A geometric view of Krylov subspace methods on singular systems

Numerical Linear Algebra with Applications

Volumn 18, Issue 3, 2011, Pages 449-469

  Hayami, Ken ^a,b   Sugihara, Masaaki ^c  

^a NATIONAL INSTITUTE OF INFORMATICS   (Japan)

^b GRADUATE UNIVERSITY FOR ADVANCED STUDIES   (Japan)

^c UNIVERSITY OF TOKYO   (Japan)

Author keywords

GCR(k) method; GMRES method; Krylov subspace method; Least squares problems; Singular systems

Indexed keywords

EID: 79954536988     PISSN: 10705325     EISSN: 10991506     Source Type: Journal    
DOI: 10.1002/nla.737     Document Type: Article

References (44)

1. Fletcher R. Conjugate gradient methods for indefinite systems. In Numerical Analysis Dundee 1975, Watson G (ed.). Lecture Notes in Mathematics, vol. 506. Springer: Berlin, 1976; 73-89.
2. Sonneveld P. CGS: a fast Lanczos-type solver for nonsymmetric linear systems. SIAM Journal on Statistical and Scientific Computing 1989; 10:36-52.
3. van der Vorst HA. Bi-CGSTAB: a more smoothly converging variant of CG-S for the solution of nonsymmetric linear systems. SIAM Journal on Statistical and Scientific Computing 1992; 13:631-644.
4. Freund RW, Nachtigal NM. QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numerische Mathematik 1991; 60:315-339.
5. Freund RW. A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems. SIAM Journal on Scientific Computing 1993; 14:425-448.
6. Saad Y, Schultz MH. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Statistical and Scientific Computing 1986; 7:856-869.
7. Eisenstat SC, Elman HC, Schultz MH. Variational iterative methods for nonsymmetric systems of linear equations. SIAM Journal on Numerical Analysis 1983; 20:345-357.
8. Biro O, Preis K, Richter KR. On the use of the magnetic vector potential in the nodal and edge finite element analysis of 3-D magnetostatic problems. IEEE Transactions on Magnetics 1996; 32:651-654.
9. Ren Z. Influence of the R.H.S. on the convergence behaviour of the curl-curl equation. IEEE Transactions on Magnetics 1996; 32:655-658.
10. Igarashi H. On the reconstruction of magnetic source in cylindrical permanent magnets. In Inverse Problems in Engineering Mechanics II, Tanaka M, Dulikravich GS (eds). Elsevier: Amsterdam, 2000; 467-476.
11. Strouboulis T, Babuška I, Copps K. The design and analysis of the generalized finite element method. Computer Methods in Applied Mechanics and Engineering 2000; 181:43-69.
12. Tanabe K. The conjugate gradient method for computing all the extremal stationary probability vectors of a stochastic matrix. Annals of the Institute of Statistical Mathematics 1985; 37 Part B:173-187.
13. Freund RW, Hochbruck M. On the use of two QMR algorithms for solving singular systems and applications in Markov chain modeling. Numerical Linear Algebra with Applications 1994; 1:403-420.
14. Dayar T, Stewart WJ. Comparison of partitioning techniques for two-level iterative solvers on large, sparse Markov chains. SIAM Journal on Scientific Computing 2000; 21:1691-1705.
15. Tanabe K. Characterization of linear stationary processes for solving singular system of linear equations. Numerische Mathematik 1974; 22:349-359.
16. Dax A. The convergence of linear stationary iterative processes for solving singular unstructured systems of linear equations. SIAM Review 1990; 32:611-635.
17. Eiermann M, Marek I, Niethammer W. On the solution of singular linear systems of algebraic equations by semiiterative methods. Numerische Mathematik 1988; 53:265-283.
18. Hanke M, Hochbruck M. A Chebyshev-like semiiteration for inconsistent linear systems. Electronic Transactions on Numerical Analysis 1993; 1:89-103.
19. Sidi A. Orthogonal polynomials and semi-iterative methods for the Drazin-inverse solution of singular linear systems. Numerische Mathematik 2003; 93:563-581.
20. Kammerer WJ, Nashed MZ. On the convergence of the conjugate gradient method for singular linear operator equations. SIAM Journal on Numerical Analysis 1972; 9:165-181.
