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20 our dropping rule is to drop the i-th element of k j , i.e., k j,i when the following two inequalities holds. |k j,i |‖a i ‖ 2 < τ d and |k j,i | < τ d ′ max l=1,...,i−1 |k l,i|. (8.1) Here we have another tolerance τ d ′. In the following examples, we simply fix it to 10.0 ∗ τ d . We use ‖u i ‖ 2 ≤ τ s‖A i−1 ‖ F ‖a i ‖ 2 , (8.2) as the criterion to judge if column a i is linearly dependent on the previous columns or not, where τ s is our switching tolerance. The stopping rule for BA-GMRES is ‖A T (b − Ax)‖ 2 ≤ 10 −8 · ‖A T b‖ 2 . (8.3) In the following tables, we use our preconditioner to precondition GMRES and compare with GMRES preconditioned by RIF, which is denoted by “RIF-GMRES”, CGLS [4] preconditioned by Incomplete Modified Gram-Schmidt orthogonalization preconditioner (IMGS) [15], which is denoted by “IMGS-CGLS”, CGLS preconditioned by using the diagonal scaling preconditioner, which is denoted by “D-CGLS” and CGLS preconditioned by the Incomplete Cholesky Factorization preconditioner [4], which is denoted by “IC-CGLS”, to solve the original least squares problem min x∈R n‖b−Ax‖ 2. The columns τ d and τ s list the dropping tolerance and the switching tolerance we used in our preconditioning algorithm. The column “ITS” gives the number of iterations. The columns “Pre. T” and “Tol. T” give the preconditioning time and the total time in seconds, respectively. For “RIF-GMRES”, “IMGS-CGLS”, “D-CGLS” and “D-CGSL”, in each cell we give the numbers of iterations and the total CPU time. For “RIF-GMRES” and “IMGS-CGLS”, we give the parameters we used in brackets. In the “ nnzP nnzA ” column, we give the ratio of the number of nonzero elements in our preconditioner and A, where nnzP is the number of nonzero elements in K plus the number of the nonzero elements in diagonal matrix F . The ∗ indicates the fastest method for each problem. Table 2 Numerical Results for Full Rank Matrices matrix τ d nnzP nnzA ITS Pre. T Tot. T RIF-GMRES IMGS-CGLS D-CGLS IC-CGLS vtp base 1.e-6 18.0 4 0.03 *0.03 5, 0.04 (1.e-9) 9, 0.10 (1.e-5) 3667, 0.29 1348, 0.12 capri 1.e-3 10.0 17 0.03 0.04 9, 0.07 (1.e-4) 371, 0.04 (1.0) 496, 0.06 195, *0.03 scfxm1 1.e-3 13.3 17 0.05 0.06 7, *0.05 (1.e-5) 116, 0.08 (1.e-1) 761, 0.11 318, 0.06 scfxm2 1.e-3 16.6 27 0.12 *0.18 12, 0.26 (1.e-5) 734, 0.28 (1.0) 1332, 0.42 680, 0.23 perold 1.e-2 12.4 17 0.17 0.21 45, 0.36 (1.e-4) 134, 0.26 (1.e-1) 1058, 0.36 394, *0.14 scfxm3 1.e-4 27.0 7 0.30 *0.34 12, 0.46 (1.e-5) 878, 0.53 (1.0) 1583, 0.72 847, 0.41 maros 1.e-6 28.8 3 1.02 *1.05 10, 1.13 (1.e-7) 383, 1.47 (1.e-1) 13714, 6.71 3602, 1.8 pilot we 1.e-1 4.5 41 0.10 0.20 12, 0.4 (1.e-5) 536, 0.37 (1.0) 736, 0.38 287, *0.14 stocfor2 1.e-5 152.2 5 10.55 10.68 16, 11.5 (1.e-7) 138, 3.14 (1.e-1) 2147, 1.59 1362, *1.12 bnl2 1.e-1 3.0 114 0.4 1.22 127, 2.79 (1.e-2) 598, 1.24 (1.0) 657, 0.64 240, *0.36 pilot 1.e-2 2.9 18 1.54 1.72 359, 4.57 (1.e-1) 741, 1.49 (1.0) 822, 1.24 269, *0.75 d2q06c 1.e-3 18.7 32 4.27 5.15 19, 10.68 (1.e-5) 1209, 3.6 (1.0) 2743, 4.05 1077, *1.63 80bau3b 5.e-1 0.5 28 0.46 0.52 26, 0.54 (5.e-1) 47,1.34 (1.0) 56, *0.09 21, 0.21 For full rank matrices, we did not list τ s because we simply set τ s to be a very small number and do not try to detect any linearly dependent columns in the coefficient matrices. From the numerical results in Table 2, we conclude that our preconditioner performs competitively when the matrices are ill-conditioned. We can also see that our
21 preconditioner is usually much denser than the original matrix A when the density of A is larger than 0.5%. When the density of A is less than 0.3%, the density of our preconditioner is usually similar to or less than that of A. In Table 3 we show the numerical results for rank deficient matrices. Column “D” gives the number of linearly dependent columns that we detected. Table 3 Numerical Results for Rank Deficient Matrices matrix τ d , τ s nnzP nnzA D ITS Pre. T Tot. T beaflw 1.e-5, 1.e-10 2.25 26 209 1.22 *2.25 beause 1.e-5, 1.e-11 2.71 31 20 1.16 *1.22 lp cycle 1.e-4, 1.e-7 33.5 15 27 2.17 *2.68 Pd rhs 1.e-4, 1.e-8 0.75 3 3 0.79 0.80 model09 1.e-2, 1.e-6 2.92 0 12 0.97 *1.12 landmark 1.e-1, 1.e-8 0.05 0 143 2.83 10.37 matrix RIF-GMRES IMGS-CGLS D-CGLS IC-CGLS beaflw † † † † beause † † † † lp cycle † 5832, 4.87(100.0) 6799, 6.74 † Pd rhs † 25, 0.93(0.1) 242, *0.29 89, 0.54 model09 46, 2.53(1.e-3) † 1267, 3.09 412, 1.22 landmark 239, 38.43(10.0) 252, 36.68(10.0) 252, *8.77 † †: GMRES did not converge in 2, 000 steps or D-CGLS did not converge in 25, 000 steps. From the results in Table 3 we can conclude that our preconditioner works for all the tested problems, and performed competitively with other preconditioners. The density of our preconditioner is about 3 times the original matrix A or less except for lp cycle. For the matrix lp cycle, we know that the rank deficient columns are, 182 184 216 237 253 717 754 961 1221 1239 1260 1261 1278 1640 1859, (8.4) 15 columns in all. As we know that the rank deficient columns are not unique, the above columns we list are the columns which are linearly dependent on their previous columns. In the following example, we can see that our preconditioning algorithm can detect many of them. Table 4 Numerical Results for lp cycle τ d τ s deficiency detected ITS Pre. T Its. T Tot. T 1.e-6 1.e-10 −1239, −1261, −1278 6 3.65 0.16 3.81 1.e-6 1.e-7 detect all exactly 4 3.7 0.10 3.8 1.e-6 1.e-5 detected 95 col † 4.7 In the above table, we tested our preconditioning algorithm with fixed dropping tolerance τ d = 10 −6 and different switching tolerances τ s. For this problem lp cycle,
Page 1 and 2: Noname manuscript No. (will be inse
Page 3 and 4: 3 Greville’s method. In Section 3
Page 5 and 6: 5 we can express A † i in a unifi
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: 19 Table 1 Information on the test
Page 23: 23 problems are equivalent to the o

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