Sufficient conditions for the convergence of the Magnus expansion

Abstract

Two different sufficient conditions are given for the convergence of the Magnus expansion arising in the study of the linear differential equationThe first one provides a bound on the convergence domain based on the norm of the operator A(t). The second condition links the convergence of the expansion with the structure of the spectrum of Y (t), thus yielding a more precise characterization. Several examples are proposed to illustrate the main issues involved and the information on the convergence domain provided by both conditions.

Solution of the quasi-one-dimensional linearized Euler equations using flow invariants and the Magnus expansion

“…is satisfied (see Blanes et al (2009);Casas (2007); Moan & Niesen (2006) for more details), but may diverge for larger values of the integral. In the above, || · || 2 is the 2-norm of the matrix, which for a square matrix is equal to the spectral norm (the largest singular value of the matrix), and can be calculated as…”

New analytic approximations based on the Magnus expansion

“…Several improved bounds to the actual radius of convergence in terms of A have been obtained along the years [18,1,15,17]. In this respect, the following result is proved in [6]: This theorem, in fact, provides the optimal convergence domain, in the sense that π is the largest constant for which the result holds without any further restrictions on the operator A(t). Nevertheless, it is quite easy to construct examples for which the bound estimate r c = π is still conservative: the Magnus series converges indeed for a larger time interval than that given by the theorem.…”

“…More specifically, in [6] the connection between the convergence of the Magnus series and the existence of multiple eigenvalues of the fundamental solution Y (t) has been analyzed. Let us introduce a new parameter ε ∈ C and denote by Y t (ε) the fundamental matrix of Y ′ = εA(t)Y .…”

Controlling the dipole-dipole interaction using NMR composite rf pulses

“…In the case of magic composite pulses, the pulse sequence not beeing periodic, assumption (8) is not garrantied. Casas et al have shown that criterion (12) is sufficient to have the existence of (8) and thus the convergence and validity of the truncation for the Magnus expansion 17,18 . However, in the case of a dipole-dipole Hamiltonian, criterion (12) is experimmentally tremendously difficult to meet due to large eigenvalues in the Hamiltonian.…”