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## Composition constants for raising the order of unconventional schemes for ordinary differential equations (1997)

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Venue: | Math. Comp |

Citations: | 14 - 1 self |

### Citations

732 |
Solving Ordinary Differential Equations I,
- Hairer, Nørsett, et al.
- 1987
(Show Context)
Citation Context ...1 involves only the approximation yn to y(tn), but not approximations at previous sample times; it is a multi-step method otherwise. Many conventional methods (linear multi-step, Runge-Kutta methods) =-=[1, 2, 7]-=- are in use. To achieve generality, they have evolved into complicated programs thousands of lines long, and have become highly refined and relatively efficient solvers of a wide range of differential... |

265 | Construction of higher order symplectic integrators
- Yoshida
(Show Context)
Citation Context ... order approximations. Surprisingly the determining equations in this general context are equivalent to those that would be otherwise derived from special cases like for separable Hamiltonian systems =-=[13]-=-, the implicit mid-point rule Q [10], and decompositions of exponential operators [12]. An explanation of such equivalence resides in Lie Algebra Tools [8]. In Yoshida [13], order conditions for order... |

251 | Numerical Methods for Ordinary Differential Systems, - Lambert - 1991 |

118 |
Solving Ordinary Differential Equations II,
- Hairer, Wanner
- 2010
(Show Context)
Citation Context ...1 involves only the approximation yn to y(tn), but not approximations at previous sample times; it is a multi-step method otherwise. Many conventional methods (linear multi-step, Runge-Kutta methods) =-=[1, 2, 7]-=- are in use. To achieve generality, they have evolved into complicated programs thousands of lines long, and have become highly refined and relatively efficient solvers of a wide range of differential... |

95 | On the numerical integration of ordinary differential equations by symmetric composition methods,
- McLachlan
- 1995
(Show Context)
Citation Context ...ions.1096 WILLIAM KAHAN AND REN-CANG LI Appendix A. Palindromic schemes for orders 6 and higher Among schemes that follow, s7odr6 appeared in Yoshida [13], and s15odr8 was also obtained by McLachlan =-=[9]-=- under different circumstances. s7odr6 c1 0.78451361047755726382 δ1 = δ7 0.78451361047755726382 c2 1.0200868238369153975 δ2 = δ6 0.23557321335935813368 c3 -0.15759316034195560944 δ3 = δ5 -1.1776799841... |

66 |
Solving Ordinary Di$erential Equations.
- Hairer, Wanner
- 1996
(Show Context)
Citation Context ...1 involves only the approximation yn to y(tn), but not approximations at previous sample times; it is a multi-step method otherwise. Many conventional methods (linear multi-step, Runge-Kutta methods) =-=[1, 2, 7]-=- are in use. To achieve generality, they have evolved into complicated programs thousands of lines long, and have become highly rened and relatively ecient solvers of a wide range of dierential equ... |

54 | The accuracy of floating point summation,
- Higham
- 1993
(Show Context)
Citation Context ...E PRECISION, and compensated summation technique is used. We briefly describe what we did with compensated summation technique. (For more discussion of compensated summation, see Kahan [5] and Higham =-=[3]-=-.) The idea of the technique is to represent a number by two double precision floating point numbers such that the number is correctly represented to roughly 30 decimal digits. Take y1 for an example.... |

52 |
General theory of fractal path integrals with applications to many-body theories and statistical physics,
- Suzuki
- 1991
(Show Context)
Citation Context ...ule by Sanz-Serna and Abia [10], and in its most general context by Kahan [6]. j=1RAISING THE ORDERS OF UNCONVENTIONAL SCHEMES 1093 2. s5odr4: m =5andδ1=δ2=δ4=δ5= 1 4− 3√ 4 ,δ3=− 3√ 4 4− 3√ . Suzuki =-=[11]-=- had 4 this scheme for exponential approximations. It has been known to the first author for quite a while, but as the minimizer to both quantities in (5) it is due to [8]. 3. s5odr4a and s5odr4b: m =... |

29 | General theory of higher-order decomposition of exponential operators and symplectic integrators Phys - Suzuki - 1992 |

