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B.3. KARATSUBA’S METHOD 225
additions at each step. While the “rounding up” is not important in O-theory,
it matters in practice, and [Mon05] shows various other formulae, e.g.
(aX 2 + bX + c)(dX 2 + eX + f) =
cf(1 − X) + be(−X + 2X 2 − X 3 ) + ad(−X 3 + X 4 ) + (b + c)(e + f)(X − X 2 )
+(a + b)(d + e)(X 3 − X 2 ) + (a + b + c)(d + e + f)X 2 ,
requiring six coefficient multiplications rather than the obvious nine, or eight if
we write it as
(
aX 2 + (bX + c) ) ( dX 2 + (eX + f) ) =
adX 4 + aX 2 (eX + f) + dX 2 (bX + c) + (bX + c)(eX + f)
and use (B.4) on the last summand (asymptotics would predict 3 log 2 3 ≈ 5.7, so
we are much nearer the asymptoic behaviour). Cascading this formula rather
than (B.4) gives O(n log 3 6 ), which as log 3 6 ≈ 1.63 is not as favorable asymptotically.
His most impressive formula describes the product of two six-term
polynomials in 17 coefficient multiplications, and log 6 17 ≈ 1.581, a slight improvement.
We refer the reader to the table in [Mon05], which shows that his
armoury of formulae can get us very close to the asymptotic costs.
Theorem 38 We can multiply two (dense) polynomials with m and n terms
respectively in O ( max(m, n) min(m, n) (log 2 3)−1) coefficient operations.
Let the polynomials be f = a m−1 Y m−1 + · · · and f = b n−1 Y n−1 + · · · Without
loss of generality, we may assume m ≥ n (so we have to prove O(mn log 2 3−1 ),
and write k = ⌈ m n
⌉. We can then divide f into blocks with n terms (possibly
fewer for the most significant one) each: f = ∑ k−1
i=1 f iY in . Then
k−1
∑
fg = (f i g) Y in .
i=1
Each f i g can be computed in time O(n log 2 3 ), and the addition merely takes
time O(m − n) (since there is one addition to be performed for each power of
Y where overlap occurs). Hence the total time is O(kn log 2 3 ), and the constant
factor implied by the O-notation allows us to replace k = ⌈ m n ⌉ by m n , which
gives the desired result.
B.3.1
Karatsuba’s method in practice
When it comes to multiplying numbers of n digits (in whatever base we are
actually using, generally a power of 2), the received wisdom is to use O(n 2 )
methods for small n (say n < 16), Karatsuba-style methods for intermediate
n (say 16 ≤ n < 4096 and Fast Fourier Transform methods (section B.3.4 or
[SS71]) for larger n. However, it is possible for the Fast Fourier Transform
to require too much memory, and [Tak10] was forced to use these methods on
numbers one quarter the required size, and then nested Karatsuba to combine
the results.