The Magnus expansion and some of its applications

Physics Reports

Volumn 470, Issue 5-6, 2009, Pages 151-238

  Blanes, S ^a   Casas, F ^b   Oteo, J A ^c   Ros, J ^c,d  

^a UNIVERSITAT POLITÈCNICA DE VALÈNCIA   (Spain)

^b UNIVERSITAT JAUME I   (Spain)

^c UNIVERSITY OF VALENCIA   (Spain)

^d INSTITUTO DE FÍSICA CORPUSCULAR IFIC   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 58149396539     PISSN: 03701573     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physrep.2008.11.001     Document Type: Review

References (246)

1. In: Abikoff W., Birman J.S., and Kuiken K. (Eds). The Mathematical Legacy of Wilhelm Magnus. Groups, Geometry and Special Functions (1994), American Mathematical Society
2. Magnus W., Karrass A., and Solitar D. Combinatorial Group Theory (2004), Dover
3. Erdelyi A., Magnus W., Oberhettinger F., and Tricomi F.G. Higher Transcendental Functions, Vols. I, II, III (1953), McGraw Hill
4. Magnus W. On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. VII (1954) 649-673
5. Friedrichs K.O. Mathematical aspects of the quantum theory of fields, part V. Commun. Pure Appl. Math. VI (1953) 1-72
6. Feynman R.P. An operator calculus having applications in Quantum Electrodynamics. Phys. Rev. 84 (1951) 108-128
7. Baker H.F. On the integration of linear differential equations. Proc. London Math. Soc. 35 (1902) 334-378
8. Ince E.L. Ordinary Differential Equations (1956), Dover
9. Baker H.F. Alternant and continuous groups. Proc. London Math. Soc. (Second series) 3 (1905) 24-47
10. Campbell J.E. On a law of combination of operators. Proc. London Math. Soc. 29 (1898) 14-32
11. Hausdorff F. Die symbolische Exponential Formel in der Gruppen theorie. Leipziger Ber. 58 (1906) 19-48
12. Dynkin E.B. Evaluation of the coefficients of the Campbell-Hausdorff formula. Dokl. Akad. Nauk. SSSR 57 (1947) 323-326
13. Dyson F.J. The radiation theorem of Tomonaga, Swinger and Feynman. Phys. Rev. 75 (1949) 486-502
14. W. Magnus, Algebraic aspects in the theory of systems of linear differential equations, Technical Report BR-3, New York University, June 1953
15. Pechukas P., and Light J.C. On the exponential form of time-displacement operators in quantum mechanics. J. Chem. Phys. 44 (1966) 3897-3912
16. Wei J. Note on global validity of the Baker-Hausdorff and Magnus theorems. J. Math. Phys. 4 (1963) 1337-1341
17. Wei J., and Norman E. Lie algebraic solution of linear differential equations. J. Math. Phys. 4 (1963) 575-581
18. Robinson D.W. Multiple Coulomb excitations of deformed nuclei. Helv. Phys. Acta 36 (1963) 140-154
19. Wilcox R.M. Exponential operators and parameter differentiation in quantum physics. J. Math. Phys. 8 (1967) 962-982
20. Fer F. Résolution de l'equation matricielle over(U, ̇) = p U par produit infini d'exponentielles matricielles. Bull. Cl. Sci. Acad. R. Belg. 44 (1958) 818-829
21. Chang D., and Light J.C. Exponential solution of the Schrödinger equation: Potential scattering. J. Chem. Phys. 50 (1969) 2517-2525
22. Baye D., and Heenen P.H. A theoretical study of fast proton-atomic hydrogen scattering. J. Phys. B 6 (1973) 105-113
23. Schek I., Jortner J., and Sage M.L. Application of the Magnus expansion for high-order multiphoton excitation. Chem. Phys. 59 (1981) 11-27
24. Ernst R.R., Bodenhausen G., and Wokaun A. Principles of NMR in One and Two Dimensions (1986), Clarendon
25. Waugh J.S. Theory of broadband spin decoupling. J. Magn. Reson. 50 (1982) 30-49
26. Dahmen H.D., Scholz B., and Steiner F. Infrared dynamics of QED and the asymptotic behavior of the electron form factor. Nuclear Phys. B 202 (1982) 365-381
27. D'olivo J.C., and Oteo J.A. Magnus expansion and the two-neutrino oscillations in matter. Phys. Rev. D 42 (1990) 256-259
28. Klarsfeld S., and Oteo J.A. Analytic properties for the Magnus operator for two solvable Hamiltonians. Phys. Lett. A 142 (1989) 393-397
29. Klarsfeld S., and Oteo J.A. Recursive generation of higher-order terms in the Magnus expansion. Phys. Rev. A 39 (1989) 3270-3273