23 |
Solving Ordinary Dierential Equations II
- HAIRER, WANNER
- 1991
(Show Context)
Citation Context ...1 involves only the approximation yn to y(tn), but not approximations at previous sample times; it is a multi-step method otherwise. Many conventional methods (linear multi-step, Runge-Kutta methods) =-=[1, 2, 7]-=- are in use. To achieve generality, they have evolved into complicated programs thousands of lines long, and have become highly rened and relatively ecient solvers of a wide range of dierential equ... |

20 | Unconventional numerical methods for trajectory calculation,” Unpublished Lecture Notes, - Kahan - 1993 |

19 |
Order conditions for canonical Runge-Kutta schemes
- Sanz-Serna, Abia
- 1991
(Show Context)
Citation Context ...the determining equations in this general context are equivalent to those that would be otherwise derived from special cases like for separable Hamiltonian systems [13], the implicit mid-point rule Q =-=[10]-=-, and decompositions of exponential operators [12]. An explanation of such equivalence resides in Lie Algebra Tools [8]. In Yoshida [13], order conditions for orders up to 8 are given; while Suzuki [1... |

18 | Numerical methods for ordinary dierential systems, the initial value problem, - Lambert - 1991 |

6 |
On the numerical integration of ordinary dierential equations by symmetric composition methods
- McLachlan
- 1995
(Show Context)
Citation Context ...f-use 1096 WILLIAM KAHAN AND REN-CANG LI Appendix A. Palindromic schemes for orders 6 and higher Among schemes that follow, s7odr6 appeared in Yoshida [13], and s15odr8 was also obtained by McLachlan =-=[9]-=- under dierent circumstances. s7odr6 c1 0.78451361047755726382 δ1 = δ7 0.78451361047755726382 c2 1.0200868238369153975 δ2 = δ6 0.23557321335935813368 c3 -0.15759316034195560944 δ3 = δ5 -1.17767998417... |

3 |
Raising the Orders of Unconventional Schemes for Ordinary Dierential Equations
- Li
- 1995
(Show Context)
Citation Context ... already overworked, so we prefer( the word) reflexive.) One yn+yn+1 example is the Implicit Mid-point Rule: yn+1 = yn + θnf 2 .Aconsistent and reflexive formula has at least second order convergence =-=[1,2,4,8]-=-andhasother properties which allow efficient constructions of higher order approximations. One such construction composes Q(θj, · ) with specially correlated step-sizes θj; details will be given in th... |

2 |
Relaxation methods for solving systems of ordinary dierential equations
- Kahan
- 1977
(Show Context)
Citation Context ... already overworked, so we prefer( the word) reflexive.) One yn+yn+1 example is the Implicit Mid-point Rule: yn+1 = yn + θnf 2 .Aconsistent and reflexive formula has at least second order convergence =-=[1,2,4,8]-=-andhasother properties which allow efficient constructions of higher order approximations. One such construction composes Q(θj, · ) with specially correlated step-sizes θj; details will be given in th... |

2 | Analysis and application of simply compensated summation. Work in progress - Kahan - 1993 |

2 |
numerical methods for trajectory calculations, lectures notes
- Unconventional
- 1993
(Show Context)
Citation Context ...al error behaves like y(T ) − yN = O(max n θp n). An updating formula Q(θ, g) isReflexive if Q(−θ, Q(θ, g)) = g. (It has been called Symmetric, Reversible, andSelf-Adjoint too but, as argued by Kahan =-=[6]-=-, these terms are already overworked, so we prefer( the word) reflexive.) One yn+yn+1 example is the Implicit Mid-point Rule: yn+1 = yn + θnf 2 .Aconsistent and reflexive formula has at least second o... |

1 |
theory of higher-order decomposition of exponential operators and symplectic integrators
- General
- 1992
(Show Context)
Citation Context ...are equivalent to those that would be otherwise derived from special cases like for separable Hamiltonian systems [13], the implicit mid-point rule Q [10], and decompositions of exponential operators =-=[12]-=-. An explanation of such equivalence resides in Lie Algebra Tools [8]. In Yoshida [13], order conditions for orders up to 8 are given; while Suzuki [12] attempted to give order conditions for orders1 ... |