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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.
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
Complexity theory for Liegroup solvers
, 1999
"... Commencing with a brief survey of Liegroup theory and differential equations evolving on Lie groups, we describe a number of numerical algorithms designed to respect Liegroup structure: RungeKuttaMuntheKaas schemes, Fer and Magnus expansions. This is followed by complexity analysis of Fer and ..."
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Cited by 5 (0 self)
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Commencing with a brief survey of Liegroup theory and differential equations evolving on Lie groups, we describe a number of numerical algorithms designed to respect Liegroup structure: RungeKuttaMuntheKaas schemes, Fer and Magnus expansions. This is followed by complexity analysis of Fer and Magnus expansions, whose conclusion is that for order four, six and eight an appropriately discretized Magnus method is always cheaper than a Fer method of the same order. Each Liegroup method of the kind surveyed in this paper requires the computation of a matrix exponential. Classical methods, e.g. Krylovsubspace and rational approximants, may fail to map elements in a Lie algebra to a Lie group. Therefore we survey a number of approximants based on the splitting approach and demonstrate that their cost is compatible (and often superior) to classical methods. 1 Introduction A central goal of classical numerical analysis is to design, implement and analyse computational algorithms that ...
Convergence of the Magnus series
 Found. Comput. Math
"... The Magnus series is an infinite series which arises in the study of linear ordinary differential equations. If the series converges, then the matrix exponential of the sum equals the fundamental solution of the differential equation. The question considered in this paper is: When does the series co ..."
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Cited by 4 (0 self)
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The Magnus series is an infinite series which arises in the study of linear ordinary differential equations. If the series converges, then the matrix exponential of the sum equals the fundamental solution of the differential equation. The question considered in this paper is: When does the series converge? The main result establishes a sufficient condition for convergence, which improves on several earlier results. 1
The Magnus expansion and some of its applications
, 2008
"... Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an ..."
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Cited by 4 (0 self)
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Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as TimeDependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial resummation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related nonperturbative
Evaluating the Evans function: Order reduction in numerical methods
 MATH. COMP
, 2008
"... We consider the numerical evaluation of the Evans function, a Wronskianlike determinant that arises in the study of the stability of travelling waves. Constructing the Evans function involves matching the solutions of a linear ordinary differential equation depending on the spectral parameter. The ..."
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Cited by 3 (2 self)
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We consider the numerical evaluation of the Evans function, a Wronskianlike determinant that arises in the study of the stability of travelling waves. Constructing the Evans function involves matching the solutions of a linear ordinary differential equation depending on the spectral parameter. The problem becomes stiff as the spectral parameter grows. Consequently, the Gauss–Legendre method has previously been used for such problems; however more recently, methods based on the Magnus expansion have been proposed. Here we extensively examine the stiff regime for a general scalar Schrödinger operator. We show that although the fourthorder Magnus method suffers from order reduction, a fortunate cancellation when computing the Evans matching function means that fourthorder convergence in the end result is preserved. The Gauss–Legendre method does not suffer from order reduction, but it does not experience the cancellation either, and thus it has the same order of convergence in the end result. Finally we discuss the relative merits of both methods as spectral tools.
A Magnus expansion for the equation . . .
, 2000
"... The subject matter of this paper is the representation of the solution of the linear differential equation Y 0 = AY \Gamma Y B, Y (0) = Y0 , in the form Y (t) = e\Omega\Gamma t) Y0 and the representation of the function\Omega as a generalisation of the classical Magnus expansion. An immediate a ..."
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Cited by 3 (0 self)
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The subject matter of this paper is the representation of the solution of the linear differential equation Y 0 = AY \Gamma Y B, Y (0) = Y0 , in the form Y (t) = e\Omega\Gamma t) Y0 and the representation of the function\Omega as a generalisation of the classical Magnus expansion. An immediate application is a new recursive algorithm for the derivation of the BakerCampbellHausdorff formula and its symmetric generalisation. 1 Introduction This paper is concerned with the solution of the linear ordinary differential system Y 0 = AY \Gamma Y B; t 0; Y (0) = Y 0 ; (1.1) where both A and B are Lipschitz functions that map [0; 1) into Mm , the set of m \Theta m matrices, and Y 0 2 Mm . The equation (1.1) features in numerous applications and the approximation of its solution is of interest. Moreover, solutions of this equation often display interesting geometry. For example, B = A results in the isospectral flow Y 0 = AY \Gamma Y A; t 0; Y (0) = Y 0 ; (1.2) whose invariant...
Numerical analysis in Lie groups
, 2000
"... There is growing recognition in the last few years that Lie groups and homogeneous spaces are often the right configuration space for the discretization of timedependent differential equations. In this paper we review briefly recent advances in Liegroup calculations, concentrating mainly on approx ..."
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There is growing recognition in the last few years that Lie groups and homogeneous spaces are often the right configuration space for the discretization of timedependent differential equations. In this paper we review briefly recent advances in Liegroup calculations, concentrating mainly on approximation methods that advance a trivialised version of the differential equation in a Lie algebra in terms of either Magnus or Cayley expansions.
unknown title
, 804
"... Efficient computation of high index SturmLiouville eigenvalues for problems in physics ..."
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Efficient computation of high index SturmLiouville eigenvalues for problems in physics