30. Blanes S., Casas F., Oteo J.A., and Ros J. Magnus and Fer expansions for matrix differential equations: The convergence problem. J. Phys. A: Math. Gen. 22 (1998) 259-268
31. Casas F. Sufficient conditions for the convergence of the Magnus expansion. J. Phys. A: Math. Theor. 40 (2007) 15001-15017
32. Moan P.C., Oteo J.A., and Ros J. On the existence of the exponential solution of linear differential systems. J. Phys. A: Math. Gen. 32 (1999) 5133-5139
33. Moan P.C., and Oteo J.A. Convergence of the exponential Lie series. J. Math. Phys. 42 (2001) 501-508
34. Casas F., Oteo J.A., and Ros J. Floquet theory: Exponential perturbative treatment. J. Phys. A: Math. Gen. 34 (2001) 3379-3388
35. Burrage K., and Burrage P.M. High strong order methods for non-commutative-stochastic ordinary differential equation systems and the Magnus formula. Physica D 133 (1999) 34-48
36. ′ = A Y - Y B. J. Comput. Math. 19 (2001) 15-26
37. Iserles A., and Nørsett S.P. On the solution of linear differential equations in Lie groups. Phil. Trans. R. Soc. A 357 (1999) 983-1019
38. Budd C.J., and Iserles A. Geometric integration: Numerical solution of differential equations on manifolds. Phil. Trans. R. Soc. A 357 (1999) 945-956
39. Blanes S., Casas F., and Ros J. Improved high order integrators based on the Magnus expansion. BIT 40 (2000) 434-450
40. Blanes S., Casas F., and Ros J. High order optimized geometric integrators for linear differential equations. BIT 42 (2002) 262-284
41. Postnikov M. Lie Groups and Lie Algebras. Semester V of Lectures in Geometry (1994), URSS Publishers
42. Golub G.H., and Van Loan C.F. Matrix Computations (1996), Johns Hopkins University Press
43. Tyrtyshnikov E.E. A Brief Introduction to Numerical Analysis (1997), Birkhäuser
44. Horn R.A., and Johnson C.R. Matrix Analysis (1985), Cambridge University Press
45. Abramowitz M., and Stegun I.A. Handbook of Mathematical Functions (1965), Dover
46. Rossmann W. Lie Groups. An Introduction Through Linear Groups (2002), Oxford University Press
47. S. Sternberg, Lie algebras, Book online: www.math.harvard.edu/~shlomo/, 2004
48. Schur F. Neue Begründung der Theorie der endlichen Transformationsgruppen. Math. Ann. 35 (1890) 161-197
49. Prato D., and Lamberti P. A note on Magnus formula. J. Chem. Phys. 106 (1997) 4640-4643
50. Bialynicki-Birula I., Mielnik B., and Plebański J. Explicit solution of the continuous Baker-Campbell-Hausdorff problem and a new expression for the phase operator. Ann. Phys. 51 (1969) 187-200
51. Mielnik B., and Plebański J. Combinatorial approach to Baker-Campbell-Hausdorff exponents. Ann. Inst. Henri Poincaré XII (1970) 215-254
52. Sáenz L., and Suárez R. Combinatorial approach to the generalized Baker-Campbell-Hausdorff-Dynkin formula. Systems Control Lett. 45 (2002) 357-370
53. Strichartz R.S. The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations. J. Funct. Anal. 72 (1987) 320-345
54. Suárez R., and Sáenz L. Lie algebras associated with the exponential solutions of nonautonomous linear differential equations. J. Math. Phys. 42 (2001) 4582-4605
55. Salzman W.R. New criterion for convergence of exponential perturbation theory in the Schrödinger representation. Phys. Rev. A 36 (1987) 5074-5076
56. Burum D.P. Magnus expansion generator. Phys. Rev. B 24 (1981) 3684-3692
57. Lam C.S. Decomposition of time-ordered products and path-ordered exponentials. J. Math. Phys. 39 (1998) 5543-5558
58. Oteo J.A., and Ros J. From time-ordered products to Magnus expansion. J. Math. Phys. 41 (2000) 3268-3277
59. Iserles A., Munthe-Kaas H.Z., Nørsett S.P., and Zanna A. Lie-group methods. Acta Numer. 9 (2000) 215-365
60. K. Ebrahimi-Fard, D. Manchon, A Magnus- and Fer-type formula in dendriform algebras. Found. Comput. Math. (2008), doi:10.1007/s10208-008-9023-3
61. Iserles A., Nørsett S.P., and Rasmussen A.F. Time symmetry and high-order Magnus methods. Appl. Numer. Math. 39 (2001) 379-401
62. Munthe-Kaas H., and Owren B. Computations in a free Lie algebra. Phil. Trans. R. Soc. A 357 (1999) 957-981
