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(gmp.info.gz) Toom 3-Way Multiplication

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 Toom 3-Way Multiplication
 -------------------------
 
 The Karatsuba formula is the simplest case of a general approach to
 splitting inputs that leads to both Toom and FFT algorithms.  A
 description of Toom can be found in Knuth section 4.3.3, with an
 example 3-way calculation after Theorem A.  The 3-way form used in GMP
 is described here.
 
    The operands are each considered split into 3 pieces of equal length
 (or the most significant part 1 or 2 limbs shorter than the other two).
 
       high                         low
      +----------+----------+----------+
      |    x2    |    x1    |    x0    |
      +----------+----------+----------+
      
      +----------+----------+----------+
      |    y2    |    y1    |    y0    |
      +----------+----------+----------+
 
 These parts are treated as the coefficients of two polynomials
 
      X(t) = x2*t^2 + x1*t + x0
      Y(t) = y2*t^2 + y1*t + y0
 
    Let b equal the power of 2 which is the size of the x0, x1, y0 and
 y1 pieces, ie. if they're k limbs each then b=2^(k*mp_bits_per_limb).
 With this x=X(b) and y=Y(b).
 
    Let a polynomial W(t)=X(t)*Y(t) and suppose its coefficients are
 
      W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0
 
    The w[i] are going to be determined, and when they are they'll give
 the final result using w=W(b), since x*y=X(b)*Y(b)=W(b).  The
 coefficients will be roughly b^2 each, and the final W(b) will be an
 addition like,
 
       high                                        low
      +-------+-------+
      |       w4      |
      +-------+-------+
             +--------+-------+
             |        w3      |
             +--------+-------+
                     +--------+-------+
                     |        w2      |
                     +--------+-------+
                             +--------+-------+
                             |        w1      |
                             +--------+-------+
                                      +-------+-------+
                                      |       w0      |
                                      +-------+-------+
 
    The w[i] coefficients could be formed by a simple set of cross
 products, like w4=x2*y2, w3=x2*y1+x1*y2, w2=x2*y0+x1*y1+x0*y2 etc, but
 this would need all nine x[i]*y[j] for i,j=0,1,2, and would be
 equivalent merely to a basecase multiply.  Instead the following
 approach is used.
 
    X(t) and Y(t) are evaluated and multiplied at 5 points, giving
 values of W(t) at those points.  In GMP the following points are used,
 
      Point    Value
      t=0      x0 * y0, which gives w0 immediately
      t=1      (x2+x1+x0) * (y2+y1+y0)
      t=-1     (x2-x1+x0) * (y2-y1+y0)
      t=2      (4*x2+2*x1+x0) * (4*y2+2*y1+y0)
      t=inf    x2 * y2, which gives w4 immediately
 
    At t=-1 the values can be negative and that's handled using the
 absolute values and tracking the sign separately.  At t=inf the value
 is actually X(t)*Y(t)/t^4 in the limit as t approaches infinity, but
 it's much easier to think of as simply x2*y2 giving w4 immediately
 (much like x0*y0 at t=0 gives w0 immediately).
 
    Each of the points substituted into W(t)=w4*t^4+...+w0 gives a
 linear combination of the w[i] coefficients, and the value of those
 combinations has just been calculated.
 
      W(0)   =                              w0
      W(1)   =    w4 +   w3 +   w2 +   w1 + w0
      W(-1)  =    w4 -   w3 +   w2 -   w1 + w0
      W(2)   = 16*w4 + 8*w3 + 4*w2 + 2*w1 + w0
      W(inf) =    w4
 
    This is a set of five equations in five unknowns, and some
 elementary linear algebra quickly isolates each w[i].  This involves
 adding or subtracting one W(t) value from another, and a couple of
 divisions by powers of 2 and one division by 3, the latter using the
 special `mpn_divexact_by3' ( Exact Division).
 
    The conversion of W(t) values to the coefficients is interpolation.
 A polynomial of degree 4 like W(t) is uniquely determined by values
 known at 5 different points.  The points are arbitrary and can be
 chosen to make the linear equations come out with a convenient set of
 steps for quickly isolating the w[i].
 
    Squaring follows the same procedure as multiplication, but there's
 only one X(t) and it's evaluated at the 5 points, and those values
 squared to give values of W(t).  The interpolation is then identical,
 and in fact the same `toom3_interpolate' subroutine is used for both
 squaring and multiplying.
 
    Toom-3 is asymptotically O(N^1.465), the exponent being
 log(5)/log(3), representing 5 recursive multiplies of 1/3 the original
 size each.  This is an improvement over Karatsuba at O(N^1.585), though
 Toom does more work in the evaluation and interpolation and so it only
 realizes its advantage above a certain size.
 
    Near the crossover between Toom-3 and Karatsuba there's generally a
 range of sizes where the difference between the two is small.
 `MUL_TOOM3_THRESHOLD' is a somewhat arbitrary point in that range and
 successive runs of the tune program can give different values due to
 small variations in measuring.  A graph of time versus size for the two
 shows the effect, see `tune/README'.
 
    At the fairly small sizes where the Toom-3 thresholds occur it's
 worth remembering that the asymptotic behaviour for Karatsuba and
 Toom-3 can't be expected to make accurate predictions, due of course to
 the big influence of all sorts of overheads, and the fact that only a
 few recursions of each are being performed.  Even at large sizes
 there's a good chance machine dependent effects like cache architecture
 will mean actual performance deviates from what might be predicted.
 
 Multiplication::) has an equivalent for Toom-3 involving only five
 multiplies, but this would be complicated and unenlightening.
 
    An alternate view of Toom-3 can be found in Zuras (
 References), using a vector to represent the x and y splits and a
 matrix multiplication for the evaluation and interpolation stages.  The
 matrix inverses are not meant to be actually used, and they have
 elements with values much greater than in fact arise in the
 interpolation steps.  The diagram shown for the 3-way is attractive,
 but again doesn't have to be implemented that way and for example with
 a bit of rearrangement just one division by 6 can be done.
 
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