University of Cambridge Approximating the Exponential from a Lie Algebra to a Lie Group Approximating the Exponential from a Lie Algebra to a Lie Group
نویسنده
چکیده
Consider a diierential equation y 0 = A(t; y)y; y(0) = y0 with y0 2 G and A : R + G ! g, where g is a Lie algebra of the matricial Lie group G. Every B 2 g can be mapped to G by the matrix exponential map exp (tB) with t 2 R. Most numerical methods for solving ordinary diierential equations (ODEs) on Lie groups are based on the idea of representing the approximation yn of the exact solution y(tn), tn 2 R + , by means of exact exponentials of suitable elements of the Lie algebra, applied to the initial value y0 1], 8], 15], 6]. This ensures that yn 2 G. When the exponential is diicult to compute exactly, as is the case when the dimension is large, an approximation of exp (tB) plays an important role in the numerical solution of ODEs on Lie groups. In some cases rational or polynomial approximants are unsuitable and we consider alternative techniques, whereby exp (tB) is approximated by a product of simpler exponentials. In this paper we present some ideas based on the use of the Strang splitting for the approximation of matrix exponentials. Several cases of g and G are considered, in tandem with general theory. Order conditions are discussed and a number of numerical experiments conclude the paper.
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