63. Coddington E.A., and Levinson N. Theory of Ordinary Differential Equations (1955), McGraw Hill
64. Gantmacher F.R. The Theory of Matrices (1959), Chelsea Publishing
65. Dixmier J. L'application exponentielle dans les groupes de Lie résolubles. Bull. Soc. Math. France 85 (1957) 113-121
66. Saito M. Sur certain groupes de Lie résolubles, I; II. Sci. Papers Coll. Gen. Educ. Univ. Tokyo 7 (1957) 1-11
67. 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
68. Karasev M.V., and Mosolova M.V. Infinite products and T products of exponentials. J. Theor. Math. Phys. 28 (1977) 721-730
69. Chacon R.V., and Fomenko A.T. Recursion formulas for the Lie integral. Adv. Math. 88 (1991) 200-257
70. P.C. Moan, Efficient approximation of Sturm-Liouville problems using Lie-group methods, Technical Report 1998/NA11, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England, 1998
71. Moan P.C., and Niesen J. Convergence of the Magnus series. Found. Comput. Math. 8 (2008) 291-301
72. B.S. Mityagin, Unpublished notes, 1990
73. Gronwall T.H. Note on the derivatives with respect to a parameter of the solutions of a system of differential equation. Ann. of Math. 20 (1919) 292-296
74. Hunter J.K., and Nachtergaele B. Applied Analysis (2001), World Scientific
75. Dunford N., and Schwartz J.T. Linear Operators. Part I: General Theory (1958), Wiley-Interscience
76. Yakubovich V.A., and Starzhinskii V.M. Linear Differential Equations with Periodic Coefficients (1975), John Wiley & Sons
77. M.M. Veshtort, Numerical simulations in Nuclear Magnetic Resonance: Theory and applications, Ph.D. Thesis, Massachusetts Institute of Technology, 2003
78. Gorbatsevich V.V., Onishchik A.L., and Vinberg E.B. Foundations of Lie Theory and Lie Transformation Groups (1997), Springer
79. Hairer E., Lubich Ch., and Wanner G. Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. second ed. (2006), Springer-Verlag
80. Kumar K. On expanding the exponential. J. Math. Phys. 6 (1965) 1928-1934
81. Weiss G.H., and Maradudin A.A. The Baker-Hausdorff formula and a problem in Crystal Physics. J. Math. Phys. 3 (1962) 771-777
82. Berry M.V. Quantal phase factors accompanying adiabatic changes. Proc. Roy. Soc. Lond. Ser. A 392 (1984) 45-57
83. Garrido L.M. Generalized adiabatic invariance. J. Math. Phys. 5 (1964) 355-362
84. Berry M.V. Quantum phase corrections from adiabatic iteration. Proc. Roy. Soc. Lond. Ser. A 414 (1987) 31-46
85. Bellman R. Introduction to Matrix Analysis (1960), McGraw-Hill
86. Klarsfeld S., and Oteo J.A. Exponential infinite-product representations of the time displacement operator. J. Phys. A: Math. Gen. 22 (1989) 2687-2694
87. Wei J., and Norman E. On global representations of the solutions of linear differential equations as a product of exponentials. Proc. Amer. Math. Soc. 15 (1964) 327-334
88. Belinfante J.G.F., and Kolman B. A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods (1972), SIAM
89. Casas F. Solution of linear partial differential equations by Lie algebraic methods. J. Comput. Appl. Math. 76 (1996) 159-170
90. Kawski M. The combinatorics of nonlinear controllability and noncommuting flows. Mathematical Control Theory. ICTP Lect. Notes vol. VIII (2002), Abdus Salam Int. Cent. Theoret. Phys., Trieste 223-311
91. Murua A. The Hopf algebra of rooted trees, free Lie algebras, and Lie series. Found. Comput. Math. 6 (2006) 387-426
92. Voslamber D. On exponential approximations for the evolution operator. J. Math. Anal. Appl. 37 (1972) 403-411
93. Oteo J.A., and Ros J. Iterative approach to the exponential representation of the time-displacement operator. J. Phys. A: Math. Gen. 38 (2005) 1-11
94. Bloch F. Über die Quantenmechanik der Electronen in Kristallgittern. Z. Physik 52 (1928) 555-600
95. Galindo A., and Pascual P. Quantum Mechanics (1990), Springer
96. Hale J.K. Ordinary Differential Equations (1980), Krieger Publishing Company
