An efficient algorithm for computing the Baker-Campbell-Hausdorff series and some of its applications. (English) Zbl 1202.17004
Summary: We provide a new algorithm for generating the Baker–Campbell–Hausdorff (BCH) series \(Z = \log(e^X e^Y)\) in an arbitrary generalized Hall basis of the free Lie algebra \(\mathcal{L}(X,Y)\) generated by \(X\) and \(Y\). It is based on the close relationship of \(\mathcal{L}(X,Y)\) with a Lie algebraic structure of labeled rooted trees. With this algorithm, the computation of the BCH series up to degree 20 (111013 independent elements in \(\mathcal{L}(X,Y)\)) takes less than 15 minutes on a personal computer and requires 1.5 GBytes of memory. We also address the issue of the convergence of the series, providing an optimal convergence domain when \(X\) and \(Y\) are real or complex matrices.
Editorial remark: No review copy delivered
Editorial remark: No review copy delivered
MSC:
| 17B01 | Identities, free Lie (super)algebras |
| 22E60 | Lie algebras of Lie groups |
| 17-08 | Computational methods for problems pertaining to nonassociative rings and algebras |
References:
| [1] | Abramowitz M., Handbook of Mathematical Functions (1965) |
| [2] | DOI: 10.1112/plms/s2-3.1.24 · JFM 36.0225.01 · doi:10.1112/plms/s2-3.1.24 |
| [3] | DOI: 10.1016/j.laa.2003.09.010 · Zbl 1054.17005 · doi:10.1016/j.laa.2003.09.010 |
| [4] | DOI: 10.1088/0305-4470/31/1/023 · Zbl 0946.34014 · doi:10.1088/0305-4470/31/1/023 |
| [5] | DOI: 10.1016/j.physrep.2008.11.001 · doi:10.1016/j.physrep.2008.11.001 |
| [6] | DOI: 10.1063/1.528242 · Zbl 0676.17004 · doi:10.1063/1.528242 |
| [7] | Bourbaki N., Lie Groups and Lie Algebras (1989) · Zbl 0672.22001 |
| [8] | Campbell J. E., Proc. London Math. Soc. 29 pp 14– (1898) |
| [9] | DOI: 10.1088/1751-8113/40/50/006 · Zbl 1131.34008 · doi:10.1088/1751-8113/40/50/006 |
| [10] | DOI: 10.1063/1.522868 · Zbl 0343.70011 · doi:10.1063/1.522868 |
| [11] | DOI: 10.1007/BFb0077472 · Zbl 0592.05006 · doi:10.1007/BFb0077472 |
| [12] | Dynkin E. B., Dokl. Akad. Nauk SSSR 57 pp 323– (1947) |
| [13] | DOI: 10.1215/S0012-7094-56-02302-X · Zbl 0070.25203 · doi:10.1215/S0012-7094-56-02302-X |
| [14] | Gorbatsevich V. V., Foundations of Lie Theory and Lie Transformation Groups (1997) · Zbl 0999.17500 |
| [15] | DOI: 10.1016/0021-8693(89)90328-1 · Zbl 0717.16029 · doi:10.1016/0021-8693(89)90328-1 |
| [16] | Hairer E., Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2. ed. (2006) · Zbl 1094.65125 |
| [17] | Hausdorff F., Ber. Verh. Saechs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 58 pp 19– (1906) |
| [18] | DOI: 10.1090/S0002-9947-03-03317-8 · Zbl 1048.16023 · doi:10.1090/S0002-9947-03-03317-8 |
| [19] | DOI: 10.1017/S0962492900002154 · doi:10.1017/S0962492900002154 |
