# pyHamSys
pyHamSys is a Python package for scientific computing involving Hamiltonian systems
Installation:
```
pip install pyhamsys
```
## Symplectic Integrators
pyHamSys includes a class SymplecticIntegrator containing the following symplectic splitting integrators:
- `Verlet` (order 2, all purpose), also referred to as Strang or Störmer-Verlet splitting
- From [Forest, Ruth, Physica D 43, 105 (1990)](https://doi.org/10.1016/0167-2789(90)90019-L):
- `FR` (order 4, all purpose)
- From [Yoshida, Phys. Lett. A 150, 262 (1990)](https://doi.org/10.1016/0375-9601(90)90092-3):
- `Yo#`: # should be replaced by an even integer, e.g., `Yo6` for 6th order symplectic integrator (all purpose)
- `Yos6`: (order 6, all purpose) optimized symplectic integrator (solution A from Table 1)
- From [McLachlan, SIAM J. Sci. Comp. 16, 151 (1995)](https://doi.org/10.1137/0916010):
- `M2` (order 2, all purpose)
- `M4` (order 4, all purpose)
- From [Omelyan, Mryglod, Folk, Comput. Phys. Commun. 146, 188 (2002)](https://doi.org/10.1016/S0010-4655(02)00451-4):
- `EFRL` (order 4) optimized for *H* = *A* + *B*
- `PEFRL` and `VEFRL` (order 4) optimized for *H* = *A*(*p*) + *B*(*q*). For `PEFRL`, *chi* should be exp(*h*A)exp(*h*B). For `VEFRL`, *chi* should be exp(*h*B)exp(*h*A).
- From [Blanes, Moan, J. Comput. Appl. Math. 142, 313 (2002)](https://doi.org/10.1016/S0377-0427(01)00492-7):
- `BM4` (order 4, all purpose) refers to S<sub>6</sub>
- `BM6` (order 6, all purpose) refers to S<sub>10</sub>
- `RKN4b` (order 4) refers to SRKN<sub>6</sub><sup>*b*</sup> optimized for *H* = *A*(*p*) + *B*(*q*). Here *chi* should be exp(*h*B)exp(*h*A).
- `RKN6b` (order 6) refers to SRKN<sub>11</sub><sup>*b*</sup> optimized for *H* = *A*(*p*) + *B*(*q*). Here *chi* should be exp(*h*B)exp(*h*A).
- `RKN6a` (order 6) refers to SRKN<sub>14</sub><sup>*a*</sup> optimized for *H* = *A*(*p*) + *B*(*q*). Here *chi* should be exp(*h*A)exp(*h*B).
- From [Blanes, Casas, Farrés, Laskar, Makazaga, Murua, Appl. Numer. Math. 68, 58 (2013)](http://dx.doi.org/10.1016/j.apnum.2013.01.003):
- `ABA104` (order (10,4)) optimized for *H* = *A* + ε *B*. Here *chi* should be exp(*h*A)exp(*h*B).
- `ABA864` (order (8,6,4)) optimized for *H* = *A* + ε *B*. Here *chi* should be exp(*h*A)exp(*h*B).
- `ABA1064` (order (10,6,4)) optimized for *H* = *A* + ε *B*. Here *chi* should be exp(*h*A)exp(*h*B).
All purpose integrators are for any splitting of the Hamiltonian *H*=∑<sub>*k*</sub> *X*<sub>*k*</sub> in any order of the functions *X*<sub>*k*</sub>. Otherwise, the order of the operators is specified for each integrator.
Usage: *integrator* = SymplecticIntegrator(*name*, *step*)
where *name* is one of the names listed above and *step* is the time step of the integrator (float).
The function *integrator*.`_integrate` integrates the Hamiltonian flow by one step.
The function *integrator*.`integrate` integrates the Hamiltonian flow from the initial conditions specified by the initial state vector *y* using *integrator*, one of the selected symplectic splitting integrators. It returns the value of *y* at times defines by the float, list or array *times*.
Parameters:
- *chi* : function of (*h*, *y*), y being the state vector.
Function returning exp(*h* X<sub>*n*</sub>)...exp(*h* X<sub>1</sub>) *y*. If the selected integrator is not all purpose, refer to the list above for the specific ordering of the operators. The operator X<sub>*k*</sub> is the Liouville operator associated with the function *X*<sub>*k*</sub>, i.e., X<sub>*k*</sub> = {*X*<sub>*k*</sub>, ·}.
- *chi_star* : function of (*h*, *y*).
Function returning exp(*h* X<sub>1</sub>)...exp(*h* X<sub>*n*</sub>) *y*.
- *y* : initial state vector (numpy array)
- *times* : times at which the values of the state vector are computed
- *command* : function of (*t*, *y*).
Function to be run at each time step (e.g., plotting an observable associated with the state vector, or register specific events).
- *autonomous* : boolean.
If autonomous is False, the state vector y should be of the form *y* = [*t*, *x*], where the first coordinate is time.
Returns:
- If *times* is a float of integer, the output is a tuple (*t*, *y* or *x*) where *y* is the value of the state vector and *y* = [*t*, *x*] if autonomous is False.
- If *times* is a list or array, returns the times and values of *y* or *x* at *times*.
- If *times* is a list or array with a single element, returns the times and values of *y* or *x* at all computed times.
