PRIMME: Preconditioned iterative multimethod eigensolver-methods and software description

ACM Transactions on Mathematical Software

Volumn 37, Issue 2, 2010, Pages

  Stathopoulos, Andreas ^a   McCombs, James R ^b  

^a The College of William and Mary   (United States)

^b Cobham Inc ^*  (United States)

Author keywords

Block; Conjugate gradient; Davidson; Eigenvalues; Eigenvectors; Hermitian; Iterative; Jacobi Davidson; Lanczos; Locking; Preconditioning; Software package

Indexed keywords

BLOCK CONJUGATE GRADIENTS; DAVIDSON; EIGENVALUES; EIGENVECTORS; HERMITIANS; ITERATIVE; JACOBI-DAVIDSON; LANCZOS;

CONJUGATE GRADIENT METHOD; PACKAGING; SOFTWARE PACKAGES; TUNING;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 77951916980     PISSN: 00983500     EISSN: 15577295     Source Type: Journal    
DOI: 10.1145/1731022.1731031     Document Type: Article

References (64)

1. ABSIL, P.-A., BAKER, C. G., AND GALLIVAN, K. A. 2006. A truncated-CG style method for symmetric generalized eigenvalue problems. J. Comput. Appl. Math. 189, 1-2, 274-285.
2. ABSIL, P.-A., MAHONY, R., SEPULCHRE, R., AND DOOREN, P. V. 2002. A Grassmann-Rayleigh quotient iteration for computing invariant subspaces. SIAM Review 44, 1, 57-73.
3. ADAMS, M. F. 2002. Evaluation of three unstructured multigrid methods on 3D finite element problems in solid mechanics. Int. J. Numer. Meth. Eng. 55, 519-534.
4. ARBENZ, P., HETMANIUK, U. L., LEHOUCQ, R. B., AND TUMINARO, R. S. 2005. A comparison of eigensolvers for large-scale 3D modal analysis using AMG-preconditioned iterative methods. Int. J. Numer. Meth. Eng. 64, 204-236.
5. BAGLAMA, J., CALVETTI, D., AND REICHEL, L. 2003. IRBLEIGS: A MATLAB program for computing a few eigenpairs of a large sparse Hermitian matrix. ACM Trans. Math. Softw. 29, 5, 337-348.
6. BAI, Z., DEMMEL, J., DONGARRA, J., RUHE, A., AND VAN DER VORST, H., Eds. 2000. Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. Software Environ. Tools 11. SIAM, Philadelphia, PA.
7. BAKER, C. G., HETMANIUK, U. L., LEHOUCQ, R. B., AND THORNQUIST, H. K. 2009. Anasazi software for the numerical solution of large-scale eigenvalue problems. ACM Trans. Math. Softw. 36, 3, Article 13, 23 pages. http://trilinos.sandia.gov/packages/anasazi.
8. BALAY, S., GROPP, W. D., MCINNES, L. C., AND SMITH, B. F. 1999. PETSc home page. http://www.mcs.anl.gov/petsc.
9. BARRETT, R., BERRY, M., CHAN, T., DEMMEL, J., DONATO, J., DONGARRA, J., EIJKHOUT, V., POZO, R., ROMINE, C., AND VAN DER VORST, H. 1994. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia, PA.
10. BOLLHÖFER, M. AND NOTAY, Y. 2007. JADAMILU: A software code for computing selected eigenvalues of large sparse symmetric matrices. Comput. Phys. Commun. 177, 12, 951-964.
11. BOLLHÖFER,M. AND SAAD, Y. 2006. Multilevel preconditioners constructed from inverse-based ilus. SIAM J. Sci. Comput. 27, 5, 1627-1650.
12. CLINT, M. AND JENNINGS, A. 1970. The evaluation of eigenvalues and eigenvectors of a real symmetric matrix by simultaneous iteration. Comput. J. 13, 76-80.
13. COHEN, M. L. AND CHELIKOWSKY, J. R. 1989. Electronic Structure and Optical Properties of Semiconditors, 2nd ed. Springer-Verlag, NY, Berlin, Germany.
14. CULLUM, J. AND DONATH, W. 1974. A block Lanczos algorithm for computing the q algebraically largest eigenvalues and a corresponding eigenspace of large, sparse, symmetric matrices. In Procedings of the IEEE Conference on Decision and Control. 505-509.
15. CULLUM, J. AND WILLOUGHBY, R. A. 1985. Lanczos Algorithms for Large Symmetric Eigenvalue Computations. Progress in Scientific Computing, vol.2: Programs. Birkhauser, Boston, MA.
16. DANIEL, J. W., GRAGG, W. B., KAUFMAN, L., AND STEWART, G. W. 1976. Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization. Math. Comp. 30, 136 (Oct.), 772-795.