97. Levante T.O., Baldus M., Meier B.H., and Ernst R.R. Formalized quantum mechanical Floquet theory and its applications to sample spinning in nuclear magnetic resonance. Mol. Phys. 86 (1995) 1195-1212
98. Munthe-Kaas H., and Zanna A. Numerical integration of differential equations on homogeneous manifolds. In: Cucker F., and Shub M. (Eds). Foundations of Computational Mathematics (1997), Springer Verlag 305-315
99. Casas F., and Iserles A. Explicit Magnus expansions for nonlinear equations. J. Phys. A: Math. Gen. 39 (2006) 5445-5461
100. Calvo M.P., Iserles A., and Zanna A. Numerical solution of isospectral flows. Math. Comput. 66 (1997) 1461-1486
101. Brockett R. Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems. Linear Algebra Appl. 146 (1991) 79-91
102. Chu M.T., and Driessel K.R. The projected gradient method for least squares matrix approximations with spectral constraints. SIAM J. Numer. Anal. 27 (1990) 1050-1060
103. Moore J.B., Mahony R.E., and Helmke U. Numerical gradient algorithms for eigenvalue and singular value calculations. SIAM J. Matrix Anal. Appl. 15 (1994) 881-902
104. Iserles A. On the discretization of double-bracket flows. Found. Comput. Math. 2 (2002) 305-329
105. Casas F. Numerical integration methods for the double-bracket flow. J. Comput. Appl. Math. 166 (2004) 477-495
106. Agrachev A.A., and Gamkrelidze R.V. Chronological algebras and nonstationary vector fields. J. Sov. Math. 17 (1979) 1650-1675
107. Agrachev A.A., and Gamkrelidze R.V. The exponential representation of flows and the chronological calculus. Math. USSR-Sb. 35 (1979) 727-785
108. Gamkrelidze R.V. Exponential representation of solutions of ordinary differential equations. Lecture Notes in Math. vol. 703 (1979), Springer-Verlag 118-129
109. Oteo J.A., and Ros J. The Magnus expansion for classical Hamiltonian systems. J. Phys. A: Math. Gen. 24 (1991) 5751-5762
110. Blanes S., and Moan P.C. Splitting methods for non-autonomous Hamiltonian equations. J. Comput. Phys. 170 (2001) 205-230
111. Olver P.J. Applications of Lie Groups to Differential Equations. second ed. (1993), Springer-Verlag
112. Arnold V.I. Mathematical Methods of Classical Mechanics. second ed. (1989), Springer-Verlag
113. Thirring W. Classical Dynamical Systems (1992), Springer-Verlag
114. Marcus R.A. High-order time-dependent perturbation theory for classical mechanics and for other systems of first-order ordinary differential equations. J. Chem. Phys. 52 (1970) 4803-4807
115. Spirig F. Algebraic aspects of perturbation theories. Celest. Mech. 20 (1979) 343-354
116. Tani S. Canonical transformation governing temporal development in classical mechanics. Amer. J. Phys. 36 (1968) 29-32
117. Chen K.-T. Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. of Math. 65 (1957) 163-178
118. Fliess M. Fonctionnelles causales non linéaires et indéterminées non commutatives. Bull. Soc. Math. France 109 (1981) 3-40
119. M. Kawski, Calculating the logarithm of the Chen-Fliess series, Proc. MTNS, Perpignan, 2000
120. Klarsfeld S., and Oteo J.A. Driven harmonic oscillators in the adiabatic Magnus approximation. Phys. Rev. A 47 (1993) 1620-1624
121. Rosen N., and Zener C. Double Stern-Gerlach experiment and related collision phenomena. Phys. Rev. 40 (1932) 502-507
122. Klarsfeld S., and Oteo J.A. Collisional models in a nonperturbative approach. Phys. Lett. A 164 (1992) 38-42
123. Klarsfeld S., and Oteo J.A. Magnus approximation in the adiabatic picture. Phys. Rev. A 45 (1992) 3229-3332
124. Lefebvre R., and Palma A. The harmonic Floquet oscillator. J. Mol. Struct. (Theochem) 390 (1997) 23-32
125. Butcher J.C. The Numerical Analysis of Ordinary Differential Equations (1987), John Wiley
126. Hairer E., Nørsett S.P., and Wanner G. Solving Ordinary Differential Equations I, Nonstiff Problems. Second revised edition (1993), Springer-Verlag
127. Leimkuhler B., and Reich S. Simulating Hamiltonian Dynamics (2004), Cambridge University Press
128. Mclachlan R.I., and Quispel R. Splitting methods. Acta Numer. 11 (2002) 341-434
