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extensisq-0.4.0
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مشاهده بیشترتوضیحات
Extend scipy.integrate with various methods for solve_ivp
| ویژگی | مقدار |
|---|---|
| سیستم عامل | - |
| نام فایل | extensisq-0.4.0 |
| نام | extensisq |
| نسخه کتابخانه | 0.4.0 |
| نگهدارنده | [] |
| ایمیل نگهدارنده | [] |
| نویسنده | W.R. Kampinga |
| ایمیل نویسنده | wrkampi@tuta.io |
| آدرس صفحه اصلی | https://github.com/WRKampi/extensisq |
| آدرس اینترنتی | https://pypi.org/project/extensisq/ |
| مجوز | - |
# extensisq
This package extends scipy.integrate with various methods (OdeSolver classes) for the solve_ivp function.
[](https://github.com/WRKampi/extensisq/blob/main/LICENSE)
[](https://pypistats.org/packages/extensisq)
[](https://anaconda.org/conda-forge/extensisq)
[](https://github.com/WRKampi/extensisq/releases)
## Installation
You can install extensisq from [PyPI](https://pypi.org/project/extensisq/):
pip install extensisq
Or, if you'd rather use [conda](https://anaconda.org/conda-forge/extensisq):
conda install -c conda-forge extensisq
## Example
Borrowed from the the scipy documentation:
from scipy.integrate import solve_ivp
from extensisq import BS5
def exponential_decay(t, y): return -0.5 * y
sol = solve_ivp(exponential_decay, [0, 10], [2, 4, 8], method=BS5)
print(sol.t)
print(sol.y)
Notice that the class `BS5` is passed to `solve_ivp`, not the string `"BS5"`. The other methods (`SWAG`, `CK5`, `Ts5`, `Pr7`, `Pr8`, `Pr9`, `CKdisc`, `CFMR7osc` and `SSV2stab`) can be used in a similar way.
More examples are available as notebooks (update needed):
1. [Integration with Scipy's `solve_ivp` function](https://github.com/WRKampi/extensisq/blob/main/docs/Demo_solve_ivp.ipynb)
2. [About `BS5` and its interpolants](https://github.com/WRKampi/extensisq/blob/main/docs/Demo_BS5.ipynb)
3. [Higher order Prince methods `Pr7`, `Pr8` and `Pr9`](https://github.com/WRKampi/extensisq/blob/main/docs/Prince.ipynb)
4. [Special method `CKdisc` for non-smooth problems](https://github.com/WRKampi/extensisq/blob/main/docs/Demo_CKdisc.ipynb)
5. [Special method `CFMR7osc` for oscillatory problems](https://github.com/WRKampi/extensisq/blob/main/docs/Demo_CFMR7osc.ipynb)
6. [Special method `SSV2stab` for large, mildly stiff problems](https://github.com/WRKampi/extensisq/blob/main/docs/Demo_SSV2stab.ipynb)
7. [Fifth order methods compared](https://github.com/WRKampi/extensisq/blob/main/docs/all_methods.ipynb)
8. [Van der Pol's equation, Shampine Gordon Watts method](https://github.com/WRKampi/extensisq/blob/main/docs/Shampine_Gordon_Watts.ipynb)
9. [Sensitivity analysis](https://github.com/WRKampi/extensisq/blob/main/docs/Demo_sensitivity.ipynb)
## Methods
Currently, several explicit methods (for non-stiff problems) are provided.
One multistep method is implemented:
* `SWAG`: the variable order Adams-Bashforth-Moulton predictor-corrector method of Shampine, Gordon and Watts [5-7]. This is a translation of the Fortran code `DDEABM`. Matlab's method `ode113` is related.
Three explicit Runge Kutta methods of order 5 are implemented:
* `BS5`: efficient fifth order method by Bogacki and Shampine [1,A]. Three interpolants are included: the original accurate fifth order interpolant, a lower cost fifth order one, and a 'free' fourth order one.
* `CK5`: fifth order method with the coefficients from [2], for general use.
* `Ts5`: relatively new solver (2011) by Tsitouras, optimized with fewer simplifying assumptions [3].
One fourth order method:
* `Me4`: Merson's method, the first embedded RK method [14]. The embedded method for error estimation is 5th order for linear problems and 3rd order for general problems. A 3rd order interpolant is added. This method has a large stability region. It may be useful as alternative to 'RK23' for solving problems to lower accuracy.
Three higher order explicit Runge Kutta methods by Prince [4] are implemented:
* `Pr7`: a seventh order discrete method with fifth order error estimate, derived from a sixth order continuous method.
* `Pr8`: an eighth order discrete method with sixth order error estimate, derived from a seventh order continuous method.
* `Pr9`: a ninth order discrete method with seventh order error estimate, derived from an eighth order continuous method.
The numbers in the names refer to the discrete methods, while the orders in [4] refer to the continuous methods. These methods are relatively efficient when dense output is needed, because the interpolants are free. (Other high-order methods typically need several additional function evaluations for dense output.)
Three methods for specific types of problems are available:
* `CKdisc`: variable order solver by Cash and Karp, tailored to solve non-smooth problems efficiently [2].
* `CFMR7osc`: explicit Runge Kutta method, with algebraic order 7, dispersion order 10 and dissipation order 9, to efficiently and accurately solve problems with oscillating solutions [12]. A free 5th order interpolant for dense output is added.
* `SSV2stab`: second order stabilized Runge Kutta Chebyshev method [13,C], to explicity and efficiently solve large systems of mildly stiff ordinary differential equations up to low to moderate accuracy. Equations arising from semi-discretization of parabolic PDEs are a typical use case.