17. DE STURLER, E. AND DER VORST, H. V. Oct 1995. Reducing the effect of global communication in GMRES(m) and CG on parallel distributed memory computers. Appl. Numer. Math. 18, 4, 441-459
18. DYAKONOV, E. G. 1983. Iteration methods in eigenvalue problems. Math. Notes 34, 945-953.
19. EDELMAN, A., ARIAS, T. A., AND SMITH, S. T. 1998. The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20, 2, 303-353.
20. FISCHER, C. F. 1977. The Hartree-FockMethod for Atoms: A Numerical Approach. J.Wiley&Sons, New York, NY.
21. FOKKEMA, D. R., SLEIJPEN, G. L. G., AND VAN DER VORST, H. A. 1998. Jacobi-Davidson style QR and QZ algorithms for the reduction of matrix pencils. SIAM J. Sci. Comput. 20, 1, 94-125.
22. FOLEY, J., JUGE, K. J., OCAIS, A., PEARDON, M., RYAN, S., AND SKULLERUD, J.-I. 2005. Practical allto- all propagators for lattice QCD. Comput. Phys. Comm. 172, 145-162.
23. FREUND, R. W. AND NACHTIGAL, N. M. 1994. A new Krylov-subspace method for symmetric indefinite linear systems. Tech. rep. AT&T Bell Laboratories, Murray Hill, NJ.
24. GEUS, R. 2002. The Jacobi-Davidson algorithm for solving large sparse symmetric eigenvalue problems with application to the design of accelerator cavities. Ph.D. dissertation. No.14734. ETH, Zurich, Switzerland.
25. GILL, P. H., MURRAY, W., AND WRIGHT, M. H. 1986. Practical Optimization. Academic Press, New York, NY.
26. GOLUB, G. H. AND UNDERWOOD, R. 1977. The block Lanczos method for computing eigenvalues. In Mathematical Software III, J. R. Rice, Ed. Academic Press, New York, NY, 361-377.
27. GOLUB, G. H. AND YE, Q. 2002. An inverse free preconditioned Krylov subspace methods for symmetric generalized eigenvalue problems. SIAM J. Sci. Comput. 24, 312-334.
28. GRIMES, R. G., LEWIS, J. G., AND SIMON, H. D. 1994. A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems. SIAM J. Matrix Anal. Appl. 15, 1, 228-272.
29. GUTKNECHT, M. H. 2005. Block Krylov space solvers: A survey. http://www.sam.math.ethz.ch/~mhg/talks/bkss.pdf.
30. HERNANDEZ, V., ROMAN, J. E., AND TOMAS, A. 2007. Parallel arnoldi eigensolvers with enhanced scalability via global communications rearrangement. Parall. Comput. 33, 521-540.
31. HERNANDEZ, V., ROMAN, J. E., TOMAS, A., AND VIDAL, V. October, 2006. A survey of software for sparse eigenvalue problems. Tech. rep. SLEPc STR-6. Unversidad Politecnica de Valencia, Valencia, Spain.
32. HERNANDEZ, V., ROMAN, J. E., AND VIDAL, V. 2005. SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw. 31, 3, 351-362.
33. HETMANIUK, U. AND LEHOUCQ, R. B. 2006. Basis selection in LOBPCG. J. Comp. Phys. 218, 324-332.
34. JIA, Z. 1998. A refined iterative algorithm based on the block Arnoldi process for large unsymmetric eigenproblems. Lin. Alg. Appl. 270, 171-189. (Pubitemid 128388303)
35. JOHNSON, S. G. AND JOANNOPOULOS, J. D. 2001. Block-iterative frequency-domain methods for Maxwells equations in a planewave basis. Opt. Express 8, 3, 173-190.
36. KNYAZEV, A. V. 1991. Convergence rate estimates for iterative methods for symmetric eigenvalue problems and its implementation in a subspace. Int. Ser. Numer. Math. 96, 143-154. Eigenwertaufgaben in Natur- und Ingenieurwissenschaften und ihre numerische Behandlung, Oberwolfach, 1990.
37. KNYAZEV, A. V. 1998. Preconditioned eigensolvers - an oxymoron? Electr. Trans. Numer. Anal. 7, 104-123.
38. KNYAZEV, A. V. 2001. Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput. 23, 2, 517-541.
39. LEHOUCQ, R. B., SORENSEN, D. C., AND YANG, C. 1998. ARPACK Users Guide: Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia, PA.
40. LIU, B. 1978. Numerical algorithms in chemistry: Algebraic methods, C. Moler and I. Shavitt, Eds. Tech. rep. LBL-8158. Lawrence Berkeley Laboratory, Berkeley, CA.
41. LUNDSTRÖM, E. AND ELD́EN, L. 2002. Adaptive eigenvalue computations using Newtons method on the Grassmann manifold. SIAM J. Matrix Anal. Appl. 23, 3, 819-839.