129. Mclachlan R.I., and Quispel R. Geometric integrators for ODEs. J. Phys. A: Math. Gen. 39 (2006) 5251-5285
130. Chan S.-K., Light J.C., and Lin J.-L. Inelastic molecular collisions: Exponential solution of coupled equations for vibration-translation energy transfer. J. Chem. Phys. 49 (1968) 86-97
131. Devries P.L. A high order method for the solution of the single channel Schrödinger equation. Mol. Phys. 54 (1985) 733-740
132. Blanes S., and Casas F. Optimization of Lie group methods for differential equations. Future Gener. Comput. Syst. 19 (2003) 331-339
133. Iserles A., Marthinsen A., and Nørsett S.P. On the implementation of the method of Magnus series for linear differential equations. BIT 39 (1999) 281-304
134. Zanna A. The Fer expansion and time symmetry: A Strang-type approach. Appl. Numer. Math. 39 (2001) 435-459
135. Iserles A. On Cayley-transform methods for the discretization of Lie-group equations. Found. Comput. Math. 1 (2001) 129-160
136. Diele F., Lopez L., and Peluso R. The Cayley transform in the numerical solution of unitary differential systems. Adv. Comput. Math. 8 (1998) 317-334
137. Marthinsen A., and Owren B. Quadrature methods based on the Cayley transform. Appl. Numer. Math. 39 (2001) 403-413
138. Moler C.B., and Van~Loan C.F. Nineteen dubious ways to compute the exponential of a matrix. SIAM Review 20 (1978) 801-836
139. Moler C.B., and Van~Loan C.F. Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review 45 (2003) 3-49
140. R.B. Sidje, Expokit software, http://www.maths.uq.edu.au/expokit
141. Sidje R.B. Expokit: Software package for computing matrix exponentials. ACM Trans. Math. Software 24 (1998) 130-156
142. Dragt A.J. Computation of maps for particle and light optics by scaling, splitting, and squaring. Phys. Rev. Lett. 75 (1995) 1946-1948
143. Najfeld I., and Havel T.F. Derivatives of the matrix exponential and their computation. Adv. Appl. Math. 16 (1995) 321-375
144. Tal-Ezer H., and Kosloff R. An accurate and efficient scheme for propagating the time dependent Schrödinger equation. J. Chem. Phys. 81 (1984) 3967-3971
145. Süli E., and Mayers D. An Introduction to Numerical Analysis (2003), Cambridge University Press
146. P. Nettesheim, Mixed quantum-classical dynamics: A unified approach to mathematical modelling and numerical simulation, Ph.D. Thesis, Freie Universität Berlin, 2000
147. Leforestier C., Bisseling R.H., Cerjan C., Feit M.D., Friesner R., Guldberg A., Hammerich A., Jolicard G., Karrlein W., Meyer H.-D., Lipkin N., Roncero O., and Kosloff R. A comparison of different propagation schemes for the time dependent Schrödinger equation. J. Comput. Phys. 94 (1991) 59-80
148. Druskin V.L., and Knizhnerman L.A. Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmetics. Numer. Linear Algebra 2 (1995) 205-217
149. Park T.J., and Light J.C. Unitary quantum time evolution by iterative lanczos reduction. J. Chem. Phys. 85 (1986) 5870-5876
150. Watkins D.S. Fundamentals of Matrix Computations. 2nd ed. (2002), John Wiley & Sons
151. Hochbruck M., and Lubich C. Exponential integrators for quantum-classical molecular dynamics. BIT 39 (1999) 620-645
152. Hochbruck M., Lubich C., and Selhofer H. Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19 (1998) 1552-1574
153. Hochbruck M., and Lubich Ch. On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34 (1997) 1911-1925
154. Blanes S., and Moan P.C. Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nyström methods. J. Comput. Appl. Math 142 (2002) 313-330
155. Feng K., and Shang Z.-J. Volume-preserving algorithms for source-free dynamical systems. Numer. Math. 71 (1995) 451-463
156. Celledoni E., and Iserles A. Approximating the exponential from a Lie algebra to a Lie group. Math. Comput. 69 (2000) 1457-1480
157. Celledoni E., and Iserles A. Methods for the approximation of the matrix exponential in a Lie-algebraic setting. IMA J. Numer. Anal. 21 (2001) 463-488
158. Munthe-Kaas H., Quispel R.G.W., and Zanna A. Generalized polar decompositions on Lie groups with involutive automorphisms. Found. Comput. Math. 1 (2001) 297-324
