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236 lines
11 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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<!-- Created by GNU Texinfo 6.4, http://www.gnu.org/software/texinfo/ -->
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<head>
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<title>Toom 3-Way 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="Toom 3-Way 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="Toom-4_002dWay-Multiplication.html#Toom-4_002dWay-Multiplication" rel="next" title="Toom 4-Way Multiplication">
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<link href="Karatsuba-Multiplication.html#Karatsuba-Multiplication" rel="prev" title="Karatsuba Multiplication">
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</head>
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<body lang="en">
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<a name="Toom-3_002dWay-Multiplication"></a>
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<div class="header">
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<p>
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Next: <a href="Toom-4_002dWay-Multiplication.html#Toom-4_002dWay-Multiplication" accesskey="n" rel="next">Toom 4-Way Multiplication</a>, Previous: <a href="Karatsuba-Multiplication.html#Karatsuba-Multiplication" accesskey="p" rel="prev">Karatsuba 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="Toom-3_002dWay-Multiplication-1"></a>
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<h4 class="subsection">15.1.3 Toom 3-Way Multiplication</h4>
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<a name="index-Toom-multiplication"></a>
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<p>The Karatsuba formula is the simplest case of a general approach to splitting
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inputs that leads to both Toom and FFT algorithms. A description of
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Toom can be found in Knuth section 4.3.3, with an example 3-way
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calculation after Theorem A. The 3-way form used in GMP is described here.
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</p>
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<p>The operands are each considered split into 3 pieces of equal length (or the
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most significant part 1 or 2 limbs shorter than the other two).
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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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| x2 | x1 | x0 |
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+----------+----------+----------+
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+----------+----------+----------+
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| y2 | y1 | y0 |
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+----------+----------+----------+
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</pre></div>
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<p>These parts are treated as the coefficients of two polynomials
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</p>
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<div class="display">
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<pre class="display"><em>X(t) = x2*t^2 + x1*t + x0</em>
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<em>Y(t) = y2*t^2 + y1*t + y0</em>
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</pre></div>
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<p>Let <em>b</em> equal the power of 2 which is the size of the x0, x1,
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y0 and y1 pieces, i.e. if they’re <em>k</em> limbs each then
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<em>b=2^(k*mp_bits_per_limb)</em>.
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With this <em>x=X(b)</em> and <em>y=Y(b)</em>.
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</p>
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<p>Let a polynomial <em>W(t)=X(t)*Y(t)</em> and suppose its coefficients
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are
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</p>
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<div class="display">
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<pre class="display"><em>W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0</em>
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</pre></div>
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<p>The <em>w[i]</em> are going to be determined, and when they are they’ll give
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the final result using <em>w=W(b)</em>, since
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<em>x*y=X(b)*Y(b)=W(b)</em>. The coefficients will be roughly
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<em>b^2</em> each, and the final <em>W(b)</em> will be an addition like,
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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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| w4 |
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+-------+-------+
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+--------+-------+
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| w3 |
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+--------+-------+
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+--------+-------+
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| w2 |
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+--------+-------+
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+--------+-------+
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| w1 |
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+--------+-------+
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+-------+-------+
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| w0 |
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+-------+-------+
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</pre></div>
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<p>The <em>w[i]</em> coefficients could be formed by a simple set of cross
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products, like <em>w4=x2*y2</em>, <em>w3=x2*y1+x1*y2</em>,
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<em>w2=x2*y0+x1*y1+x0*y2</em> etc, but this would need all
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nine <em>x[i]*y[j]</em> for <em>i,j=0,1,2</em>, and would be equivalent merely
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to a basecase multiply. Instead the following approach is used.
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</p>
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<p><em>X(t)</em> and <em>Y(t)</em> are evaluated and multiplied at 5 points, giving
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values of <em>W(t)</em> at those points. In GMP the following points are used,
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</p>
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<blockquote>
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<table>
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<tr><td>Point</td><td>Value</td></tr>
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<tr><td><em>t=0</em></td><td><em>x0 * y0</em>, which gives w0 immediately</td></tr>
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<tr><td><em>t=1</em></td><td><em>(x2+x1+x0) * (y2+y1+y0)</em></td></tr>
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<tr><td><em>t=-1</em></td><td><em>(x2-x1+x0) * (y2-y1+y0)</em></td></tr>
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<tr><td><em>t=2</em></td><td><em>(4*x2+2*x1+x0) * (4*y2+2*y1+y0)</em></td></tr>
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<tr><td><em>t=inf</em></td><td><em>x2 * y2</em>, which gives w4 immediately</td></tr>
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</table>
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</blockquote>
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<p>At <em>t=-1</em> the values can be negative and that’s handled using the
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absolute values and tracking the sign separately. At <em>t=inf</em> the
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value is actually <em>X(t)*Y(t)/t^4 in
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the limit as t approaches infinity</em>, but it’s much easier to think of as
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simply <em>x2*y2</em> giving w4 immediately (much like
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<em>x0*y0</em> at <em>t=0</em> gives w0 immediately).