| [20] | DOI: 10.1098/rsta.1999.0362 · Zbl 0958.65080 · doi:10.1098/rsta.1999.0362 |
| [21] | Jacobson N., Lie Algebras (1979) |
| [22] | DOI: 10.1063/1.530381 · Zbl 0772.17004 · doi:10.1063/1.530381 |
| [23] | Koseleff, P.V. ”Calcul formel pour les méthodes de Lie en mécanique Hamiltonienne,” Ph.D. thesis, École Polytechnique, 1993. |
| [24] | DOI: 10.1063/1.1704742 · Zbl 0137.23904 · doi:10.1063/1.1704742 |
| [25] | Lothaire M., Combinatorics on Words (1983) · Zbl 0514.20045 |
| [26] | DOI: 10.1002/cpa.3160070404 · Zbl 0056.34102 · doi:10.1002/cpa.3160070404 |
| [27] | DOI: 10.1017/S0962492902000053 · Zbl 1105.65341 · doi:10.1017/S0962492902000053 |
| [28] | Michel J., Séminaire Dubreil. Algèbre 27 pp 1– (1974) |
| [29] | Moan, P. C. ”On backward error analysis and Nekhoroshev stability in the numerical analysis of conservative systems of ODEs,” Ph.D. thesis, University of Cambridge, 2002. |
| [30] | DOI: 10.1007/s10208-003-0111-0 · Zbl 1116.17004 · doi:10.1007/s10208-003-0111-0 |
| [31] | DOI: 10.1080/03081088908817923 · Zbl 0713.22007 · doi:10.1080/03081088908817923 |
| [32] | DOI: 10.2307/2007889 · doi:10.2307/2007889 |
| [33] | DOI: 10.1063/1.529428 · Zbl 0725.47052 · doi:10.1063/1.529428 |
| [34] | Postnikov M., Lie Groups and Lie Algebras. Semester V of Lectures in Geometry (1994) |
| [35] | DOI: 10.1063/1.533250 · Zbl 0974.22015 · doi:10.1063/1.533250 |
| [36] | Reutenauer C., Free Lie Algebras (1993) |
| [37] | DOI: 10.1002/cpa.3160180111 · Zbl 0156.26303 · doi:10.1002/cpa.3160180111 |
| [38] | Rouvière F., Ann. Sci. Ec. Normale Super. 19 pp 553– (1986) |
| [39] | DOI: 10.1007/978-1-4899-3093-4 · doi:10.1007/978-1-4899-3093-4 |
| [40] | DOI: 10.1103/PhysRevA.60.1956 · doi:10.1103/PhysRevA.60.1956 |
| [41] | DOI: 10.1007/BF01614161 · Zbl 0366.47016 · doi:10.1007/BF01614161 |
| [42] | DOI: 10.2307/2044385 · Zbl 0497.17002 · doi:10.2307/2044385 |
| [43] | DOI: 10.1080/03081088608817715 · Zbl 0596.15025 · doi:10.1080/03081088608817715 |
| [44] | DOI: 10.1016/0024-3795(88)90251-0 · Zbl 0655.15024 · doi:10.1016/0024-3795(88)90251-0 |
| [45] | DOI: 10.1016/0024-3795(89)90688-5 · Zbl 0678.22003 · doi:10.1016/0024-3795(89)90688-5 |
| [46] | DOI: 10.1137/S0036144502410427 · Zbl 1084.17002 · doi:10.1137/S0036144502410427 |
| [47] | DOI: 10.1007/978-1-4612-1126-6 · doi:10.1007/978-1-4612-1126-6 |
| [48] | DOI: 10.1007/BFb0067950 · doi:10.1007/BFb0067950 |
| [49] | DOI: 10.1063/1.1724280 · Zbl 0108.44804 · doi:10.1063/1.1724280 |
| [50] | DOI: 10.1063/1.1705306 · Zbl 0173.29604 · doi:10.1063/1.1705306 |
| [51] | Yakubovich V. A., Linear Differential Equations with Periodic Coefficients (1975) · Zbl 0308.34001 |
| [52] | DOI: 10.1016/0375-9601(90)90092-3 · doi:10.1016/0375-9601(90)90092-3 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