159. Iserles A., and Zanna A. Efficient computation of the matrix exponential by generalized polar decomposition. SIAM J. Numer. Anal. 42 (2005) 2218-2256
160. Zanna A., and Munthe-Kaas H.Z. Generalized polar decompositions for the approximation of the matrix exponential. SIAM J. Matrix Anal. Appl. 23 (2001) 840-862
161. Iserles A. On the global error of discretization methods for highly-oscillatory ordinary differential equations. BIT 42 (2002) 561-599
162. Iserles A. Think globally, act locally: Solving highly-oscillatory ordinary differential equations. Appl. Numer. Math. 43 (2002) 145-160
163. Iserles A. On the numerical quadrature of highly-oscillating integrals I: Fourier transform. IMA J. Numer. Anal. 24 (2004) 365-391
164. Iserles A., and Nørsett S.P. On quadrature methods for highly oscillatory integrals and their implementation. BIT 44 (2004) 755-772
165. Degani I., and Schiff J. RCMS: Right correction Magnus series approach for oscillatory ODEs. J. Comput. Appl. Math 193 (2006) 413-436
166. Jahnke T. Long-time-step integrators for almost-adiabatic quantum dynamics. SIAM J. Sci. Comput. 25 (2004) 2145-2164
167. Jahnke T., and Lubich C. Numerical integrators for quantum dynamics close to the adiabatic limit. Numer. Math. 94 (2003) 289-314
168. Lorenz K., Jahnke T., and Lubich C. Adiabatic integrators for highly oscillatory second-order linear differential equations with time-varying eigendecomposition. BIT 45 (2005) 91-115
169. Aparicio N.D., Malham S.J.A., and Oliver M. Numerical evaluation of the Evans function by Magnus integration. BIT 45 (2005) 219-258
170. Malham S.J.A., and Niesen J. Evaluating the Evans function: Order reduction in numerical methods. Math. Comput. 77 (2008) 159-179
171. Blanes S., and Moan P.C. Splitting methods for the time-dependent Schrödinger equation. Phys. Lett. A 265 (2000) 35-42
172. Baye D., Goldstein G., and Capel P. Fourth-order factorization of the evolution operator for time-dependent potentials. Phys. Lett. A 317 (2003) 337-342
173. Lu Y.Y. Fourth order conservative exponential methods for linear evolution equations. BIT 40 (2000) 762-774
174. Thalhammer M. A fourth-order commutator-free exponential integrator for nonautonomous differential equations. SIAM J. Numer. Anal. 44 (2006) 851-864
175. Blanes S., and Moan P.C. Fourth- and sixth-order commutator-free Magnus integrators for linear and non-linear dynamical systems. Appl. Numer. Math. 56 (2006) 1519-1537
176. Sanz-Serna J.M., and Portillo A. Classical numerical integrators for wave-packet dynamics. J. Chem. Phys. 104 (1996) 2349-2355
177. Blanes S., and Casas F. Splitting methods for non-autonomous separable dynamical systems. J. Phys. A: Math. Gen. 39 (2006) 5405-5423
178. Blanes S., Casas F., and Murua A. Splitting methods for non-autonomous linear systems. Int. J. Comput. Math. 6 (2007) 713-727
179. Munthe-Kaas H. Runge-Kutta methods on Lie groups. BIT 38 (1998) 92-111
180. Munthe-Kaas H. High order Runge-Kutta methods on manifolds. Appl. Numer. Math. 29 (1999) 115-127
181. Zanna A. Collocation and relaxed collocation for the Fer and the Magnus expansions. SIAM J. Numer. Anal. 36 (1999) 1145-1182
182. Lubich C. Integrators for quantum dynamics: A numerical analyst's brief review. In: Grotendorst J., Marx D., and Muramatsu A. (Eds). Quantum Simulations of Many-Body Systems: From Theory to Algorithms. NIC Series vol. 10 (2002), Jülich 459-466
183. Hochbruck M., and Lubich C. On Magnus integrators for time-dependent Schrödinger equations. SIAM J. Numer. Anal. 41 (2003) 945-963
184. Gray S., and Verosky J.M. Classical Hamiltonian structures in wave packet dynamics. J. Chem. Phys. 100 (1994) 5011-5022
185. Jódar L., and Marletta M. Solving ODEs arising from non-selfadjoint Hamiltonian eigenproblems. Adv. Comput. Math. 13 (2000) 231-256
186. Pryce J.D. Numerical Solution of Sturm-Liouville Problems (1993), Oxford University Press
187. Zettl A. Sturm-Liouville Theory (2005), American Mathematical Society
