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

18
10. f i = ‖u‖ 2 2
11. v i = u
12. else
13. f i = ‖k i ‖ 2 2 + 1
14. v i = (A † ∑i−1
1
i−1 )T k i = v p(e p − k p) T k i
f p
p=1
15. end if
16. end for
17. K = [k 1 , . . . , k n], F = Diag {f 1 , . . . , f n}, V = [v 1 , . . . , v n].
In Algorithm 2, all the columns k j , j = i, . . . , n are updated in one outer loop. In
Algorithm 5, all the updates for each k i column are finished at a time. This left-looking
approach is suggested in [1,5]. From [1], a left-looking version is sometimes more robust
and efficient. In Section 8 our preconditioners are computed by Algorithm 5.
In the next section, we will use Algorithm 5 as our preconditioning algorithm.
Since we cannot control the sparsity of V , and V can be very dense, we try not to
store V . Note that when column a j is recognized as a linearly independent column,
the corresponding column becomes v j = A(e j − k j ). Hence, we do not have to store
v j . When we need v j in the 4th line in Algorithm 5, we can compte θ = a T i A(e j − k j ),
where a T i A is used for i − 1 times and discarded. For the v j corresponding to a linearly
dependent column, we need to store it.
8 Numerical Examples
In this section, we use matrices from the Florida University Sparse Matrices Collection
[11] to test our algorithms, where zero rows are omitted and if the original matrix
has more columns than rows we use its transpose. All computations were run on a
Dell Precision 690, where the CPU is 3 GHz and the memory is 16 GB, and the
programming language and compiling environment was GNU C/C++ 3.4.3 in Redhat
Linux. Detailed information of the matrices is given in Table 1.