The Magnus expansion and some of its applications

By Sergio Blanes, Fernando Casas, José Ángel Oteo, José Ros. 2008

1,289 citations recorded by the source corpus.

Abstract Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem, shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to build up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial re-summation of infinite terms with the important additional property of preserving, at any order, certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related non-perturbative expansions. Second, to provide a bridge with its implementation as generator of especial purpose numerical integration methods, a field of intense activity during the last decade. Third, to illustrate with examples the kind of results one can expect from Magnus expansion, in comparison with those from both perturbative schemes and standard numerical integrators. We buttress this issue with a revision of the wide range of physical applications found by Magnus expansion in the literature.

Subfields: Computational Theory and Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/3f7da6287b6ec8ee28e92697c4910238ce07f4d0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Modeling and Simulation

In Numerical Analysis

Citations in the corpus

Solution of the quasi-one-dimensional linearized Euler equations using flow invariants and the Magnus expansion (2013, methodology): “The series converges fast when the condition given by∫ ξF 0 ‖A(ξ)‖2 dξ < π (4.12) is satisfied (see Moan & Niesen (2006), Casas (2007) and Blanes et al. (2009) for more details), but may diverge for larger values of the integral.”
The Quantum Harmonic Otto Cycle (2016, methodology): “The Magnus expansion [96] is employed to obtain the l period propagator U3a(X = 2lπ), where the periods are of the adiabatic propagator Ua of Equation (38): U3a(X = 2lπ) ≈ e12 , (55)”
Conditional Markov chains: Properties, construction and structured dependence (2017, methodology): “The above Kolmogorov equations admit unique solution provided that Γ satis es (4.10) (see formulae given by Peano-Baker series or by Magnus expansion see e.g. Blanes, Casas, Oteo and Ros Blanes et al. (2009)).”
Magnus and Dyson series for Master Integrals (2014, background): “3 Magnus series expansion Consider a generic linear matrix differential equation [18] ∂xY (x) = A(x)Y (x) , Y (x0) = Y0 .”
Many-body energy localization transition in periodically driven systems (2012, background): “The corresponding ME [23, 24] is guaranteed to converge for finite-dimensional Hamiltonians and sufficiently short periods.”
GYRE: An open-source stellar oscillation code based on a new Magnus Multiple Shooting Scheme (2013, methodology): “This constant-Jacobian case is the only one having a closed-form expression for the Magnus matrix; however, as discussed by Blanes et al. (2009) it is relatively straightforward to construct approximations to eqn.”
Observing the Space- and Time-Dependent Growth of Correlations in Dynamically Tuned Synthetic Ising Models with Antiferromagnetic Interactions (2017, background): “For the linear ramps considered here, a Magnus expansion [71] of the propagator is appropriate.”
The In-Medium Similarity Renormalization Group: A novel ab initio method for nuclei (2015, methodology): “[63] that utilizes the Magnus expansion from the theory of matrix differential equations [62, 105].”