42. MCCOMBS, J. R. AND STATHOPOULOS, A. 2006. Iterative validation of eigensolvers: A scheme for improving the reliability of Hermitian eigenvalue solvers. SIAM J. Sci. Comput. 28, 6, 2337-2358.
43. MORGAN, R. B. 1991. Computing interior eigenvalues of large matrices. Lin. Alg. Appl. 154-156, 289-309.
44. MURRAY, C.W., RACINE, S. C., AND DAVIDSON, E. R. 1992. Improved algorithms for the lowest eigenvalues and associated eigenvectors of large matrices. J. Comput. Phys. 103, 2, 382-389.
45. NOTAY, Y. 2002. Combination of Jacobi-Davidson and conjugate gradients for the partial symmetric eigenproblem. Numer. Lin. Alg. Appl. 9, 21-44.
46. NOTAY, Y. 2005. Is Jacobi-Davidson faster than Davidson? SIAM J. Matrix Anal. Appl. 26, 2, 522-543.
47. OLSEN, J., JÖRGENSEN, P., AND SIMONS, J. 1990. Passing the one-billion limit in full configurationinteraction (FCI) calculations. Chem. Phys. Lett. 169, 6, 463-472.
48. PAIGE, C. C., PARLETT, B. N., AND VAN DER VORST. 1995. Approximate solutions and eigenvalue bounds from Krylov spaces. Num. Lin. Alg. Appl. 2, 115-133.
49. PARLETT, B. N. 1998. The Symmetric Eigenvalue Problem. SIAM, Philadelphia, PA.
50. SAMEH, A. AND TONG, Z. 2000. The trace minimization method for the symmetric generalized eigenvalue problem. J. Comput. Appl. Math. 123, 155-175.
51. SIMONCINI, V. AND ELD́EN, L. 2002. Inexact Rayleigh quotient-type methods for eigenvalue computations. BIT 42, 1, 159-182.
52. SLEIJPEN, G. L. G. AND VAN DER VORST, H. A. 1996. A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17, 2, 401-425.
53. SORENSEN,D. C. 1992. Implicit application of polynomial filters in a K-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13, 1, 357-385.
54. STATHOPOULOS, A. 2005. Locking issues for finding a large number of eigenvectors of Hermitian matrices. Tech. rep. WM-CS-2005-2009
55. STATHOPOULOS, A. 2007. Nearly optimal preconditioned methods for Hermitian eigenproblems under limited memory. Part I: Seeking one eigenvalue. SIAM J. Sci. Comput. 29, 2, 481-514.
56. STATHOPOULOS, A. AND FISCHER, C. F. 1993. Reducing synchronization on the parallel Davidson method for the large,sparse, eigenvalue problem. In Proceedings of the Supercomputing93. IEEE Computer Society Press, Los Alamitos, CA, 172-180.
57. STATHOPOULOS, A. AND FISCHER, C. F. 1994. A Davidson program for finding a few selected extreme eigenpairs of a large, sparse, real, symmetric matrix. Comput. Phys. Comm. 79, 2, 268-290.
58. STATHOPOULOS, A. AND MCCOMBS, J. R. 2006. PRIMME: PReconditioned Iterative Multimethod Eigensolver. http://www.cs.wm.edu/?andreas/software/.
59. STATHOPOULOS, A. AND MCCOMBS, J. R. 2007. Nearly optimal preconditioned methods for Hermitian eigenproblems under limited memory. Part II: Seeking many eigenvalues. SIAM J. Sci. Comput. 29, 5, 2162-2188.
60. STATHOPOULOS, A. AND ORGINOS, K. 2008. Computing and deflating eigenvalues while solving multiple right hand side linear systems with an application to quantum chromodynamics. Tech. rep. arXiv.org 0707.0131v2. June 12 (Original version July 1, 2007). http://arxiv.org/pdf/0707.0131v2. SIAM J. Sci. Comput. To appear.
61. STATHOPOULOS, A. AND SAAD, Y. 1998. Restarting techniques for (Jacobi-)Davidson symmetric eigenvalue methods. Electr. Trans. Numer. Anal. 7, 163-181. (Pubitemid 39017175)
62. STATHOPOULOS, A., SAAD, Y., AND FISCHER, C. F. 1995. Robust preconditioning of large, sparse, symmetric eigenvalue problems. J. Comput. Appl. Math. 64, 197-215.
63. STATHOPOULOS, A., SAAD, Y., AND WU, K. 1998. Dynamic thick restarting of the Davidson, and the implicitly restarted Arnoldi methods. SIAM J. Sci. Comput. 19, 1, 227-245.
64. YANG, C., PEYTON, B. W., NOID, D. W., SUMPTER, B. G., AND TUZUN, R. E. 2001. Large-scale normal coordinate analysis for molecular structures. SIAM J. Sci. Comput. 23, 2, 563-582.