
162
C. - T. Pan and P. T. P. Tang
Moreover, ak(A) and ak+l(A) can be estimated by the quantities
r(R11)
and
y(R22):
<_ <_ +
and
+ k)) _<
_<
k
In other words, the conclusions (5), (6), (7) and (8) of Lemmas 2 and 3 are all satisfied
with f = 1. More generally, if Rlx is an .f-pivoted block, 0 < .f < 1, then (5), (6), (7)
and (8) are all satisfied with the relaxation factor f.
In particular, if A has numerical rank k, we have a rank-revealing QR factorization with
O'min(.~ll )
~>
(f/~/k(n- k + 1))ak(A)
and O'max(R22 )
<: (~/(n-
k)(k + 1)/f)ak+l(A).
(See [6] for a proof).
Now we ...