Acceleration techniques for approximating the matrix exponential operator

SIAM Journal on Matrix Analysis and Applications

Volumn 30, Issue 2, 2008, Pages 657-683

  Popolizio, M ^a   Simoncini, V ^b,c  

^a UNIVERSITY OF BARI   (Italy)

^b UNIVERSITY OF BOLOGNA   (Italy)

^c CIRSA   (Italy)

Author keywords

Iterative methods; Krylov Subspace; Large matrices; Matrix exponential; Rational functions

Indexed keywords

ACCELERATION PARAMETERS; ACCELERATION TECHNIQUE; APRIORI; KRYLOV SUBSPACE; LARGE MATRICES; MATRIX EXPONENTIAL; NEGATIVE SEMI-DEFINITE; NEW RESULTS; NUMERICAL APPROACHES; NUMERICAL EXPERIMENTS; NUMERICAL RESULTS; PARTIAL FRACTION EXPANSION; PERFORMANCE EVALUATION; PRECONDITIONING TECHNIQUES; RATIONAL FUNCTION; RATIONAL FUNCTIONS; SPECTRAL TRANSFORMATIONS; SUBSPACE PROJECTION;

FUNCTION EVALUATION; NUMERICAL METHODS;

ITERATIVE METHODS;

EID: 49749108595     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/060672856     Document Type: Article

References (43)

