## On Cayley-transform methods for the discretization of Lie-group equations (1999)

Venue: | FOUND. COMPUT. MATH |

Citations: | 16 - 4 self |

### BibTeX

@TECHREPORT{Iserles99oncayley-transform,

author = {Arieh Iserles},

title = {On Cayley-transform methods for the discretization of Lie-group equations},

institution = {FOUND. COMPUT. MATH},

year = {1999}

}

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### Abstract

In this paper we develop in a systematic manner the theory of time-stepping methods based on the Cayley transform. Such methods can be applied to discretise differential equations that evolve in some Lie groups, in particular in the orthogonal group and the symplectic group. Unlike many other Lie-group solvers, they do not require the evaluation of matrix exponentials. Similarly to the theory of Magnus expansions in (Iserles & Nørsett 1999), we identify terms in a Cayley expansion with rooted trees, which can be constructed recursively. Each such term is an integral over a polytope but all such integrals can be evaluated to high order by using special quadrature formulae similar to the construction in (Iserles & Nrsett 1999). Truncated Cayley expansions (with exact integrals) need not be time-symmetric, hence the method does not display the usual advantages associated with time symmetry, e.g. even order of approximation. However, time symmetry (with its attendant benefits) is attained when exact integrals are replaced by certain quadrature formulae. 1 Quadratic Lie groups The theme of this paper is geometric integration: numerical discretization of differential equations that respects their underlying geometry. It is increasingly recognised by numerical analysts and users of computational methods alike that geometric integration often represents a highly efficient and precise means toward obtaining a numerical solution, whilst retaining important qualitative attributes of the differential system (Budd & Iserles 1999). Large number of differential equations with a wide range of practical applications evolve on Lie groups G = fA 2 GLn (R) : AJA where GLn (R) is the group of all n \Theta n nonsingular real matrices and J 2 GLn (R) is given. (We refer the reader to (Cart...