Magnus Expansion - Magnus Generator

Magnus Generator

It is possible to design a recursive procedure to generate all the terms in the Magnus expansion. Specifically, with the matrices defined recursively through

one has

Here is a shorthand for an iterated commutator,

and are the Bernoulli numbers.

When this recursion is worked out explicitly, it is possible to express as a linear combination of -fold integrals of nested commutators containing matrices ,

$\Omega_n(t) = \sum_{j=1}^{n-1} \frac{B_j}{j!} \, \sum_{ k_1 + \cdots + k_j = n-1 \atop k_1 \ge 1, \ldots, k_j \ge 1} \, \int_0^t \, \mathrm{ad}_{\Omega_{k_1}(\tau )} \, \mathrm{ad}_{\Omega_{k_2}(\tau )} \cdots \, \mathrm{ad}_{\Omega_{k_j}(\tau )} A(\tau ) \, d\tau \qquad n \ge 2,$

an expression that becomes increasingly intricate with .

Read more about this topic: Magnus Expansion

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