(gmp.info.gz) Karatsuba Multiplication
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Karatsuba Multiplication
------------------------
The Karatsuba multiplication algorithm is described in Knuth section
4.3.3 part A, and various other textbooks. A brief description is
given here.
The inputs x and y are treated as each split into two parts of equal
length (or the most significant part one limb shorter if N is odd).
high low
+----------+----------+
| x1 | x0 |
+----------+----------+
+----------+----------+
| y1 | y0 |
+----------+----------+
Let b be the power of 2 where the split occurs, ie. if x0 is k limbs
(y0 the same) then b=2^(k*mp_bits_per_limb). With that x=x1*b+x0 and
y=y1*b+y0, and the following holds,
x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0
This formula means doing only three multiplies of (N/2)x(N/2) limbs,
whereas a basecase multiply of NxN limbs is equivalent to four
multiplies of (N/2)x(N/2). The factors (b^2+b) etc represent the
positions where the three products must be added.
high low
+--------+--------+ +--------+--------+
| x1*y1 | | x0*y0 |
+--------+--------+ +--------+--------+
+--------+--------+
add | x1*y1 |
+--------+--------+
+--------+--------+
add | x0*y0 |
+--------+--------+
+--------+--------+
sub | (x1-x0)*(y1-y0) |
+--------+--------+
The term (x1-x0)*(y1-y0) is best calculated as an absolute value,
and the sign used to choose to add or subtract. Notice the sum
high(x0*y0)+low(x1*y1) occurs twice, so it's possible to do 5*k limb
additions, rather than 6*k, but in GMP extra function call overheads
outweigh the saving.
Squaring is similar to multiplying, but with x=y the formula reduces
to an equivalent with three squares,
x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2
The final result is accumulated from those three squares the same
way as for the three multiplies above. The middle term (x1-x0)^2 is now
always positive.
A similar formula for both multiplying and squaring can be
constructed with a middle term (x1+x0)*(y1+y0). But those sums can
exceed k limbs, leading to more carry handling and additions than the
form above.
Karatsuba multiplication is asymptotically an O(N^1.585) algorithm,
the exponent being log(3)/log(2), representing 3 multiplies each 1/2
the size of the inputs. This is a big improvement over the basecase
multiply at O(N^2) and the advantage soon overcomes the extra additions
Karatsuba performs. `MUL_KARATSUBA_THRESHOLD' can be as little as 10
limbs. The `SQR' threshold is usually about twice the `MUL'.
The basecase algorithm will take a time of the form M(N) = a*N^2 +
b*N + c and the Karatsuba algorithm K(N) = 3*M(N/2) + d*N + e, which
expands to K(N) = 3/4*a*N^2 + 3/2*b*N + 3*c + d*N + e. The factor 3/4
for a means per-crossproduct speedups in the basecase code will
increase the threshold since they benefit M(N) more than K(N). And
conversely the 3/2 for b means linear style speedups of b will increase
the threshold since they benefit K(N) more than M(N). The latter can
be seen for instance when adding an optimized `mpn_sqr_diagonal' to
`mpn_sqr_basecase'. Of course all speedups reduce total time, and in
that sense the algorithm thresholds are merely of academic interest.
Info Catalog
(gmp.info.gz) Basecase Multiplication
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