Greville's Method for Preconditioning Least Squares ... - Projects

7
Remark 4 If k i is sparse, when we update M i−1 , we only need to update the rows
which correspond to the nonzero elements in k i . Hence, the rank-one update can be
efficient.
Remark 5 When A is nonsingular, the inverse of A can also be computed by the
Sherman-Morrison formula, which is also a rank-one update algorithm. Following this
Sherman-Morrison formula method, an incomplete factorization preconditioner can
also be developed, we refer readers to [7,8].
If we want to construct the matrix K, F and V without forming M i explicitly,
we can use a vector-wise version of the above algorithm. In Algorithm 1, the column
vectors of K are constructed one column at a step, and u i , f i , v i are defined according
to k i . Hence, it is possible to rewrite Algorithm 1 into a vector-wise form.
Since u i can be computed from a i −A i−1 k i , which does not refer to M i−1 explicitly,
to vectorize Algorithm 1, we only need to form k i and v i = Mi−1 T k i when linear
dependence occurs, without using M i−1 explicitly.
Consider the case when numerical droppings are not used. Since we already know
that
A † = (I − K)F −1 V T
⎡
f1
−1
⎢
= (I − [ k 1 . . . k n ]) ⎣
=
n∑
(e i − k i ) 1 vi T ,
f i
i=1
. ..
f −1 n
⎤ ⎡
v1 T
⎥ ⎢ ...
⎦ ⎣
for any integer 1 ≤ p ≤ n, it is easy to see that
p∑
A † p = (e i − k i ) 1 vi T . (3.26)
f i
Therefore, we have
and
i=1
v i = (A † i−1 )T k i (3.27)
∑i−1
= ( (e p − k 1 p) vp T ) T k i
f p
(3.28)
=
p=1
∑i−1
1
v p(e p − k p) T k i (3.29)
f p
p=1
k i = A † i−1 a i (3.30)
∑i−1
= (e p − k 1 p) vp T a i
f p
(3.31)
p=1
∑i−2
= (e p − k 1 p) v T 1
p a i + (e i−1 − k i−1 ) v
f p f
i−1a T i
p=1
i−1
(3.32)
= A † i−2 a 1
i + (e i−1 − k i−1 ) v
f
i−1a T i .
i−1
(3.33)
v T n
⎤
⎥
⎦