21. Kaasschieter EF. Preconditioned conjugate gradients for solving singular systems. Journal of Computational and Applied Mathematics 1988; 24:265-275.
22. Fischer B, Hanke M, Hochbruck M. A note on conjugate-gradient type methods for indefinite and/or inconsistent linear systems. Numerical Algorithms 1996; 11:181-189.
23. Wei Y, Wu H. Convergence properties of Krylov subspace methods for singular linear systems with arbitrary index. Journal of Computational and Applied Mathematics 2000; 114:305-318.
24. Abe K, Ogata H, Sugihara M, Zhang S-L, Mitsui T. Convergence theory of the CR method for linear singular systems. Transactions of the Japan Society for Industrial and Applied Mathematics 1999; 9:1-13 (in Japanese).
25. Hayami K. On the behaviour of the conjugate residual method for singular systems. In Proceedings of Fifth China-Japan Seminar on Numerical Mathematics, Shanghai, 2000. Shi Z-C, Kawarada H (eds). Science Press: Beijing/New York, 2002; 117-126.
26. Hayami K. On the convergence of the conjugate residual method for singular systems. Transactions of the Japan Society for Industrial and Applied Mathematics 2003; 13:1-33 (in Japanese).
27. Zhang S-L, Oyanagi Y, Sugihara M. Necessary and sufficient conditions for the convergence of Orthomin(k) on inconsistent linear systems. Numerische Mathematik 2000; 87:391-405.
28. Sidi A. A unified approach to Krylov subspace methods for the Drazin-inverse solution of singular nonsymmetric linear systems. Linear Algebra and its Applications 1999; 298:99-113.
29. Sidi A. DGMRES: a GMRES-type algorithm for Drazin-inverse solution of singular nonsymmetric linear systems. Linear Algebra and its Applications 2001; 335:189-204.
30. Brown P, Walker HF. GMRES on (nearly) singular systems. SIAM Journal on Matrix Analysis and Applications 1997; 18:37-51.
31. Smoch L. Some results about GMRES in the singular case. Numerical Algorithms 1999; 22:193-212.
32. Ipsen ICF, Meyer CD. The idea behind Krylov methods. American Mathematical Monthly 1998; 105:889-899.
33. Calvetti D, Lewis B, Reichel L. GMRES-type methods for inconsistent systems. Linear Algebra and its Applications 2000; 316:157-169.
34. Schneider O. Krylov subspace methods and their generalizations for solving singular operator equations with applications to continuous time Markov chains. Doctoral Dissertation, Fakultät für Mathematik und Informatik, der Technischen Universität Bergakademie Freiberg, November 2005.
35. Washio T, Doi S. A solution of Neumann problem on curvilinear coordinate systems. In Matrix Analysis and Parallel Computing, Proceedings of PCG'94, Natori M, Nodera T (eds), Keio University, March 1994; 75-87.
36. Saad Y. Iterative Methods for Sparse Linear Systems (2nd edn). SIAM: Philadelphia, PA, 2003.
37. Elman HC. Iterative methods for large, sparse, nonsymmetric systems of linear equations. Ph.D. Thesis, Research Report {music sharp sign}229, Department of Computer Science, Yale University, 1982.
38. Mori M, Sugihara M, Murota K. Linear Computation. Iwanami-shoten: Tokyo, 1994 (in Japanese).
39. van der Vorst HA, Vuik C. GMRESR: a family of nested GMRES methods. Numerical Linear Algebra with Applications 1994; 1:369-386.
40. de Sturler E. Nested Krylov methods based on GCR. Journal of Computational and Applied Mathematics 1996; 67:15-41.
41. de Sturler E. Truncation strategies for optimal Krylov subspace methods. SIAM Journal on Numerical Analysis 1999; 36:864-889.
42. Langou J. GCR vs flexible GMRES. Book of Abstracts, GAMM-SIAM Conference on Applied Linear Algebra, Düsseldorf, 2006; 118.
43. Hayami K, Sugihara M. On the convergence of the GCR(k) method for singular systems, NII Technical Reports, NII-2004-009E, National Institute of Informatics, Tokyo, 2004; 1-24. Available from.
44. Hayami K, Sugihara M. A geometric view of Krylov subspace methods on singular systems, NII Technical Reports, NII-2009-007E, National Institute of Informatics, Tokyo, 2009; 1-28. Available from.