Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

• In step 4, we have to take care that with S 1 . (a) If k is odd, then copy the
first column of A 21 into W mk . (b) Complete S 1 .
• In step 10, we have to take care that with S 4 . (a) If k is odd, then pretend the
first column of A 21 = 0 in W mk . (b) Complete S 4 .
• In step 11, we have to take care that with M 6 . (a) If m is odd, then save the
first row of M 5 . (b) Calculate most of M 6 . (c) Complete M 6 using (a) based
on whether or not m is odd.
• In step 21, we have to take care that with M 3 . (a) Caluclate M 3 using an
index shift.
This all sounds very complicated. However, the code GEMMW that is readily
available on the Web effectively is implemented in 27 calls to subroutines that
do the matrix operations and actually implements
C = ?Cop(A)op(B) + DCC,
where op(X) is either X, X transpose, X conjugate, or X conjugate transpose.
145
What is the total cost?
• There are 7 submatrix-submatrix multiplies and 15 submatrix-submatrix
adds or subtracts. So the cost is f(n) = 7f(n/2) + 15n 2 /4 when m=k=n. This is
actually an O(n 2.807 logn) algorithm, where log 2 7 = 2.807.
• The work area W mk needs 7((m+1)max(k,n)+m+4)/48 space.
• The work area W kn needs 7((k+1)n+n+4)/48 space.
• If C overlaps A or B in memory, an additional mn space is needed to save C
before calculating DCC when D-0.
• The maximum amount of extra memory is bounded by
(mCmax(k,n)+kn)/3+(m+max(k,n)+k+3n)/2+32+mn. Hence, the overall
extra storage is cN 2 /3, where c,{2,5}.
• Typical memory usage when m=k=n is
o D-0 or A or B overlap with C: 1.67N 2 .
o D=0 and A and B do not overlap with C: 0.67N 2 .
146