Anderson model of localization: a challenge for modern eigenvalue methods

SIAM Journal of Scientific Computing

Volumn 20, Issue 6, 1999, Pages 2089-2102

  Elsner, Ulrich ^a   Mehrmann, Volker ^a   Milde, Frank ^a   Römer, Rudolf A ^a   Schreiber, Michael ^a  

^a CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; BOUNDARY CONDITIONS; CONVERGENCE OF NUMERICAL METHODS; MATHEMATICAL MODELS; MATRIX ALGEBRA; POLYNOMIALS; QUANTUM THEORY;

ANDERSON MODEL OF LOCALIZATION; ARNOLDI ALGORITHM; LANCZOS ALGORITHM; SHIFT AND INVERT CONVERGENCE ACCELERATORS; TRIDIAGONALIZATION METHOD;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 0033293107     PISSN: 10648275     EISSN: None     Source Type: None    
DOI: 10.1137/S1064827598332217     Document Type: Article

References (36)

1. P. W. ANDERSON, Absence of diffusion in certain random lattices, Phys. Rev., 109 (1958), pp. 1492-1505.
2. R. BARRETT, M. BERRY, T. F. CHAN, J. DEMMEL, J. DONATO, J. DONGARRA, V. EIJKHOUT, R. POZO, C. ROMINE, AND H. VAN DER VORST, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1994.
3. J. G. L. BOOTEN, H. A. VAN DER VORST, P. M. MEIJER, AND H. J. J. TE RIELE, A Preconditioned Jacobi-Davidson Method for Solving Large Generalized Eigenvalue Problems, Tech. Report CWI-NM-R9414, Centrum voor Wiskunde en Informatica, Amsterdam, July 1994.
4. I. A. CAVERS, A hybrid tridiagonalization algorithm for symmetric sparse matrices, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 1363-1380.
5. 2 via basis Lanczos algorithm, Chem. Phys. Lett., 277 (1997), pp. 191-198.
6. J. CULLUM AND R. A. WILLOUGHBY, Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Volume 1: Theory, Birkhäuser, Boston, 1985.
7. J. CULLUM AND R. A. WILLOUGHBY, Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Volume 2: Programs, Birkhäuser, Boston, 1985. Available online at http://www.netlib.org/lanczos
8. J. DEMMEL, J. GILBERT, AND X. LI, SuperLU Users' Guide, 1997. Available online at http://www.netlib.org/scalapack/prototype/superlu_ug.ps
9. A. EILMES, R. A. RÖMER, AND M. SCHREIBER, The Anderson model of localization with random hopping, European Phys. J. B, 1 (1998), pp. 29-38.
10. R. FLETCHER, Conjugate gradient methods for indefinite systems, in Proc. Dundee Biennial Conf. Numerical Analysis 1974, G. Watson, ed., Springer-Verlag, Heidelberg, 1975, pp. 73-89.
11. R. FREUND AND N. NACHTIGAL, QMRPACK. Available online at http://www.netlib.org/linalg
12. R. FREUND AND N. NACHTIGAL, QMR: A quasi-minimal residual method for non-hermitian linear systems, Numer. Math., 60 (1991), pp. 315-339.
13. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, 1996.
14. H. GRUSSBACH AND M. SCHREIBER, Determination of the mobility edge in the Anderson model of localization in three dimensions by multifractal analysis, Phys. Rev. B, 57 (1995), pp. 663-666.
15. W. HACKBUSCH, Iterative Solution of Large Sparse Systems of Equations, Appl. Math. Sci. 95, Springer-Verlag, Berlin, 1994.
16. Harwell subroutine library. Available online at http://www.dci.clrc.ac.uk/Activity.asp?HSL
17. B. HENDRICKSON AND R. LELAND, The Chaco User's Guide Version 2, Sandia National Laboratories, Albuquerque, 1995. Available online at ftp://ftp.cs.sandia.gov/pub/papers/ bahendr/guide.ps.Z
18. E. HOFSTETTER AND M. SCHREIBER, Statistical properties of the eigenvalue spectrum of the three-dimensional Anderson hamiltonian, Phys. Rev. B, 48 (1993), pp. 16979-16985.
19. R. B. LEHOUCQ AND J. A. SCOTT, An Evaluation of Software for Computing Eigenvalues of Sparse Nonsymmetric Matrices, Tech. Report MCS-P547-1195, Argonne National Library, Argonne, IL, 1996. ACM Trans. Math. Software, submitted. Available online at ftp://info.mcs.anl.gov/pub/tech_reports/reports/P547.ps.Z
20. R. B. LEHOUCQ, D. SORENSEN, AND C. YANG, ARPACK Users Guide: Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM, Philadelphia, 1998.
21. F. MILDE, R. A. RÖMER, AND M. SCHREIBER, Multifractal analysis of the metal-insulator transition in anisotropic systems, Phys. Rev. B, 55 (1997), pp. 9463-9469.
22. B. N. PARLETT, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, 1980.
23. B. N. PARLETT AND D. S. SCOTT, The Lanczos algorithm with selective orthogonalization, Math. Comp., 33 (1979), pp. 217-238.
24. W. H. PRESS, B. P. FLANNERY, S. A. TEUKOLSKY, AND W. T. VETTERLING, Numerical Recipes in FORTRAN, 2nd ed., Cambridge University Press, Cambridge, 1992.
25. Y. SAAD, Numerical Methods for Large Eigenvalue Problems: Theory and Algorithms, John Wiley, New York, 1992.
26. Y. SAAD AND M. SCHULTZ, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856-869.
27. M. SCHREIBER, Fractal eigenstates in disordered systems, Phys. A, 167 (1990), pp. 188-198.
28. M. SCHREIBER, Multifractal characteristics of electronic wave functions in disordered systems, in Computational Physics, K. H. Hoffmann and M. Schreiber, eds., Springer-Verlag, Berlin, 1996, pp. 147-165.
29. M. SCHREIBER AND H. GRUSSBACH, Multifractal wave functions at the Anderson transition, Phys. Rev. Lett., 67 (1991), pp. 607-610.
30. G. SLEIJPEN AND H. VAN DER VORST, A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 401-425.
31. P. SONNEVELD, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 10 (1989), pp. 36-52.
32. D. SORENSEN AND C. SUN, private communication, Dept. of Computer Science, Rice University, Houston, TX 1998.
33. D. E. STEWART AND Z. LEYK, Meschach. Available online at http://www.netlib.org/c/ meschach/
34. H. A. VAN DER VORST, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 13 (1992), pp. 613-644.
35. I. K. ZHAREKESHEV AND B. KRAMER, private communication, Universitäet Hamburg, 1997.
36. I. K. ZHAREKESHEV AND B. KRAMER, Asymptotics of universal probability of neighboring level spacings at the Anderson transition, Phys. Rev. Lett., 79 (1997), pp. 717-720.