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In this paper we explore quadrature methods for highly oscillatory integrals. Generalizing the method of stationary phase, we expand such integrals into asymptotic series in inverse powers of the frequency. The outcome are two families of methods , one based on a truncation of the asymptotic series and the other extending an approach implicit in the work of(More)
The subject matter of this paper is the solution of the linear diierential equation y 0 = aty, y0 = y0, where y0 2 G, a : R + ! g and g is a Lie algebra of the Lie group G. By building upon an earlier work of Wilhelm Magnus 166, we represent the solution as an innnite series whose terms are indexed by binary trees. This relationship between the innnite(More)
Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Lie-group structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry,(More)
Consider a differential equation y = A(t, y)y, y(0) = y 0 with y 0 ∈ G and A : R + × G → g, where g is a Lie algebra of the matricial Lie group G. Every B ∈ g can be mapped to G by the matrix exponential map exp (tB) with t ∈ R. Most numerical methods for solving ordinary differential equations (ODEs) on Lie groups are based on the idea of representing the(More)
Highly-oscillatory integrals are allegedly difficult to calculate. The main assertion of this paper is that that impression is incorrect. As long as appropriate quadrature methods are used, their accuracy increases when oscillation becomes faster and suitable choice of quadrature points renders this welcome phenomenon more pronounced. We focus our analysis(More)
In this paper we explore the computation of the matrix exponential in a manner that is consistent with Lie-group structure. Our point of departure is the method of generalized polar decompositions, which we modify and combine with similarity transformations that bring the underlying matrix to a form more amenable to efficient computation. We develop(More)
O Lord, how manifold are thy works! In wisdom hast thou made them all: the earth is full of thy riches. Psalms 104:24 Since their introduction by Sir Isaac Newton, diierential equations have played a decisive role in mathematical study of natural phenomena. An important and widely-acknowledged lesson of the last three centuries is that critical information(More)
The main theme of this paper is the construction of efficient, reliable and affordable error bounds for two families of quadrature methods for highly oscillatory integrals. We demonstrate, using asymptotic expansions, that the error can be bounded very precisely indeed at the cost of few extra derivative evaluations. Moreover, in place of derivatives it is(More)
Let A be a banded matrix of bandwidth s 1. The exponential of A is usually dense. Yet, its elements decay very rapidly away from the diagonal and, in practical computation, can be set to zero away from some bandwidth r > s which depends on the required accuracy threshold. In this paper we investigate this phenomenon and present sharp, computable bounds on(More)