Parallelizing the Divide and Conquer Algorithm - Innovative ...

2.2 Deation
Dongarra and Sorensen showed [14] that the problem (2.5) can potentially
be reduced in size. If z i = 0 for some i, we see from (2.3) that d i is an
eigenvalue of D + zz T
with eigenvector e i . Moreover, if D has an eigenvalue
d i of multiplicity m>1, we can rotate the eigenvector basis in order
to zero out the component ofzcorresponding to the repeated eigenvalues.
Then, we can remove rows and columns from D + zz T
zero components of z.
corresponding to
When working in nite precision arithmetic, we must deal with the
problem of the z i 's nearly equal to zero and nearly equal d i 's.
Then,
in order to precisely describe when we can deate, we need to dene a
tolerance . Let = "kD + zz T k2 where " is the machine precision.
We say that the rst type of deation arises when jz i j .
assume that kzk2 =1. Wehave
k(D+zz T )e i , d i e i k2 = jz i jkzk2 :
Now
Then (d i ;e i )may be considered as an approximate eigenpair for D + zz T
and z i is set to zero.
The second type of deation comes from a Givens rotation applied to
D + zz T
jz i z j jjd i , d j j=
q
in order to set a component ofzequal to zero. Suppose that
with c = z i =r; s=z j =r; r=
z 2 i + z2 j . Let G ij be the Givens rotation dened by
[e i ;e j ] T G ij [e i ;e j ]=
q
z 2 i +z2 j
c
s
,s c
!
and c2 + s 2 = 1. Then
G i (D + zz T )G T i = e D + ~z~z T + E ij ; kE ij k2 ;
where ~z i = r 2 ; ~z j =0; ~ d i =d i c 2 +d j s 2 ; ~ d j =d i s 2 +d j c 2 and E ij =(d i ,d j )cs.
The result of recognizing all these deations is to replace the rankone
update problem D + zz T with one of smaller size. Hence, if G is
the product of all the rotations used to zero out certain components of z
and if P is the accumulation of permutations used to translate the zero
components of z to the bottom of z, the result is
PG(D+zz T )G T P T =
eD + ~z~z T 0
0
6
!
+ E; kEk2 c