188. Bailey P.B., Gordon M.K., and Shampine L.F. Automatic solution of the Sturm-Liouville problem. ACM Trans. Math. Software 4 (1978) 193-208
189. Greenberg L., and Marletta M. Oscillation theory and numerical solution of sixth order Sturm-Liouville problems. SIAM J. Numer. Anal. 35 (1998) 2070-2098
190. Naimark M.A. Linear Differential Operators, Part I (1968), Ungar
191. Dieci L. Numerical integration of the differential Riccati equation and some related issues. SIAM J. Numer. Anal. 29 (1992) 781-815
192. Dieci L., Osborne M.R., and Russell R.D. A Riccati transformation method for solving linear BVP I: Theoretical aspects, II: Computational aspects. SIAM J. Numer. Anal. 25 (1988) 1055-1092
193. In: Bittanti S., Laub A.J., and Willems J.C. (Eds). The Riccati Equation (1991), Springer
194. Reid W.T. Riccati Differential Equations (1972), Academic Press
195. Kenney C.S., and Leipnik R.B. Numerical integration of the differential Riccati equation. IEEE Trans. Automat. Control AC-30 (1985) 962-970
196. Schiff J., and Shnider S. A natural approach to the numerical integration of Riccati differential equations. SIAM J. Numer. Anal. 36 (1999) 1392-1413
197. Burrage K., Burrage P.M., and Tian T. Numerical methods for strong solutions of stochastic differential equations: An overview. Proc. Roy. Soc. Lond. Ser. A 460 (2004) 373-402
198. Kloeden P.E., and Platen E. Numerical Solution of Stochastic Differential Equations (1992), Springer Verlag
199. Platen E. An introduction to numerical methods for stochastic differential equations. Acta Numer. 8 (1999) 197-246
200. Maruyama G. Continuous Markov processes and stochastic equations. Rend. Circ. Mat. Palermo 4 (1955) 48-90
201. Lord G., Malham S.J.A., and Wiese A. Efficient strong integrators for linear stochastic systems. SIAM J. Numer. Anal. (2006)
202. Malham S.J.A., and Wiese A. Stochastic Lie group integrators. SIAM J. Sci. Comput. 30 (2008) 597-617
203. Misawa T. A Lie algebraic approach to numerical integration of stochastic differential equations. SIAM J. Sci. Comput. 23 (2001) 866-890
204. Eichler J. Magnus approximation for K-shell ionization by heavy-ion impact. Phys. Rev. A 15 (1977) 1856-1862
205. Wille U. Magnus expansion for rotationally induced inner-shell excitation. Phys. Lett. A 82 (1981) 389-392
206. Wille U., and Hippler R. Mechanisms of inner-shell vacancy production in slow ion-atom collisions. Phys. Rep. 132 (1986) 129-260
207. Hyman H.A. Dipole Magnus approximation for electron-atom collisions: Excitation of the resonance transitions of Li, Na and K. Phys. Rev. A 31 (1985) 2142-2148
208. Cady W.A. Rotational spectral line broadening of OCS by noble gases. J. Chem. Phys. 60 (1974) 3318-3323
209. Milfeld K.F., and Wyatt R.E. Study, extension, and application of Floquet theory for quantum molecular systems in an oscillating field. Phys. Rev. A 27 (1983) 72-94
210. Evans W. On some applications of Magnus expansion in Nuclear Magnetic Resonance. Ann. Phys. 48 (1968) 72-93
211. Haeberlen U., and Waugh J.S. Coherent averaging effects in Magnetic resonance. Phys. Rev. 175 (1968) 453-467
212. Vandersypen L.M.K., and Chuang I.L. NMR techniques for quantum control and computation. Rev. Mod. Phys. 76 (2004) 1037-1069
213. Veshtort M., and Griffin R.G. Spin evolution: A powerful tool for the simulation of solid and liquid state NMR experiments. J. Magn. Reson. 178 (2006) 248-282
214. Stefanovich E.V. Quantum field theory without infinities. Ann. Phys. 292 (2001) 139-156
215. Dahmen H.D., Krzyzanowski W., and Larsen M.L. Graphical rules for functions of the time-evolution operator. Phys. Rev. D 33 (1986) 1726-1744
216. Dahmen H.D., Mannel T., and Manakos P. Practical application of the unitary approximation to infrared singularities. Phys. Rev. D 38 (1988) 1176-1182
217. Dahmen H.D., Manakos P., Mannel T., and Ohl T. Unitary approximation for radiative high energy scattering processes: Application to Bhabha scattering. Z. Phys. C 50 (1991) 75-84
