Node:Toom-Cook 3-Way Multiplication, Next:FFT Multiplication, Previous:Karatsuba Multiplication, Up:Multiplication Algorithms
The Karatsuba formula is the simplest case of a general approach to splitting inputs that leads to both Toom-Cook and FFT algorithms. A description of Toom-Cook 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 others).
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
Again 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. The points used can be chosen in various ways, but in GMP the following are used
Point Value t=0 x0*y0, which gives w0 immediately t=2 (4*x2+2*x1+x0)*(4*y2+2*y1+y0) t=1 (x2+x1+x0)*(y2+y1+y0) t=1/2 (x2+2*x1+4*x0)*(y2+2*y1+4*y0) t=inf x2*y2, which gives w4 immediately
At t=1/2 the value calculated is actually 16*X(1/2)*Y(1/2), giving a value for 16*W(1/2), and this is always an integer. 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).
Now 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 16*W(1/2) = w4 + 2*w3 + 4*w2 + 8*w1 + 16*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], by subtracting multiples of one equation from another.
In the code the set of five values W(0),...,W(inf)
will represent those certain linear combinations. By adding or subtracting
one from another as necessary, values which are each w[i] alone are
arrived at. This involves only a few subtractions of small multiples (some of
which are powers of 2), and so is fast. A couple of divisions remain by
powers of 2 and one division by 3 (or by 6 rather), and that last uses the
special mpn_divexact_by3 (see Exact Division).
In the code the values w4, w2 and w0 are formed in the
destination with pointers E, C and A, and w3 and
w1 in temporary space D and B are added to them. There
are extra limbs tD, tC and tB at the high end of
w3, w2 and w1 which are handled separately. The final
addition then is as follows.
high low
+-------+-------+-------+-------+-------+-------+
| E | C | A |
+-------+-------+-------+-------+-------+-------+
+------+-------++------+-------+
| D || B |
+------+-------++------+-------+
-- -- --
|tD| |tC| |tB|
-- -- --
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 can be chosen to make the linear equations come out with a convenient set of steps for isolating the w[i].
In mpn/generic/mul_n.c the interpolate3 routine performs the
interpolation. The open-coded one-pass version may be a bit hard to
understand, the steps performed can be better seen in the USE_MORE_MPN
version.
Squaring follows the same procedure as multiplication, but there's only one
X(t) and it's evaluated at 5 points, and those values squared to give
values of W(t). The interpolation is then identical, and in fact the
same interpolate3 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. This is an improvement over Karatsuba at O(N^1.585), though Toom-Cook 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.
The formula given above for the Karatsuba algorithm 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 (see 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.