalanced Multiplication</a>, Previous: <a href="FFT-Multiplication.html#FFT-Multiplication" accesskey="p" rel="prev">FFT 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="Other-Multiplication-1"></a>
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<h4 class="subsection">15.1.7 Other Multiplication</h4>
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<a name="index-Toom-multiplication-3"></a>
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<p>The Toom algorithms described above (see <a href="Toom-3_002dWay-Multiplication.html#Toom-3_002dWay-Multiplication">Toom 3-Way Multiplication</a>,
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see <a href="Toom-4_002dWay-Multiplication.html#Toom-4_002dWay-Multiplication">Toom 4-Way Multiplication</a>) generalizes to split into an arbitrary
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number of pieces, as per Knuth section 4.3.3 algorithm C. This is not
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currently used. The notes here are merely for interest.
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</p>
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<p>In general a split into <em>r+1</em> pieces is made, and evaluations and
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pointwise multiplications done at <em>2*r+1</em> points. A 4-way split does 7
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pointwise multiplies, 5-way does 9, etc. Asymptotically an <em>(r+1)</em>-way
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algorithm is <em>O(N^(log(2*r+1)/log(r+1)))</em>. Only
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the pointwise multiplications count towards big-<em>O</em> complexity, but the
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time spent in the evaluate and interpolate stages grows with <em>r</em> and has
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a significant practical impact, with the asymptotic advantage of each <em>r</em>
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realized only at bigger and bigger sizes. The overheads grow as
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<em>O(N*r)</em>, whereas in an <em>r=2^k</em> FFT they grow only as <em>O(N*log(r))</em>.
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</p>
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<p>Knuth algorithm C evaluates at points 0,1,2,…,<em>2*r</em>, but exercise 4
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uses <em>-r</em>,…,0,…,<em>r</em> and the latter saves some small
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multiplies in the evaluate stage (or rather trades them for additions), and
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has a further saving of nearly half the interpolate steps. The idea is to
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separate odd and even final coefficients and then perform algorithm C steps C7
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and C8 on them separately. The divisors at step C7 become <em>j^2</em> and the
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multipliers at C8 become <em>2*t*j-j^2</em>.
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</p>
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<p>Splitting odd and even parts through positive and negative points can be
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thought of as using <em>-1</em> as a square root of unity. If a 4th root of
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unity was available then a further split and speedup would be possible, but no
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such root exists for plain integers. Going to complex integers with
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<em>i=sqrt(-1)</em> doesn’t help, essentially because in Cartesian
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form it takes three real multiplies to do a complex multiply. The existence
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of <em>2^k'</em>th roots of unity in a suitable ring or field lets the fast
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Fourier transform keep splitting and get to <em>O(N*log(r))</em>.
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</p>
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<p>Floating point FFTs use complex numbers approximating Nth roots of unity.
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Some processors have special support for such FFTs. But these are not used in
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GMP since it’s very difficult to guarantee an exact result (to some number of
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bits). An occasional difference of 1 in the last bit might not matter to a
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typical signal processing algorithm, but is of course of vital importance to
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GMP.
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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="Unbalanced-Multiplication.html#Unbalanced-Multiplication" accesskey="n" rel="next">Unbalanced Multiplication</a>, Previous: <a href="FFT-Multiplication.html#FFT-Multiplication" accesskey="p" rel="prev">FFT 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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