On the convergence and optimization of the Baker-Campbell-Hausdorff formula

Linear Algebra and Its Applications

Volumn 378, Issue 1-3, 2004, Pages 135-158

  Blanes, Sergio ^a   Casas, Fernando ^a  

^a UNIVERSITAT JAUME I   (Spain)

Author keywords

BCH formula; Convergence; Lie algebras; Lie groups

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; DIFFERENTIAL EQUATIONS; MATHEMATICAL TRANSFORMATIONS; NUMERICAL ANALYSIS; OPTIMIZATION; PERTURBATION TECHNIQUES; POLYNOMIAL APPROXIMATION; THEOREM PROVING;

LIE ALGEBRAS; TANGENT ALGEBRA;

LINEAR ALGEBRA;

EID: 0347476982     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2003.09.010     Document Type: Article

References (29)

1. Blanes S., Casas F., Oteo J.A., Ros J. Magnus and Fer expansions for matrix differential equations: the convergence problem. J. Phys. A: Math. Gen. 31:1998;259-268.
2. Blanes S., Casas F., Ros J. High order optimized geometric integrators for linear differential equations. BIT. 42:2002;262-284.
3. Bourbaki N. Lie Groups and Lie Algebras. 1989;Springer, Berlin. (Chapters 1-3).
4. Bialynicki-Birula I., Mielnik B., Plebański J. Explicit solution of the continuous Baker-Campbell-Hausdorff problem and a new expression for the phase operator. Ann. Physics. 51:1969;187-200.
5. F. Casas, B. Owren, Cost efficient Lie group integrators in the RKMK class, preprint NTNU no. 5/2002, 2002. Accepted for publication in BIT.
6. Day J., So W., Thompson R.C. Some properties of the Campbell Baker Hausdorff series. Linear and Multilinear Algebra. 29:1991;207-224.
7. Dynkin E.B. Calculation of the coefficients in the Campbell-Hausdorff formula. Dokl. Akad. Nauk SSSR. 57:1947;323-326. English traslation in: Yushkevich A.A., Seitz G.M., Onishchik A.L. Selected Papers of E.B. Dynkin with Commentary. 2000;AMS, International Press.
8. Dynkin E.B. Calculation of the coefficients in the Campbell-Hausdorff formula. Dokl. Akad. Nauk SSSR. 57:1947;323-326. English traslation in: Yushkevich A.A., Seitz G.M., Onishchik A.L. Selected Papers of E.B. Dynkin with Commentary. 2000;AMS, International Press.
9. y) Duke Math. J. 23:1956;13-21.
10. Gorbatsevich V.V., Onishchik A.L., Vinberg E.B. Foundations of Lie Theory and Lie Transformation Groups. 1997;Springer, Berlin.
11. Hairer E., Lubich C., Wanner G. Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. 2002;Springer, Berlin.
12. Iserles A., Munthe-Kaas H.Z., Nørsett S.P., Zanna A. Lie-group methods. Acta Numer. 9:2000;215-365.
13. Kolsrud M. Maximal reductions in the Baker-Hausdorff formula. J. Math. Phys. 34:1993;270-285.
14. Kumar K. On expanding the exponential. J. Math. Phys. 6:1965;1928-1934.
15. Magnus W. On the exponential solution of differential equations for a linear operator. Comm. Pure Appl. Math. 7:1954;649-673.
16. B.S. Mityagin, unpublished notes, 1990.
17. Moan P.C., Oteo J.A., Ros J. On the existence of the exponential solution of linear differential systems. J. Phys. A: Math. Gen. 32:1999;5133-5139.
18. P.C. Moan, On backward error analysis and Nekhoroshev stability in the numerical analysis of conservative systems of ODEs, Ph.D. Thesis, University of Cambridge, 2002.
19. Moan P.C., Oteo J.A. Convergence of the exponential Lie series. J. Math. Phys. 42:2001;501-508.
20. Munthe-Kaas H., Owren B. Computations in a free Lie algebra. Philos. Trans. Roy. Soc. A. 357:1999;957-982.
21. Newman M., So W., Thompson R.C. Convergence domains for the Campbell-Baker-Hausdorff formula. Linear and Multilinear Algebra. 24:1989;301-310.
22. Oteo J.A. The Baker-Campbell-Hausdorff formula and nested commutator identities. J. Math. Phys. 32:1991;419-424.
23. Postnikov M. Lie Groups and Lie Algebras. Semester V of Lectures in Geometry. 1994;URSS Publishers, Moscow.
24. W. So, Exponential formulas and spectral indices, Ph.D. Thesis, University of California at Santa Barbara, 1991.
25. Thompson R.C. Convergence proof for Goldberg's exponential series. Linear Algebra Appl. 121:1989;3-7.
26. Varadarajan V.S. Lie Groups, Lie Algebras, and Their Representations. 1984;Springer, New York.
27. Vinokurov V.A. The logarithm of the solution of a linear differential equation, Hausdorff's formula, and conservation laws. Soviet Math. Dokl. 44:1992;200-205.
28. Wei J. Note on global validity of the Baker-Hausdorff and Magnus theorems. J. Math. Phys. 4:1963;1337-1341.
29. Wilcox R.M. Exponential operators and parameter differentiation in quantum physics. J. Math. Phys. 8:1967;962-982.