Greville's Method for Preconditioning Least Squares ... - Projects
Greville's Method for Preconditioning Least Squares ... - Projects Greville's Method for Preconditioning Least Squares ... - Projects
4 To utilize the above theorem, let A = n∑ a i e T i , (2.4) i=1 where a i is the ith column of A. Further let A i = [a 1 , . . . , a i , 0, . . . , 0] ∈ R m×n . Hence we have i∑ A i = a k e T k , i = 1, . . . , n, (2.5) and if we denote A 0 = 0 m×n , k=1 A i = A i−1 + a i e T i , i = 1, . . . , n. (2.6) Thus every A i , i = 1, . . . , n is a rank-one update of A i−1 . Noticing that A † 0 = 0 n×m, we can utilize Theorem 1 to compute the Moore-Penrose inverse of A step by step and have A † = A † n in the end. In Theorem 1, substituting a i into c and e i into d, we can rewrite Equation (2.2) as follows. a i ∈ R(A i−1 ) ⇔ u = (I − A i−1 A † i−1 )a i = 0 (2.7) No matter whether a i is linearly dependent on the previous columns or not, we have for any i = 1, 2, . . . , n β = 1 + e T i A † i−1 a i = 1. (2.8) The vector v in equation (2.3) can be written as v = e T i (I − A † i−1 A i−1) ≠ 0, (2.9) which is nonzero for any i = 1, . . . , n. Hence, we can use Case 1 and Case 3 for case a i ∉ R(A i−1 ) and case a i ∈ R(A i−1 ), respectively. Then from Theorem 1, denoting A 0 = 0 m×n , we obtain a method to compte A † i based on A † i−1 as A † i = { A † i−1 + (e i − A † i−1 a i)((I − A i−1 A † i−1 )a i) † if a i ∉ R(A i−1 ) A † i−1 + 1 σ i (e i − A † i−1 a i)(A † i−1 a i) T A † i−1 if a i ∈ R(A i−1 ) (2.10) where σ i = 1 + ‖A † i−1 a i‖ 2 2. This method was proposed by Greville in the 1960s[13]. 3 Preconditioning Algorithm In this section, we construct our preconditioning algorithm according to the Greville’s method in section 2. First of all, we notice that in Equation (2.10) the difference between case a i ∉ R(A i−1 ) and case a i ∈ R(A i−1 ) lies in the second term. If we define vectors k i , u i , v i and scalar f i for i = 1, . . . , n as k i = A † i−1 a i, (3.1) u i = a i − A i−1 k i = (I − A i−1 A † i−1 )a i, (3.2) { ‖ui ‖ 2 2 if a f i = i ∉ R(A i−1 ) 1 + ‖k i ‖ 2 2 if a i ∈ R(A i−1 ) , (3.3) { u i if a i ∉ R(A i−1 ) v i = (A † i−1 )T k i if a i ∈ R(A i−1 ) , (3.4)
5 we can express A † i in a unified form for general matrices as and we have A † i = A† i−1 + 1 f i (e i − k i )v T i , (3.5) A † = n∑ i=1 If we define matrices K ∈ R n×n , V ∈ R m×n , F ∈ R n×n as we obtain a matrix factorization of A † as follows. 1 f i (e i − k i )v T i . (3.6) K = [k 1 , . . . , k n], (3.7) V = [v 1 , . . . , v n], (3.8) ⎡ ⎤ f 1 · · · 0 ⎢ F = . ⎣ 0 .. ⎥ 0 ⎦ , (3.9) 0 · · · f n Theorem 2 Let A ∈ R m×n and rank(A) ≤ min{m, n}. Using the above notations, the Moore-Penrose inverse of A has the following factorization A † = (I − K)F −1 V T . (3.10) Here I is the identity matrix of order n, K is a strict upper triangular matrix, F is a diagonal matrix whose diagonal elements are all positive. If A is full column rank, then V = A(I − K) (3.11) A † = (I − K)F −1 (I − K) T A T . (3.12) Proof Denote Āi = [a 1 , . . . , a i ]. Then since k i = A † i−1 a i (3.13) = [a 1 , . . . , a i−1 , 0, . . . , 0] † a i (3.14) = [ Ā i−1 , 0, . . . , 0 ] † ai (3.15) [ ] Ā† = i−1 a i (3.16) 0 ⎡ ⎤ k i,1 . = k i,i−1 0 , (3.17) ⎢ ... ⎥ ⎣ ⎦ 0 K = [k 1 , . . . , k n] is a strictly upper triangular matrix. Since u i = 0 ⇔ a i ∈ R(A i−1 ), { ‖ui ‖ 2 2 if f i = 1 + ‖k i ‖ 2 2 if a i ∉ R(A i−1 ) a i ∈ R(A i−1 ) . (3.18)
