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B.4. STRASSEN’S METHOD 229
If the matrices have 2 k rows/columns, then this method requires 7 k =
(
2
k ) log 2 7
multiplications rather than the classic
(
2
k ) 3
. For arbitrary sizes n,
not necessarily a power of two, the cost of “rounding up” to a power of two is
subsumed in the O notation, and we see a cost of O(n log 2 7 ) rather than the
classic O(n 3 ) coefficient multiplications.
We note that log 2 7 ≈ 2.8074, and the number of extra coefficient additions
required is also O(n log 2 7 ), being eighteen additions at each step.
Theorem 39 (Strassen) We can multiply two (dense) n×n matrices in O(n log 2 7 ≈
n 2.8074 multiplications, and the same order of additions, of matrix entries.
B.4.1
Strassen’s method in practice
Strassen’s method, with floating-point numbers, has break-even points between
400 and 2000 rows (160,000 to 4 million elements) [DN07].
B.4.2
Further developments
Seven is in fact minimal for 2 × 2 matrices [Lan06]. 3 × 3 matrices can be multiplied
in 23 multiplications rather than the obvious 27 (but log 3 23 ≈ 2.854 >
log 2 7), and there is an approximation algorithm with 21 multiplications [Sch71],
giving a O(n log 3 21≈2.77 ) general method, but for 3 × 3 matrices the best known
lower bound is 15 [Lic84, LO11]. In general the best known lower bound is
2n 2 + O(n 3/2 ) [Lan12].
In theory one can do better than Theorem 39, O(n 2.376 ), [CW90], but these
methods require unfeasably large n to be cost-effective. The complexity of
matrix multilication has to be at least O(n 2 ) since there are that many entries.
Notation 32 It is custonary to assume that the cost of n × n matrix multiplication
is O(n ω ), where 2 ≤ ω ≤ 3.
B.4.3 Matrix Inversion
( P Q
Lemma 15 (Frobenius) Let A be a square matrix divided into blocks
R S
with P square and nonsingular. Assume that U = S − R(P −1 Q) is also nonsingular.
Then
( )
A −1 P
=
−1 + (P −1 Q)(U −1 RP −1 ) −(P −1 Q)U −1
−(U −1 RP −1 ) U −1 . (B.14)
The proof is by direct verification. Note that the computation of the righthand
side of (B.14) requires two matrix inverses (P and U) and six matrix
multiplications (viz. P −1 Q, R(P −1 Q), RP −1 , U(RP −1 ), (P −1 Q)(U −1 RP −1 )
and (P −1 Q)U −1 ). If the conditions of Lemma 15 were always satisfied, and we
took P to be precisely one quarter the size of A (half as many rows and half as
)
,