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</p>
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<p>Each of the points substituted into
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<em>W(t)=w4*t^4+…+w0</em> gives a linear combination
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of the <em>w[i]</em> coefficients, and the value of those combinations has just
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been calculated.
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</p>
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<div class="example">
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<pre class="example">W(0) = w0
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W(1) = w4 + w3 + w2 + w1 + w0
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W(-1) = w4 - w3 + w2 - w1 + w0
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W(2) = 16*w4 + 8*w3 + 4*w2 + 2*w1 + w0
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W(inf) = w4
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</pre></div>
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<p>This is a set of five equations in five unknowns, and some elementary linear
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algebra quickly isolates each <em>w[i]</em>. This involves adding or
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subtracting one <em>W(t)</em> value from another, and a couple of divisions by
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powers of 2 and one division by 3, the latter using the special
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<code>mpn_divexact_by3</code> (see <a href="Exact-Division.html#Exact-Division">Exact Division</a>).
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</p>
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<p>The conversion of <em>W(t)</em> values to the coefficients is interpolation. A
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polynomial of degree 4 like <em>W(t)</em> is uniquely determined by values known
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at 5 different points. The points are arbitrary and can be chosen to make the
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linear equations come out with a convenient set of steps for quickly isolating
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the <em>w[i]</em>.
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</p>
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<p>Squaring follows the same procedure as multiplication, but there’s only one
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<em>X(t)</em> and it’s evaluated at the 5 points, and those values squared to
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give values of <em>W(t)</em>. The interpolation is then identical, and in fact
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the same <code>toom_interpolate_5pts</code> subroutine is used for both squaring and
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multiplying.
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</p>
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<p>Toom-3 is asymptotically <em>O(N^1.465<!-- /@w -->)</em>, the exponent being
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<em>log(5)/log(3)</em>, representing 5 recursive multiplies of 1/3 the
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original size each. This is an improvement over Karatsuba at
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<em>O(N^1.585<!-- /@w -->)</em>, though Toom does more work in the evaluation and
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interpolation and so it only realizes its advantage above a certain size.
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</p>
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<p>Near the crossover between Toom-3 and Karatsuba there’s generally a range of
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sizes where the difference between the two is small.
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<code>MUL_TOOM33_THRESHOLD</code> is a somewhat arbitrary point in that range and
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successive runs of the tune program can give different values due to small
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variations in measuring. A graph of time versus size for the two shows the
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effect, see <samp>tune/README</samp>.
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</p>
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<p>At the fairly small sizes where the Toom-3 thresholds occur it’s worth
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remembering that the asymptotic behaviour for Karatsuba and Toom-3 can’t be
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expected to make accurate predictions, due of course to the big influence of
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all sorts of overheads, and the fact that only a few recursions of each are
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being performed. Even at large sizes there’s a good chance machine dependent
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effects like cache architecture will mean actual performance deviates from
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what might be predicted.
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</p>
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<p>The formula given for the Karatsuba algorithm (see <a href="Karatsuba-Multiplication.html#Karatsuba-Multiplication">Karatsuba Multiplication</a>) has an equivalent for Toom-3 involving only five multiplies,
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but this would be complicated and unenlightening.
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</p>
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<p>An alternate view of Toom-3 can be found in Zuras (see <a href="References.html#References">References</a>), using
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a vector to represent the <em>x</em> and <em>y</em> splits and a matrix
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multiplication for the evaluation and interpolation stages. The matrix
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inverses are not meant to be actually used, and they have elements with values
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much greater than in fact arise in the interpolation steps. The diagram shown
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for the 3-way is attractive, but again doesn’t have to be implemented that way
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and for example with a bit of rearrangement just one division by 6 can be
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done.
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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-4_002dWay-Multiplication.html#Toom-4_002dWay-Multiplication" accesskey="n" rel="next">Toom 4-Way Multiplication</a>, Previous: <a href="Karatsuba-Multiplication.html#Karatsuba-Multiplication" accesskey="p" rel="prev">Karatsuba 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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