General Mathematics

The Centered Magnus Exponent and Magnus Quadrature Formulas of Orders 2, 4, 6 and 8

Authors: Kenneth C. Johnson

The Magnus exponent is formulated over an integration interval [−t, t] that is centered on the origin, so that the solution at t is obtained from the solution at −t. In this centered formulation the propagator satisfies U(−t) = U(t)−1, the Magnus exponent is an odd function of t, and the Magnus theorem takes a form in which the even and odd parts of the coefficient matrix appear separately. The Magnus exponent is then expressed in Legendre moments of the coefficient matrix over the centered interval, and quadrature formulas with residual errors of order t3, t5, t7 and t9 are given, using 0, 1, 3 and 6 commutators. The moments are evaluated by Gauss-Legendre, Newton-Cotes or Gauss-Lobatto quadrature, and Gauss-Kronrod rules are shown to give embedded error estimates. The methodology and results are compared with those of Blanes, Casas, Oteo and Ros.

Comments: 15 Pages.

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[v1] 2026-11-02 10:25:30

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