Scaling, reordering, and diagonal pivoting in ILU preconditionings. (English) Zbl 1183.65044
Summary: We consider several techniques for construction of general-purpose ILU preconditioning. We also provide some new theoretical results supporting the proposed preconditioning strategy. For the preprocession stage we consider row/column norm-equalizing two-side scaling, as well as diagonal dominance improvement by unsymmetric permutations. Within the incomplete LU-factorization, we use a variant of the inverse norm-reducing complete diagonal pivoting. Numerical results are given for 67 test problems chosen from publicly available matrix sets.
MSC:
| 65F35 | Numerical computation of matrix norms, conditioning, scaling |
| 65F10 | Iterative numerical methods for linear systems |
References:
| [1] | DOI: 10.1002/(SICI)1098-2426(200001)16:1<11::AID-NUM2>3.0.CO;2-X · Zbl 0956.65018 · doi:10.1002/(SICI)1098-2426(200001)16:1<11::AID-NUM2>3.0.CO;2-X |
| [2] | O. Axelsson, Iterative solution methods. Cambridge University Press, Cambridge, 1994. · Zbl 0795.65014 |
| [3] | DOI: 10.1007/BF01389448 · Zbl 0564.65017 · doi:10.1007/BF01389448 |
| [4] | DOI: 10.1137/S1064827599361308 · Zbl 0985.65036 · doi:10.1137/S1064827599361308 |
| [5] | DOI: 10.1016/S0024-3795(01)00385-8 · Zbl 0991.65028 · doi:10.1016/S0024-3795(01)00385-8 |
| [6] | DOI: 10.1137/040608374 · Zbl 1104.65037 · doi:10.1137/040608374 |
| [7] | DOI: 10.1007/BF01733786 · Zbl 0865.65017 · doi:10.1007/BF01733786 |
| [8] | DOI: 10.1093/imamat/20.1.61 · Zbl 0364.65028 · doi:10.1093/imamat/20.1.61 |
| [9] | Int. J. Comp. Math. 40 pp 169– (1992) |
| [10] | DOI: 10.1002/(SICI)1099-1506(199811/12)5:6<483::AID-NLA156>3.0.CO;2-7 · Zbl 0969.65037 · doi:10.1002/(SICI)1099-1506(199811/12)5:6<483::AID-NLA156>3.0.CO;2-7 |
| [11] | DOI: 10.1002/nla.436 · Zbl 1164.65350 · doi:10.1002/nla.436 |
| [12] | DOI: 10.1016/0021-9991(80)90015-7 · Zbl 0442.65022 · doi:10.1016/0021-9991(80)90015-7 |
| [13] | DOI: 10.1137/S1064827502405094 · Zbl 1042.65025 · doi:10.1137/S1064827502405094 |
| [14] | O. E. Livne and G. H. Golub, Scaling by Binormalization. Numer. Algebra (2004) 35, 97-120. · Zbl 1050.65049 |
| [15] | A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications. Academic Press, New York, 1979. · Zbl 0437.26007 |
| [16] | J. Mayer, Symmetric permutation for I-matrices to delay and avoid small pivots during factorization. Preprint No. 06/08, 2004, Universitat Karlsruhe (submitted to SJSC). |
| [17] | DOI: 10.1137/S0036142993259792 · Zbl 0873.65054 · doi:10.1137/S0036142993259792 |
| [18] | DOI: 10.1007/BF01385754 · Zbl 0791.65016 · doi:10.1007/BF01385754 |
| [19] | DOI: 10.1137/0907058 · Zbl 0599.65018 · doi:10.1137/0907058 |
| [20] | Schneider M. H., Operations Research 38 pp 439– (1990) |
| [21] | V. Simoncini and D. B. Szyld, On the occurrence of superlinear convergence of exact and inexact Krylov subspace methods. Dept. Math., Temple University Report No. 03-3-13, Philadelphia, Pennsylvania, March 2003. |
| [22] | DOI: 10.2307/2314570 · Zbl 0166.03702 · doi:10.2307/2314570 |
| [23] | DOI: 10.1007/BF02165662 · Zbl 0182.49002 · doi:10.1007/BF02165662 |
| [24] | DOI: 10.1137/0913035 · Zbl 0761.65023 · doi:10.1137/0913035 |
| [25] | DOI: 10.1016/0377-0427(93)90028-A · Zbl 0797.65026 · doi:10.1016/0377-0427(93)90028-A |
| [26] | DOI: 10.1137/0717002 · Zbl 0447.65021 · doi:10.1137/0717002 |
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