Numerical recipes

in terms of which
2.11 Is Matrix Inversion an N 3 Process? 103
c11 = Q1 + Q4 − Q5 + Q7
c21 = Q2 + Q4
c12 = Q3 + Q5
c22 = Q1 + Q3 − Q2 + Q6
(2.11.4)
What’s the use of this? There is one fewer multiplication than in equation
(2.11.2), but many more additions and subtractions. It is not clear that anything
has been gained. But notice that in (2.11.3) the a’s and b’s are never commuted.
Therefore (2.11.3) and (2.11.4) are valid when the a’s and b’s are themselves matrices.
The problem of multiplying two very large matrices (of order N =2m for some
integer m) can now be broken down recursively by partitioning the matrices into
quarters, sixteenths, etc. And note the key point: The savings is not just a factor
“7/8”; it is that factor at each hierarchical level of the recursion. In total it reduces
the process of matrix multiplication to order N log2 7 instead of N 3 .
What about all the extra additions in (2.11.3)–(2.11.4)? Don’t they outweigh
the advantage of the fewer multiplications? For large N, it turns out that there are
six times as many additions as multiplications implied by (2.11.3)–(2.11.4). But,
if N is very large, this constant factor is no match for the change in the exponent
from N 3 to N log2 7 .
With this “fast” matrix multiplication, Strassen also obtained a surprising result
for matrix inversion [1]. Suppose that the matrices
� � � �
a11 a12
c11 c12
a21 a22
and
c21 c22
(2.11.5)
are inverses of each other. Then the c’s can be obtained from the a’s by the following
operations (compare equations 2.7.22 and 2.7.25):
R1 = Inverse(a11)
R2 = a21 × R1
R3 = R1 × a12
R4 = a21 × R3
R5 = R4 − a22
R6 = Inverse(R5)
c12 = R3 × R6
c21 = R6 × R2
R7 = R3 × c21
c11 = R1 − R7
c22 = −R6
(2.11.6)
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