## Sensitivity analysis
Three methods for sensitiviy analysis are available; see [15] and Example 9 above. These can be used with any of the solvers.
* `sens_forward`: to calculate the sensitivity of all solution components to (a few) parameters.
* `sens_adjoint_end`: to calculate the sensitivity of a scalar function of the solution to (many) parameters.
* `sens_adjoint_int`: to calculate the sensitivity of a scalar integral of the solution to (many) parameters.
## Other features
The initial step size, when not supplied by you, is estimated using the method of Watts [7,B]. This method analyzes your problem with a few (3 to 4) evaluations and carefully estimates a safe stepsize to start the integration with.
Most of extensisq's Runge Kutta methods have stiffness detection. If many steps fail, or if the integration needs a lot of steps, the power iteration method of Shampine [8,A] is used to test your problem for stiffness. You will get a warning if your problem is diagnosed as stiff. The kind of roots (real, complex or nearly imaginary) is also reported, such that you can select a stiff solver that better suits your problem.
Second order stepsize controllers [9-11] can be enabled for most of extensisq's Runge Kutta methods. You can set your own coefficients, or select one of the default values.
## References
[1] P. Bogacki, L.F. Shampine, "An efficient Runge-Kutta (4,5) pair", Computers & Mathematics with Applications, Vol. 32, No. 6, 1996, pp. 15-28. https://doi.org/10.1016/0898-1221(96)00141-1
[2] J. R. Cash, A. H. Karp, "A Variable Order Runge-Kutta Method for Initial Value Problems with Rapidly Varying Right-Hand Sides", ACM Trans. Math. Softw., Vol. 16, No. 3, 1990, pp. 201-222. https://doi.org/10.1145/79505.79507
[3] Ch. Tsitouras, "Runge-Kutta pairs of order 5(4) satisfying only the first column simplifying assumption", Computers & Mathematics with Applications, Vol. 62, No. 2, 2011, pp. 770 - 775. https://doi.org/10.1016/j.camwa.2011.06.002
[4] P.J. Prince, "Parallel Derivation of Efficient Continuous/Discrete Explicit Runge-Kutta Methods", Guisborough TS14 6NP U.K., September 6 2018. http://www.peteprince.co.uk/parallel.pdf
[5] L.F. Shampine and M.K. Gordon, "Computer solution of ordinary differential equations: The initial value problem", San Francisco, W.H. Freeman, 1975.
[6] H.A. Watts and L.F. Shampine, "Smoother Interpolants for Adams Codes", SIAM Journal on Scientific and Statistical Computing, Vol. 7, No. 1, 1986, pp. 334-345. https://doi.org/10.1137/0907022
[7] H.A. Watts, "Starting step size for an ODE solver", Journal of Computational and Applied Mathematics, Vol. 9, No. 2, 1983, pp. 177-191. https://doi.org/10.1016/0377-0427(83)90040-7
[8] L.F. Shampine, "Diagnosing Stiffness for Runge–Kutta Methods", SIAM Journal on Scientific and Statistical Computing, Vol. 12, No. 2, 1991, pp. 260-272. https://doi.org/10.1137/0912015
[9] K. Gustafsson, "Control Theoretic Techniques for Stepsize Selection in Explicit Runge-Kutta Methods", ACM Trans. Math. Softw., Vol. 17, No. 4, 1991, pp. 533–554. https://doi.org/10.1145/210232.210242
[10] G.Söderlind, "Automatic Control and Adaptive Time-Stepping", Numerical Algorithms, Vol. 31, No. 1, 2002, pp. 281-310. https://doi.org/10.1023/A:1021160023092
[11] G. Söderlind, "Digital Filters in Adaptive Time-Stepping", ACM Trans. Math. Softw., Vol. 29, No. 1, 2003, pp. 1–26. https://doi.org/10.1145/641876.641877
[12] M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, "Explicit Runge-Kutta methods for initial value problems with oscillating solutions", Journal of Computational and Applied Mathematics, Vol. 76, No. 1–2, 1996, pp. 195-212. https://doi.org/10.1016/S0377-0427(96)00103-3
[13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, "RKC: An explicit solver for parabolic PDEs", Journal of Computational and Applied Mathematics, Vol. 88, No. 2, 1998, pp. 315-326. https://doi.org/10.1016/S0377-0427(97)00219-7
[14] E. Hairer, G. Wanner, S.P. Norsett, "Solving Ordinary Differential Equations I", Springer Berlin, Heidelberg, 1993, https://doi.org/10.1007/978-3-540-78862-1
[15] R.Serban, A.C. Hindmarsh, "CVODES: The Sensitivity-Enabled ODE Solver in SUNDIALS", 5th International Conference on Multibody Systems Nonlinear Dynamics and Control, Vol. 6, 2005, https://doi.org/10.1115/DETC2005-85597
## Original source codes (Fortran)
[A] RKSuite, R.W. Brankin, I. Gladwell, L.F. Shampine. https://www.netlib.org/ode/rksuite/
[B] DDEABM, L.F. Shampine, H.A. Watts, M.K. Gordon. https://www.netlib.org/slatec/src/
[C] RKC, B.P. Sommeijer, L.F. Shampine, J.G. Verwer. https://www.netlib.org/ode/
زبان مورد نیاز
| مقدار | نام |
|---|---|
| >=3.8 | Python |
نحوه نصب
نصب پکیج whl extensisq-0.4.0:
pip install extensisq-0.4.0.whl
نصب پکیج tar.gz extensisq-0.4.0:
pip install extensisq-0.4.0.tar.gz