218. Connes A., and Kreimer D. Renormalization in quantum field theory and the Riemann-Hilbert problem I. The Hopf algebra structure of graphs and the main theorem. Comm. Math. Phys. 210 (2000) 249-273
219. Connes A., and Kreimer D. Renormalization in quantum field theory and the Riemann-Hilbert problem II. The β-function, diffeomorphisms and the renormalization group. Comm. Math. Phys. 216 (2001) 215-241
220. K. Ebrahimi-Fard, D. Manchon, F. Patras, A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion, Technical Report, arXiv:0705.1265v2, 19 August 2008
221. Ionescu D.C. Unitary and electron-positron pairs created by strong external fields. Phys. Rev. A 49 (1994) 3188-3195
222. D'Olivo J.C. Magnus approximation for nonadiabatic transitions and neutrino oscillations in matter. Phys. Rev. D 45 (1992) 924-929
223. D'Olivo J.C., and Oteo J.A. Nonadiabatic three-neutrino oscillations in matter. Phys. Rev. D 54 (1996) 1187-1193
224. Supanitsky A.D., D'Olivo J.C., and Medina-Tanco G. Perturbative exponential expansion and matter neutrino oscillations. Phys. Rev. D 78 (2008) 045024
225. M.A. Botchev, J.G. Verwer, Numerical integration of damped Maxwell equations, Technical Report, CWI Amsterdam, April 2008
226. Jiang L.-L., Mao J.-F., and Wu X.-L. Symplectic finite difference time-domain method for Maxwell equations. IEEE Trans. Mag. 42 (2006) 1991-1995
227. Lu Y.Y. Some techniques for computing propagation in optical waveguides. Comm. Comput. Phys. 1 (2006) 1056-1075
228. Dattoli G., Gallardo J.C., and Torre A. An algebraic view to the operatorial ordering and its applications to Optics. Riv. Nuovo Cimento 11 (1988) 1-79
229. Khan S.A. Wavelength-dependent modifications in Helmholtz Optics. Int. J. Theor. Phys. 44 (2005) 96-125
230. Lu Y.Y. A fourth-order Magnus scheme for Helmholtz equation. J. Comput. Appl. Math. 173 (2005) 247-258
231. Lu Y.Y. A fourth order derivative-free operator marching method for Helmholtz equation in waveguides. J. Comput. Math. 25 (2007) 719-729
232. Reimer M., and Yevick D. Clifford analysis of polarization-mode dispersion and polarization-dependent loss. IEEE Photon. Technol. Lett. 18 (2006) 734-736
233. Reimer M., and Yevick D. Mueller matrix description of polarization mode dispersion and polarization-dependent loss. J. Opt. Soc. Amer. A 23 (2006) 1503-1508
234. López-Ariste A. On the asymmetry of Stokes profiles. Astrophys. J. 564 (2002) 379-384
235. López-Ariste A., and Semel M. DIAGONAL: A numerical solution of the Stokes transfer equation. Astron. Astrophys. Suppl. Ser. 139 (1999) 417-424
236. Semel M., and López Ariste A. Integration of the radiative transfer equation for polarized light: The exponential solution. Astron. Astrophys. 342 (1999) 201-211
237. San Miguel A. Numerical determination of time transfer in general relativity. Gen. Relativ. Gravit. 39 (2007) 2025-2033
238. Viswanath D. Symbolic dynamics and periodic orbits of the Lorenz attractor. Nonlinearity 16 (2003) 1035-1056
239. Bullo F., and Lewis A.D. Geometric Control of Mechanical Systems (2005), Springer
240. Kalman R.E., ho Y.-C., and Narendra K.S. Controllability of linear dynamical systems. Contributions to Differential Equations 1 (1963) 189-213
241. Duleba I. Locally optimal motion planning of nonholonomic systems. Int. J. Robot. Syst. 14 (1997) 767-788
242. Duleba I. On a computationally simple form of the generalized Campbell-Baker-Hausdorff-Dynkin formula. Systems Control Lett. 34 (1998) 191-202
243. Murray R.M., Li Z., and Satry S.S. A Mathematical Introduction to Robotic Manipulation (1994), CRC Press
244. Engwerda J.C. LQ Dynamic Optimization and Differential Games (2005), Wiley
245. Blanes S., Jódar L., and Ponsoda E. Approximate solutions with a priori error bounds for continuous coefficient matrix Riccati equations. Math. Comput. Modelling 31 (2000) 1-15
246. Jódar L., and Abou-Kandil H. A resolution method for Riccati differential systems coupled in their quadratic terms. SIAM J. Matrix Anal. Appl. 19 (1988) 1425-1430