1. O. AXELSSON AND A. KUCHEROV, Real valued iterative methods for solving complex symmetric linear systems, Nutner. Linear Algebra Appl., 7 (2000), pp. 197-218.
2. Z. BAI, J. DEMMEL, J. DONGARRA, A. RUHE, and H. A. VAN DER VORST. EDS., Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, Philadelphia, 2000.
3. G. A. BAKER and P. GRAVES-MORRIS, Padé Approximants, Encyclopedia Math. Appl. 59. 2nd ed., Cambridge University Press, Cambridge, UK, 1996.
4. C. BALDWIN, R. FREUND, AND E. GALLOPOULOS, A parallel iterative method for exponential propagation, in Proceedings of the Seventh SIAM Conference on Parallel Processing for Scientific Computing, D. Bailey et al., eds., SIAM, Philadelphia, 1995, pp. 534-539.
5. M. BENZI, Preconditioning techniques for large linear systems: A survey, J. Comput. Phys., 182 (2002), pp. 418-477.
6. M. BENZI AND D. BERTACCINI, Approximate inverse preconditioning for shifted linear systems, BIT, 43 (2003), pp. 231-244.
7. M. BENZI and D. BERTACCINI, Block Preconditioning of Real-valued Iterative Algorithms for Complex Linear Systems, to appear in IMA J. Numer. Anal.
8. L. BERGAMASCHI and M. VIANELLO, Efficient computation of the exponential operator for large, sparse, symmetric matrices, Numer. Linear Algebra Appl., 7 (2000), pp. 27-45.
9. A. J. CARPENTER, A. RUTTAN, and R. S. VARGA, Extended numerical computations on the 1/9 conjecture in rational approximation theory, in Rational Approximation and Interpolation, Lecture Notes in Math. 1105, P. R. Graves-Morris, E. B. Saff, and R. S. Varga, eds., Springer-Verlag, Berlin, 1984, pp. 383-411.
10. P. CASTILLO and Y. SAAD, Preconditioning the matrix exponential operator with applications, J. Sci. Comput., 13 (1999), pp. 275-302.
11. W. J. CODY, G. MEINARDUS, and R. S. VARGA, Chebyshev rational approximations to e~x in [0,+∞) and applications to heat-conduction problems, J. Approx. Theory, 2 (1969), pp. 50-65.
12. V. DRUSKIN, A. GREENBAUM, and L. KNIZHNERMAN, Using nonorthogonal Lanczos vectors in the computation of matrix functions, SIAM J. Sci. Comput., 19 (1998), pp. 38-54.
13. V. DRUSKIN and L. KNIZHNERMAN, Two polynomial methods of calculating functions of symmetric matrices, USSR Comput. Math. Math. Phys., 29 (1989), pp. 112-121.
14. V. DRUSKIN and L. KNIZHNERMAN, Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmetic, Numer. Linear Algebra Appl., 2 (1995), pp. 205-217.
15. V. DRUSKIN AND L. KNIZHNERMAN, Extended Krylov subspaces: Approximation of the matrix square root and related functions, SIAM J. Matrix Anal. Appl., 19 (1998), pp. 755-771.
16. M. EIERMANN AND O. ERNST, A restarted Krylov subspace method for the evaluation of matrix functions, SIAM J. Numer. Anal., 44 (2006), pp. 2481-2504.
17. R. W. FREUND, On conjugate gradient type methods and polynomial preconditioned for a class of complex non-Hermitian matrices, Numer. Math., 57 (1990), pp. 285-312.
18. R. W. FREUND and N. M. NACHTIGAL, Software for simplified Lanczos and QMR algorithms, Appl. Numer. Math., 19 (1995), pp. 319-341.
19. A. FROMMER and V. SIMONCINI, Stopping criteria for rational matrix functions of Hermitian and symmetric matrices, SIAM J. Sci. Comput., 30 (2008), pp. 1387-1412
20. E. GALLOPOULOS and Y. SAAD, Efficient solution of parabolic equations by Krylov approximation methods, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 1236-1264.
21. G. GOLUB and C. F. VAN LOAN, Matrix Computations, 3rd ed., The Johns Hopkins University Press, Baltimore, 1996.
22. N. J. HIGHAM, The scaling and squaring method for the matrix exponential revisited, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 1179-1193.
23. M. HOCHBRUCK AND M. E. HOCHSTENBACH, Subspace Extraction for Matrix Functions, preprint, 2005.
24. M. HOCHBRUCK and C. LUBICH, On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 34 (1997), pp. 1911-1925.
25. M. HOCHBRUCK and C. LUBICH, Exponential integrators for quantum-classical molecular dynamics, BIT, 39 (1999), pp. 620-645.
26. A. ISERLES, How large is the exponential of a banded matrix?, New Zealand J. Math., 29 (2000), pp. 177-192.
27. L. LOPEZ and V. SIMONCINI, Analysis of projection methods for rational function approximation to the matrix exponential, SIAM J. Numer. Anal., 44 (2006), pp. 613-635.
28. THE MATHWORKS, INC., MATLAB 7, 2004.
29. C. MOLER and C. VAN LOAN, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev., 45 (2003), pp. 3-49.
30. I. MORET, On RD-rational Krylov approximations to the core-functions of exponential integrators, Numer. Linear Algebra Appl., 14 (2007), pp. 445-457.
31. I. MORET and P. NOVATI, An interpolatory approximation of the matrix exponential based on Faber polynomials, J. Comput. Appl. Math., 131 (2001), pp. 361-380.
32. I. MORET and P. NOVATI, RD-rational approximations of the matrix exponential, BIT, 44 (2004), pp. 595-615.
33. I. MORET and P. NOVATI, Interpolating functions of matrices on zeros of quasi-kernel polynomials, Numer. Linear Algebra Appl., 11 (2005), pp. 337-353.
34. Y. SAAD, Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 29 (1992), pp. 209-228.
35. Y. SAAD, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, Philadelphia, 2003.
36. -z by rational functions with real poles, Numer. Math., 25 (1976), pp. 307-322.
37. R. B. SIDJE, Expokit: A Software Package for Computing Matrix Exponentials, ACM Trans. Math. Softw., 24 (1998), pp. 130-156.
38. V. SIMONCINI and D. B. SZYLD, Theory of inexact Krylov subspace methods and applications to scientific computing, SIAM J. Sci. Comput., 25 (2003), pp. 454-477.
39. V. SIMONCINI and D. B. SZYLD, Recent computational developments in Krylov subspace methods for linear systems, Numer. Linear Algebra Appl., 14 (2007), pp. 1-59.
40. L. N. TREFETHEN, J. A. C. WEIDEMAN, and T. SCHMELZER, Talbot quadratures and rational approximations, BIT, 46 (2006), pp. 653-670.
41. J. VAN DEN ESHOF and M. HOCHBRUCK, Preconditioning Lanczos approximations to the matrix exponential, SIAM J. Sci. Comput., 27 (2006), pp. 1438-1457.
42. J. VAN DEN ESHOF and G. L. G. SLEIJPEN, Inexact Krylov subspace methods for linear systems, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 125-153.
43. H. A. VAN DER VORST, An iterative solution method for solving f(A)x=b using Krylov subspace information obtained for the symmetric positive definite matrix A, J. Comput. Appl. Math., 18 (1987), pp. 249-263.