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Many numerical algorithms involve computations in Lie algebras, like composition and splitting methods, methods involving the Baker-Campbell-Hausdor formula and the recently developed Lie group methods for integration of di erential equations on manifolds. This paper is concerned with complexity and optimization of such computations in the general case… (More)

- Brynjulf Owren
- 1995

We consider numerical integration methods for di erentiable manifolds as proposed by Crouch and Grossman. The di erential system is phrased by means of a system of frame vector elds E1; : : : ; En on the manifold. The numerical approximation is obtained by composing ows of certain vector elds in the linear span of E1; : : : ; En that are tangent to the di… (More)

We de ne the Newton iteration for solving the equation f(y) = 0, where f is a map from a Lie group to its corresponding Lie algebra. Two versions are presented, which are formulated independently of any metric on the Lie group. Both formulations reduce to the standard method in the Euclidean case, and are related to existing algorithms on certain Riemannian… (More)

- Elena Celledoni, Arne Marthinsen, Brynjulf Owren
- Future Generation Comp. Syst.
- 2003

RKMK methods and Crouch-Grossman methods are two classes of Lie group methods. The former is using flows and commutators of a Lie algebra of vector fields as a part of the method definition. The latter uses only compositions of flows of such vector fields, but the number of flows which needs to be computed is much higher than in the RKMK methods. We present… (More)

Research in explicit Runge-Kutta methods is producing continual improvements to the original algorithms, and the aim of this survey is to relate the current state-of-the-art. In drawing attention to recent advances, we hope to provide useful information for those who apply numerical methods. We describe recent work in the derivation of Runge-Kutta… (More)

Using the notion of integrating factors, Lawson developed a class of numerical methods for solving stiff systems of ordinary differential equations. However, the performance of these “Generalized Runge–Kutta processes” was demonstrably poorer when compared to the ETD schemes of Certaine and Nørsett, recently rediscovered by Cox and Matthews. The deficit is… (More)

- Brynjulf Owren, Marino Zennaro
- SIAM J. Scientific Computing
- 1992

Continuous Explicit Runge-Kutta methods with the minimal number of stages are considered. These methods are continuously diierentiable if and only if one of the stages is the FSAL evaluation. A characterization of a subclass of these methods is developed for order 3,4 and 5. It is shown how the free parameters of these methods can be used either to minimize… (More)

The mono-implicit Runge-Kutta (MIRK) schemes, a subset of the family of implicit Runge-Kutta (IRK) schemes, were originally proposed for the numerical solution of initial value ODEs more than fifteen years ago. During the last decade, a considerable amount of attention has been given to the use of these schemes in the numerical solution of boundary value… (More)

- ME T HODS, Brynjulf Owren, Kristian Seip
- 1990

The theory of positive real functions is used to provide bounds for the largest possible disk to be inscribed in the stability region of an explicit Runge-Kutta method. In particular, we show that the closed disk 1~ + rl < r can be contained in the stability region of an explicit m-stage Runge-Kutta method of order two if and only if r _< m 1. AMS… (More)