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B.4. STRASSEN’S METHOD 227
answers to the next two questions are given by the Fast Fourier Transform (FFT)
[SS71], and we can compute ˜f from f (or h from ˜h) in time O(k2 k ) = O(n log n)
(note that we are still assuming n is a power of 2). Putting these together gives
O(n log n) + O(n log n)
} {{ } } {{ }
˜f from f ˜g from g
+ O(n)
} {{ }
˜h from ˜f, ˜g
+ O(n log n) = O(n log n). (B.8)
} {{ }
h from ˜h
While this method works fine for multiplying polynomials over a finite field K
with an appropriate root of unity, or over C, it requires substantial attention to
detail to make it work over the integers. Nevertheless it is practicable, and the
point at which this FFT-based method is faster than ones based on Theorem
38 depends greatly on the coefficient domain.
TO BE COMPLETED
Notation 31 We will denote M Z (n) the asymptotic O(M Z (n)) time taken to
multiply two integers of length n, and M K[x] (n) the number of operations in K
or its associated structures needed to multiply two polynomials with n terms (or
of degree n, since in O-speak the two are the same). Most authors omit the
subscripts, and speak of M(n), when the context is clear.
For integers, the point at which this FFT-based method is faster than ones
based on Theorem 38 is tyically when the integers are of length a few thousand
words.
B.3.5
Faster division
TO BE COMPLETED
B.3.6
Faster g.c.d. computation
TO BE COMPLETED
B.4 Strassen’s method
Just as Karatsuba’s method (Excursus B.4) lets us multiply dense polynomials
of degree n − 1 (n terms) in O(n log 2 3 ≈ n 1.585 ) operations rather than O(n 2 ),
so Strassen’s method, introduced in [Str69], allows us to multiply dense n × n
matrices in O(n log 2 7 ≈ n 2.807 ) operations rather than O(n 3 ) operations. Again
like Karatsuba’s method, it is based on “divide and conquer” and an ingenious
base case for n = 2, analogous to (B.4).
is usually computed as
( )
c1,1 c 1,2
=
c 2,1 c 2,2
( )
c1,1 c 1,2
=
c 2,1 c 2,2
( ) ( )
a1,1 a 1,2 b1,1 b 1,2
a 2,1 a 2,2 b 2,1 b 2,2
( )
a1,1 b 1,1 + a 1,2 b 2,1 a 1,1 b 1,2 + a 1,2 b 2,2
a 2,1 b 1,1 + a 2,2 b 2,1 a 2,1 b 1,2 + a 2,2 b 2,2
(B.9)
(B.10)