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Liegroup methods
 ACTA NUMERICA
, 2000
"... Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having ..."
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Cited by 93 (18 self)
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Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup 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, this paper surveys the novel theory of numerical integrators that respect Liegroup structure, highlighting theory, algorithmic issues and a number of applications.
Integration Methods Based on Canonical Coordinates of the Second Kind
, 1998
"... We present a new class of integration methods for differential equations on manifolds, in the framework of Lie group actions. Canonical coordinates of the second kind is used for representing the Lie group locally by means of its corresponding Lie algebra. The coordinate map itself can, in many case ..."
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Cited by 20 (5 self)
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We present a new class of integration methods for differential equations on manifolds, in the framework of Lie group actions. Canonical coordinates of the second kind is used for representing the Lie group locally by means of its corresponding Lie algebra. The coordinate map itself can, in many cases, be computed inexpensively, but the approach also involves the inversion of its differential, a task that can be challenging. To succeed, it is necessary to consider carefully howtochoose a basis for the Lie algebra, and the ordering of the basis is important as well. For semisimple Lie algebras, one may take advantage of the root space decomposition to provide a basis with desirable properties. The problem of ordering leads us to introduce the concept of an admissible ordered basis (AOB). The existence of an AOB is established for some of the most important Lie algebras. The computational cost analysis shows that the approachmay lead to more efficient solvers for ODEs on manifolds than those ba...
On the Global Error of Discretization Methods for HighlyOscillatory Ordinary Differential Equations
, 2000
"... Commencing from a globalerror formula, originally due to Henrici, we investigate the accumulation of global error in the numerical solution of linear highlyoscillating systems of the form y 00 + g(t)y = 0, where g(t) t!1 \Gamma! 1. Using WKB analysis we derive an explicit form of the globalerror ..."
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Cited by 20 (5 self)
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Commencing from a globalerror formula, originally due to Henrici, we investigate the accumulation of global error in the numerical solution of linear highlyoscillating systems of the form y 00 + g(t)y = 0, where g(t) t!1 \Gamma! 1. Using WKB analysis we derive an explicit form of the globalerror envelope for RungeKutta and Magnus methods. Our results are closely matched by numerical experiments. Motivated by the superior performance of Liegroup methods, we present a modification of the Magnus expansion which displays even better longterm behaviour in the presence of oscillations.
On the implementation of the method of Magnus series for linear differential equations
, 1998
"... The method of Magnus series has recently been analysed by Iserles & Nørsett (1997). It approximates the solution of linear differential equations y 0 = a(t)y in the form y#t#=e ##t# , solving a nonlinear differential equation for # by means of an expansion in iterated integrals of commutators. An ap ..."
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Cited by 18 (9 self)
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The method of Magnus series has recently been analysed by Iserles & Nørsett (1997). It approximates the solution of linear differential equations y 0 = a(t)y in the form y#t#=e ##t# , solving a nonlinear differential equation for # by means of an expansion in iterated integrals of commutators. An appealing feature of the method is that, whenever the exact solution evolves in a Lie group, so does the numerical solution. The subject matter of the present paper is practical implementation of the method of Magnus series. We commence by briefly reviewing the method and highlighting its connection with graph theory. This is followed by the derivation of error estimates, a task greatly assisted by the graphtheoretical connection. These error estimates have been incorporated into a variablestep fourthorder code. The concluding section of the paper is devoted to a number of computer experiments that highlight the promise of the proposed approach even in the absence of a Liegroup structure....
Hamilton–Pontryagin integrators on Lie groups
, 2007
"... Abstract In this paper, structurepreserving timeintegrators for rigid bodytype mechanical ..."
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Cited by 17 (6 self)
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Abstract In this paper, structurepreserving timeintegrators for rigid bodytype mechanical
Improved high order integrators based on the Magnus expansion
 BIT
, 1999
"... We build high order efficient numerical integration methods for solving the linear differential equation X = A(t)X based on Magnus expansion. These methods ..."
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Cited by 17 (3 self)
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We build high order efficient numerical integration methods for solving the linear differential equation X = A(t)X based on Magnus expansion. These methods
On Cayleytransform methods for the discretization of Liegroup equations
 FOUND. COMPUT. MATH
, 1999
"... In this paper we develop in a systematic manner the theory of timestepping 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 Liegr ..."
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Cited by 16 (4 self)
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In this paper we develop in a systematic manner the theory of timestepping 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 Liegroup 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 timesymmetric, 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...
Efficient approximation of SturmLiouville problems using Liegroup methods
, 1998
"... We present a new approach to the numerical solution of SturmLiouville eigenvalue problems based on Magnus expansions. Our algorithms are closely related to Pruess' methods [Pre73], but provide for high order approximations at nearly the same cost as the secondorder Pruess method. By using Newton i ..."
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Cited by 15 (0 self)
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We present a new approach to the numerical solution of SturmLiouville eigenvalue problems based on Magnus expansions. Our algorithms are closely related to Pruess' methods [Pre73], but provide for high order approximations at nearly the same cost as the secondorder Pruess method. By using Newton iteration to solve for the eigenvalues, we are able to present an efficient algorithm for computing a range of eigenvalues and eigenfunctions. Numerical experiments display promising results, and asymptotic corrections are made to improve further the accuracy of the schemes.
On the Discretization of DoubleBracket Flows
 Found. Comput. Math
, 2001
"... This paper extends the method of Magnus series to Liealgebraic equations originating in doublebracket flows. We show that the solution of the isospectral flow Y = [[Y; N ]; Y ], Y (0) = Y0 2 Sym(n), can be represented in the form Y (t) = e \Omega\Gamma Y0e , where the Taylor expansion of\Om ..."
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Cited by 13 (4 self)
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This paper extends the method of Magnus series to Liealgebraic equations originating in doublebracket flows. We show that the solution of the isospectral flow Y = [[Y; N ]; Y ], Y (0) = Y0 2 Sym(n), can be represented in the form Y (t) = e \Omega\Gamma Y0e , where the Taylor expansion of\Omega can be constructed explicitly, termbyterm, identifying individual expansion terms with certain rooted trees with bicolour leaves. This approach is extended to other Liealgebraic equations that can be appropriately expressed in terms of a finite `alphabet'. 1