Error estimates for polynomial krylov approximations to matrix functions

SIAM Journal on Matrix Analysis and Applications

Volumn 30, Issue 4, 2008, Pages 1546-1565

  Diele, Fasma ^a   Moret, Igor ^b   Ragni, Stefania ^c  

^a CNR   (Italy)

^b UNIVERSITY OF TRIESTE   (Italy)

^c UNIVERSITY OF BARI   (Italy)

Author keywords

Error estimates; Exponential like functions; Krylov methods; Matrix functions

Indexed keywords

APPROXIMATION PROCEDURE; DIFFERENTIAL PROBLEMS; ERROR ESTIMATES; EXPONENTIAL-LIKE FUNCTIONS; KRYLOV METHODS; MATRIX FUNCTIONS; NUMERICAL EXAMPLE; POSTERIORI ERROR ESTIMATES; SQUARE MATRICES;

FUNCTION EVALUATION;

ERROR ANALYSIS;

EID: 70449371467     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/070688924     Document Type: Article

References (34)

1. M. Abramovitz and A. Stegun, Handbook of Mathematical Functions, Dover Publications Inc., New York, 1965.
2. H. Berland, B. Skaflestad, and W. Wright, Expint - A Matlab package for exponential ntegrators, ACM Trans. Math. Software, 33 (2007).
3. D. A. Bini, N. J. Higham, and B. Meini, Algorithms for the matrix pth root, Numer. Algorithms, 9 (2005), pp. 349-378.
4. M. Caliari, M. Vianello, and L. Bergamaschi, Integrating discrete advection-diffusion propagators at spectral Leja sequence, J. Comput. Appl. Math., 172 (2004), pp. 79-99.
5. E. Celledoni and I. Moret, A Krylov projection method for systems of ODEs, Appl. Numer. Math., 24 (1997), pp. 365-378.
6. P. I. Davies and N. Higham, Computing f(A)v for matrix functions f, in QCD and Numerical Analysis III, A. Borici, A. Frommer, B. Joo, A. Kennedy, and B. Pendleton, eds., Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin, 47 (2005) pp. 15-24.
7. V. Druskin and L. Knizhnerman, Two polynomial methods for calculating functions of symmetric matrices, Comput. Math. Math. Phys., 29 (1989), pp. 112-121.
8. M. Eiermann and O. G. Ernst, A restarted Krylov subspace method for the evaluation of matrix functions, SIAM J. Numer. Anal., 44 (2006), pp. 2481-2504.
9. K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000.
10. J. v. d. Eshof and M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential, SIAM J. Sci. Comput., (27) 2005, pp. 1438-1457.
11. H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North Holland, Amsterdam, 1995.
12. 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.
13. W. Gautschi, Numerical Analysis, an Introduction, Birkḧuser, Boston, 1997.
14. I. Gohberg and S. Goldberg, Basic Operator Theory, Birkḧuser, Boston, 1980.
15. E. Hairer, Ch. Lubich, and G. Wanner, Geometric Numerical Integration, Springer, Berlin, 2006.
16. M. Hochbruck and C. Lubich, Exponential integrators for quantum-classical molecular dynamics, BIT, 39 (1999), pp. 620-645.
17. M. Hochbruck and C. Lubich, On Krylov subspace approximation to the matrix exponential operator, SIAM J. Numer. Anal., 34 (1997), pp. 1911-1925.
18. M. Hochbruck, C. Lubich, and H. Selhofer, Exponential integrators for large systems of differential equations, SIAM J. Sci. Comput., 19 (1998), pp. 1552-1574.
19. M. Hochbruck and A. Ostermann, Exponential Runge-Kutta methods for parabolic problems, Appl. Numer. Math., 53 (2005), pp. 323-339.
20. M. Hochbruck and A. Ostermann, Explicit exponential Runge-Kutta methods for semilinear parabolic problems, SIAM J. Numer. Anal., 43 (2005), pp. 1069-1090.
21. A.-K. Kassam and L. N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAMJ. Sci. Comput., 26 (2005), pp. 1214-1233.
22. L. Knizhnermann, Calculation of functions of unsymmetric matrices using Arnoldi's method, U.S.S.R. Comput. Maths. Math. Phys., 31 (1991), pp. 1-9, (English Edition by Pergamon Press).
23. 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.
24. G. Meurant and Z. Strakos, The Lancsos and conjugate gradients algorithms in finite precision arithmetic, Acta Numer., 15 (2006), pp. 471-542.
25. B. V. Minchev and W. M. Wright, A review of exponential integrators for first order semilinear problems, preprint Numerics 2/2005, Norwegian University of Science and Technology, Trondheim, Norway, www math.ntnu.no/preprint/ numerics/2005/ N2-2005.ps.
26. 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.
27. I. Moret and P. Novati, Interpolating functions of matrices on zeros of quasi-kernel polynomials, Numer. Linear Algebra Appl., 12 (2005), pp. 337-353.
28. I. Moret and P. Novati, RD-rational approximations of the matrix exponential, BIT, 44 (2004), pp. 595-615.
29. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
30. M. Popolizio and V. Simoncini, Acceleration techniques for approximating the matrix exponential operator, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 657-683.
31. Y. Saad, Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 29 (1992), pp. 209-228.
32. B. A. Schmitt and R. Weiner, Matrix-free W-methods using a multiple Arnoldi iteration, Appl. Numer. Math., 18 (1995), pp. 307-320.
33. V. Simoncini, On the convergence of restarted Krylov subspace methods, SIAMJ.MatrixAnal. Appl., 22 (2000), pp. 430-452.
34. L. N. Trefethen and D. Bau, Numerical Linear Algebra, SIAM, Philadelphia, 1997.