References:
- Hairer, Lubich, Wanner, 2003, *Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations* (Springer)
- McLachlan, *Tuning symplectic integrators is easy and worthwhile*, Commun. Comput. Phys. 31, 987 (2022); [arxiv:2104.10269](https://arxiv.org/abs/2104.10269)
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"description": "# pyHamSys\npyHamSys is a Python package for scientific computing involving Hamiltonian systems\n\n\n\n\nInstallation: \n```\npip install pyhamsys\n```\n\n## Symplectic Integrators\npyHamSys includes a class SymplecticIntegrator containing the following symplectic splitting integrators:\n\n- `Verlet` (order 2, all purpose), also referred to as Strang or St\u00f6rmer-Verlet splitting \n- From [Forest, Ruth, Physica D 43, 105 (1990)](https://doi.org/10.1016/0167-2789(90)90019-L): \n - `FR` (order 4, all purpose)\n- From [Yoshida, Phys. Lett. A 150, 262 (1990)](https://doi.org/10.1016/0375-9601(90)90092-3):\n - `Yo#`: # should be replaced by an even integer, e.g., `Yo6` for 6th order symplectic integrator (all purpose)\n - `Yos6`: (order 6, all purpose) optimized symplectic integrator (solution A from Table 1)\n- From [McLachlan, SIAM J. Sci. Comp. 16, 151 (1995)](https://doi.org/10.1137/0916010):\n - `M2` (order 2, all purpose)\n - `M4` (order 4, all purpose)\n- From [Omelyan, Mryglod, Folk, Comput. Phys. Commun. 146, 188 (2002)](https://doi.org/10.1016/S0010-4655(02)00451-4): \n - `EFRL` (order 4) optimized for *H* = *A* + *B*\n - `PEFRL` and `VEFRL` (order 4) optimized for *H* = *A*(*p*) + *B*(*q*). For `PEFRL`, *chi* should be exp(*h*A)exp(*h*B). For `VEFRL`, *chi* should be exp(*h*B)exp(*h*A).\n- From [Blanes, Moan, J. Comput. Appl. Math. 142, 313 (2002)](https://doi.org/10.1016/S0377-0427(01)00492-7):\n - `BM4` (order 4, all purpose) refers to S<sub>6</sub> \n - `BM6` (order 6, all purpose) refers to S<sub>10</sub>\n - `RKN4b` (order 4) refers to SRKN<sub>6</sub><sup>*b*</sup> optimized for *H* = *A*(*p*) + *B*(*q*). Here *chi* should be exp(*h*B)exp(*h*A).\n - `RKN6b` (order 6) refers to SRKN<sub>11</sub><sup>*b*</sup> optimized for *H* = *A*(*p*) + *B*(*q*). Here *chi* should be exp(*h*B)exp(*h*A).\n - `RKN6a` (order 6) refers to SRKN<sub>14</sub><sup>*a*</sup> optimized for *H* = *A*(*p*) + *B*(*q*). Here *chi* should be exp(*h*A)exp(*h*B).\n- From [Blanes, Casas, Farr\u00e9s, Laskar, Makazaga, Murua, Appl. Numer. Math. 68, 58 (2013)](http://dx.doi.org/10.1016/j.apnum.2013.01.003):\n - `ABA104` (order (10,4)) optimized for *H* = *A* + ε *B*. Here *chi* should be exp(*h*A)exp(*h*B).\n - `ABA864` (order (8,6,4)) optimized for *H* = *A* + ε *B*. Here *chi* should be exp(*h*A)exp(*h*B).\n - `ABA1064` (order (10,6,4)) optimized for *H* = *A* + ε *B*. Here *chi* should be exp(*h*A)exp(*h*B).\n \nAll purpose integrators are for any splitting of the Hamiltonian *H*=∑<sub>*k*</sub> *X*<sub>*k*</sub> in any order of the functions *X*<sub>*k*</sub>. Otherwise, the order of the operators is specified for each integrator.\n\nUsage: *integrator* = SymplecticIntegrator(*name*, *step*)\nwhere *name* is one of the names listed above and *step* is the time step of the integrator (float). \n\nThe function *integrator*.`_integrate` integrates the Hamiltonian flow by one step.\n\nThe function *integrator*.`integrate` integrates the Hamiltonian flow from the initial conditions specified by the initial state vector *y* using *integrator*, one of the selected symplectic splitting integrators. It returns the value of *y* at times defines by the float, list or array *times*.\n\nParameters:\n - *chi* : function of (*h*, *y*), y being the state vector.\n Function returning exp(*h* X<sub>*n*</sub>)...exp(*h* X<sub>1</sub>) *y*. If the selected integrator is not all purpose, refer to the list above for the specific ordering of the operators. The operator X<sub>*k*</sub> is the Liouville operator associated with the function *X*<sub>*k*</sub>, i.e., X<sub>*k*</sub> = {*X*<sub>*k*</sub>, ·}.\n - *chi_star* : function of (*h*, *y*).\n Function returning exp(*h* X<sub>1</sub>)...exp(*h* X<sub>*n*</sub>) *y*.\n - *y* : initial state vector (numpy array)\n - *times* : times at which the values of the state vector are computed\n - *command* : function of (*t*, *y*).\n Function to be run at each time step (e.g., plotting an observable associated with the state vector, or register specific events). \n - *autonomous* : boolean.\n If autonomous is False, the state vector y should be of the form *y* = [*t*, *x*], where the first coordinate is time. \n\nReturns:\n - If *times* is a float of integer, the output is a tuple (*t*, *y* or *x*) where *y* is the value of the state vector and *y* = [*t*, *x*] if autonomous is False.\n - If *times* is a list or array, returns the times and values of *y* or *x* at *times*. \n - If *times* is a list or array with a single element, returns the times and values of *y* or *x* at all computed times. \n\nReferences:\n - Hairer, Lubich, Wanner, 2003, *Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations* (Springer)\n - McLachlan, *Tuning symplectic integrators is easy and worthwhile*, Commun. Comput. Phys. 31, 987 (2022); [arxiv:2104.10269](https://arxiv.org/abs/2104.10269)\n",
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