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We present new symmetric fourth and sixth-order symplectic partitioned Runge–Kutta and Runge–Kutta–Nystr' om methods. We studied compositions using several extra stages, optimising the e1ciency. An e2ective error, Ef, is de3ned and an extensive search is carried out using the extra parameters. The new methods have smaller values of Ef than other methods… (More)

- P. C. Moan
- 1998

We present a new approach to the numerical solution of Sturm{Liouville 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 second-order Pruess method. By using Newton iteration to solve for the eigenvalues, we are able to… (More)

- Per Christian Moan, Jitse Niesen
- Foundations of Computational Mathematics
- 2008

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 necessary condition… (More)

- S. Blanes, P. C. Moan
- 2001

We present an algorithm for numerically integrating non-autonomous Hamiltonian differential equations. Special attention is paid to the separable case and, in particular, a new fourth-order splitting method is presented which in a certain measure is optimal. In combination with a new way of handling non-autonomous problems, the schemes we present are based… (More)

We present new symmetric fourth and sixth-order symplectic Partitioned Runge{ Kutta and Runge{Kutta{Nystrr om methods. We studied compositions using several extra stages, optimising the eeciency. An eeective error, E f , is deened and an extensive search is carried out using the extra parameters. The new methods have smaller values of E f than other methods… (More)

- Colin Rowat, Jim Jordan, +5 authors Nasser Zakariya
- 2009

We add exchange to a pillage economy based on that in Jordan (2006a). We fully characterise the core and stable sets in the Edgeworth box. Jordan’s core extends naturally beyond this. For particular utility functions, his stable sets extend similarly. Generically, exchange increases the opportunities for pillage: agents engaging in pillage need not expect… (More)

A new fourth order splitting method is presented which in a certain measure is optimal. In combination with a new way of handling non-autonomous problems, the schemes we present are based on Mag-nus expansions and they show very promising results when applied to Hamiltonian ODEs.

Cheap and easy to implement fourth-order methods for the Schrr odinger equation with time-dependent Hamiltonians are introduced. The methods require evaluations of exponentials of simple unidimensional inte-grals, and can be considered an averaging technique, preserving many of the qualitative properties of the exact solution. INTRODUCTION. In this work we… (More)

We revisit an algorithm by Skeel et al. [5, 16] for computing the modified, or shadow, energy associated with symplectic discretizations of Hamiltonian systems. We amend the algorithm to use Richardson extrapolation in order to obtain arbitrarily high order of accuracy. Error estimates show that the new method captures the exponentially small drift… (More)

- P. C. Moan, Christian Moan
- 1998

We present a new approach to the numerical solution of Sturm{Liouville 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 second-order Pruess method. By using Newton iteration to solve for the eigenvalues, we are able to… (More)

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