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6 Thus f i (i = 1, . . . , n) are always positive, which implies that F is a diagonal matrix with positive diagonal elements. If A is a full rank matrix, we have V = [u 1 , . . . , u n] (3.19) ] = [(I − A 0 A † 0 )a 1, . . . , (I − A n−1 A † n−1 )an (3.20) = [a 1 − A 0 k 1 , . . . , a n − A n−1 k n] (3.21) = A − [A 0 k 1 , . . . , A n−1 k n] (3.22) = A − [A 1 k 1 , . . . , A nk n] (3.23) = A(I − K). (3.24) The second from the bottom equality follows from the fact that K is a strictly upper triangular matrix. Now, when A is full rank, A † can be decomposed as follows. A † = (I − K)F −1 V T = (I − K)F −1 (I − K) T A T ✷ (3.25) Remark 1 From the above proof, it is easy to see that when A is a full column rank matrix, (I − K) −T F (I − K) −1 is a LDL T Decomposition of A T A. Based on the Greville’s method, we can construct a preconditioning algorithm. We want to construct a sparse approximation to the Moore-Penrose inverse of A. Hence, we perform some numerical droppings in the middle of the algorithm to maintain the sparsity of the preconditioner. We call the following algorithm the global Greville preconditioning algorithm, since it forms or updates the whole matrix at a time rather than column by column. Algorithm 1 Global Greville preconditioning algorithm 1. set M 0 = 0 2. for i = 1 : n 3. k i = M i−1 a i 4. u i = a i − A i−1 k i 5. if ‖u i ‖ 2 is not small 6. f i = ‖u i ‖ 2 2 7. v i = u i 8. else 9. f i = 1 + ‖k i ‖ 2 2 10. v i = Mi−1 T k i 11. end if 12. M i = M i−1 + f 1 i (e i − k i )vi T 13. perform numerical droppings to M i 14. end for 15. Obtain M n ≈ A † . Remark 2 In Algorithm 1, we do not need to store k i , v i , f i , i = 1, . . . , n, because we form the M i explicitly. Remark 3 In Algorithm 1, we need to update M i at every step, but we do not need to update the whole matrix, since only the first i − 1 rows of M i−1 can be nonzero. Hence, to compute M i , we need to update the first i − 1 rows of M i−1 , and then add one new nonzero row to be the i-th row.
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 ⎤ ⎥ ⎦
Page 1 and 2: Noname manuscript No. (will be inse
Page 3 and 4: 3 Greville’s method. In Section 3
Page 5: 5 we can express A † i in a unifi
Page 9 and 10: 9 6. k j = k j + uT i aj f i (e i
Page 11 and 12: 11 where C is a nonsingular matrix.
Page 13 and 14: 13 Hence, [ ] M = (I − K)F −1 I
Page 15 and 16: 15 where ∆E is an error matrix. H
Page 17 and 18: 17 Theorem 10 Let A ∈ R m×n . If
Page 19 and 20: 19 Table 1 Information on the test
Page 21 and 22: 21 preconditioner is usually much d
Page 23: 23 problems are equivalent to the o

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