Parallelizing the Divide and Conquer Algorithm - Innovative ...

with W = Q(PG) T U.
When not properly implemented, the computation of W can be very
expensive. To simplify the explanation, we illustrate it with a 44 example:
we suppose that d1 = d3 and that G is the Givens rotation used to set to
zero the third component ofz. The matrix P is a permutation that moves
z3 to the bottom of z. We indicate by a\" that a value has changed.
There are two way of applying the transformation (PG) T , either on the
left, that is on Q, or on the right, that is on U. Note that Q = diag(Q1;Q2)
is block diagonal (see(2.3)) and so we would like tototake advantage of
this structure. It would halve the cost of the matrix multiplication if Q1,Q2
are of the same size. To preserve the block diagonal form of Q, we need to
apply (PG) T
on the right:
Q (PG) T U =
=
=
0
B
@
0
B
@
0
B
@
1 0
C
A (PG)T B
@
1 0
C
A G B
@
10
C
A
B
@
1
0
0
The product between the two last matrices is performed with 64 ops
instead of the 2n 3 = 128 ops of a full matrix product. However, if we apply
(PG) T on the left, we can reduce further the number of ops. Consider
again the 4 4 example:
Q(PG) T U =
0
B
@
1 0
C
A GT P T B
@
1
C
A
1
C
A
1
1
1
C
A
1
C
A
17