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Volume 16, Issue 3
The Step-Transition Operators for Multi-Step Methods of ODE's

K. Feng

J. Comp. Math., 16 (1998), pp. 193-202.

Published online: 1998-06

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  • Abstract

In this paper, we propose a new definition of symplectic multistep methods. This definition differs from the old ones in that it is given via the one step method defined directly on $M$ which is corresponding to the $m$ step scheme defined on $M$ while the old definitions are given out by defining a corresponding one step method on $M\times M \times \cdots \times M=M^m$ with a set of new variables. The new definition gives out a step-transition operator $g: M\longrightarrow M$. Under our new definition, the Leap-frog method is symplectic only for linear Hamiltonian systems. The transition operator $g$ will be constructed via continued fractions and rational approximations.

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@Article{JCM-16-193, author = {}, title = {The Step-Transition Operators for Multi-Step Methods of ODE's}, journal = {Journal of Computational Mathematics}, year = {1998}, volume = {16}, number = {3}, pages = {193--202}, abstract = {

In this paper, we propose a new definition of symplectic multistep methods. This definition differs from the old ones in that it is given via the one step method defined directly on $M$ which is corresponding to the $m$ step scheme defined on $M$ while the old definitions are given out by defining a corresponding one step method on $M\times M \times \cdots \times M=M^m$ with a set of new variables. The new definition gives out a step-transition operator $g: M\longrightarrow M$. Under our new definition, the Leap-frog method is symplectic only for linear Hamiltonian systems. The transition operator $g$ will be constructed via continued fractions and rational approximations.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9152.html} }
TY - JOUR T1 - The Step-Transition Operators for Multi-Step Methods of ODE's JO - Journal of Computational Mathematics VL - 3 SP - 193 EP - 202 PY - 1998 DA - 1998/06 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9152.html KW - Multi-step methods, Explike and loglike function, Fractional and rational approximation, Simplecticity of LMM, Nonexistence of SLMM. AB -

In this paper, we propose a new definition of symplectic multistep methods. This definition differs from the old ones in that it is given via the one step method defined directly on $M$ which is corresponding to the $m$ step scheme defined on $M$ while the old definitions are given out by defining a corresponding one step method on $M\times M \times \cdots \times M=M^m$ with a set of new variables. The new definition gives out a step-transition operator $g: M\longrightarrow M$. Under our new definition, the Leap-frog method is symplectic only for linear Hamiltonian systems. The transition operator $g$ will be constructed via continued fractions and rational approximations.

K. Feng. (1970). The Step-Transition Operators for Multi-Step Methods of ODE's. Journal of Computational Mathematics. 16 (3). 193-202. doi:
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