- Page 1 and 2: Noname manuscript No. (will be inse
- Page 3: 3 Greville’s method. In Section 3
- Page 7 and 8: 7 Remark 4 If k i is sparse, when w
- 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
4<br />
To utilize the above theorem, let<br />
A =<br />
n∑<br />
a i e T i , (2.4)<br />
i=1<br />
where a i is the ith column of A. Further let A i = [a 1 , . . . , a i , 0, . . . , 0] ∈ R m×n . Hence<br />
we have<br />
i∑<br />
A i = a k e T k , i = 1, . . . , n, (2.5)<br />
and if we denote A 0 = 0 m×n ,<br />
k=1<br />
A i = A i−1 + a i e T i , i = 1, . . . , n. (2.6)<br />
Thus every A i , i = 1, . . . , n is a rank-one update of A i−1 . Noticing that A † 0 = 0 n×m,<br />
we can utilize Theorem 1 to compute the Moore-Penrose inverse of A step by step and<br />
have A † = A † n in the end.<br />
In Theorem 1, substituting a i into c and e i into d, we can rewrite Equation (2.2)<br />
as follows.<br />
a i ∈ R(A i−1 ) ⇔ u = (I − A i−1 A † i−1 )a i = 0 (2.7)<br />
No matter whether a i is linearly dependent on the previous columns or not, we have<br />
<strong>for</strong> any i = 1, 2, . . . , n<br />
β = 1 + e T i A † i−1 a i = 1. (2.8)<br />
The vector v in equation (2.3) can be written as<br />
v = e T i (I − A † i−1 A i−1) ≠ 0, (2.9)<br />
which is nonzero <strong>for</strong> any i = 1, . . . , n. Hence, we can use Case 1 and Case 3 <strong>for</strong> case<br />
a i ∉ R(A i−1 ) and case a i ∈ R(A i−1 ), respectively.<br />
Then from Theorem 1, denoting A 0 = 0 m×n , we obtain a method to compte A † i<br />
based on A † i−1 as<br />
A † i = { A<br />
†<br />
i−1 + (e i − A † i−1 a i)((I − A i−1 A † i−1 )a i) † if a i ∉ R(A i−1 )<br />
A † i−1 + 1 σ i<br />
(e i − A † i−1 a i)(A † i−1 a i) T A † i−1 if a i ∈ R(A i−1 ) (2.10)<br />
where σ i = 1 + ‖A † i−1 a i‖ 2 2. This method was proposed by Greville in the 1960s[13].<br />
3 <strong>Preconditioning</strong> Algorithm<br />
In this section, we construct our preconditioning algorithm according to the Greville’s<br />
method in section 2. First of all, we notice that in Equation (2.10) the difference<br />
between case a i ∉ R(A i−1 ) and case a i ∈ R(A i−1 ) lies in the second term. If we define<br />
vectors k i , u i , v i and scalar f i <strong>for</strong> i = 1, . . . , n as<br />
k i = A † i−1 a i, (3.1)<br />
u i = a i − A i−1 k i = (I − A i−1 A † i−1 )a i, (3.2)<br />
{ ‖ui ‖ 2 2 if a<br />
f i =<br />
i ∉ R(A i−1 )<br />
1 + ‖k i ‖ 2 2 if a i ∈ R(A i−1 ) , (3.3)<br />
{<br />
u i if a i ∉ R(A i−1 )<br />
v i =<br />
(A † i−1 )T k i if a i ∈ R(A i−1 ) , (3.4)
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