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

9
6. k j = k j + uT i aj
f i
(e i − k i )
7. perform numerical droppings on k j
8. end for
9. end for
10. Obtain K = [k 1 , . . . , k n], F = Diag{f 1 , . . . , f n}.
In Algorithm 3,
u i = a i − A i−1 k i
⎡ ⎤
−k i,1
.
−k i,i−1
= [a 1 , . . . , a i , 0, . . . , 0]
1
0 ⎢
⎣ ... ⎥
⎦
0
= A i (e i − k i )
= A(e i − k i ).
If we denote e i − k i as z i , then u i = Az i .
Then, line 6 of Algorithm 3 can be rewritten as
k j = k j + uT i a j
‖u i ‖ 2 (e i − k i )
2
e j − k j = e j − k j − uT i a j
‖u i ‖ 2 (e i − k i )
2
z j = z j − uT i a j
‖u i ‖ 2 z i .
2
Denote d i = ‖u i ‖ 2 2 and θ = uT i a j
. Then, combining all the new notations, we can
d i
rewrite the algorithm as follows.
Algorithm 4
1. set Z = I n×n
2. for i = 1 : n
3. u i = A i z i
4. d i = (u i , u i )
5. for j = i + 1, . . . , n
6. θ = (ui,aj)
d i
7. z j = z j − θz i
8. end for
9. end for
10. Z = [z 1 , . . . , z n], D = Diag{d 1 , . . . , d n}.
Remark 6 Since z i = e i − k i , we have Z = I − K. Denoting D = Diag{d 1 , . . . , d n}, the
factorization of A † in Theorem 2 can be rewritten as
A † = ZD −1 Z T A T (4.1)