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1import numpy as np
2from .base import OdeSolver, DenseOutput
3from .common import (validate_max_step, validate_tol, select_initial_step,
4 norm, warn_extraneous, validate_first_step)
5from . import dop853_coefficients
7# Multiply steps computed from asymptotic behaviour of errors by this.
8SAFETY = 0.9
10MIN_FACTOR = 0.2 # Minimum allowed decrease in a step size.
11MAX_FACTOR = 10 # Maximum allowed increase in a step size.
14def rk_step(fun, t, y, f, h, A, B, C, K):
15 """Perform a single Runge-Kutta step.
17 This function computes a prediction of an explicit Runge-Kutta method and
18 also estimates the error of a less accurate method.
20 Notation for Butcher tableau is as in [1]_.
22 Parameters
23 ----------
24 fun : callable
25 Right-hand side of the system.
26 t : float
27 Current time.
28 y : ndarray, shape (n,)
29 Current state.
30 f : ndarray, shape (n,)
31 Current value of the derivative, i.e., ``fun(x, y)``.
32 h : float
33 Step to use.
34 A : ndarray, shape (n_stages, n_stages)
35 Coefficients for combining previous RK stages to compute the next
36 stage. For explicit methods the coefficients at and above the main
37 diagonal are zeros.
38 B : ndarray, shape (n_stages,)
39 Coefficients for combining RK stages for computing the final
40 prediction.
41 C : ndarray, shape (n_stages,)
42 Coefficients for incrementing time for consecutive RK stages.
43 The value for the first stage is always zero.
44 K : ndarray, shape (n_stages + 1, n)
45 Storage array for putting RK stages here. Stages are stored in rows.
46 The last row is a linear combination of the previous rows with
47 coefficients
49 Returns
50 -------
51 y_new : ndarray, shape (n,)
52 Solution at t + h computed with a higher accuracy.
53 f_new : ndarray, shape (n,)
54 Derivative ``fun(t + h, y_new)``.
56 References
57 ----------
58 .. [1] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential
59 Equations I: Nonstiff Problems", Sec. II.4.
60 """
61 K[0] = f
62 for s, (a, c) in enumerate(zip(A[1:], C[1:]), start=1):
63 dy = np.dot(K[:s].T, a[:s]) * h
64 K[s] = fun(t + c * h, y + dy)
66 y_new = y + h * np.dot(K[:-1].T, B)
67 f_new = fun(t + h, y_new)
69 K[-1] = f_new
71 return y_new, f_new
74class RungeKutta(OdeSolver):
75 """Base class for explicit Runge-Kutta methods."""
76 C = NotImplemented
77 A = NotImplemented
78 B = NotImplemented
79 E = NotImplemented
80 P = NotImplemented
81 order = NotImplemented
82 error_estimator_order = NotImplemented
83 n_stages = NotImplemented
85 def __init__(self, fun, t0, y0, t_bound, max_step=np.inf,
86 rtol=1e-3, atol=1e-6, vectorized=False,
87 first_step=None, **extraneous):
88 warn_extraneous(extraneous)
89 super(RungeKutta, self).__init__(fun, t0, y0, t_bound, vectorized,
90 support_complex=True)
91 self.y_old = None
92 self.max_step = validate_max_step(max_step)
93 self.rtol, self.atol = validate_tol(rtol, atol, self.n)
94 self.f = self.fun(self.t, self.y)
95 if first_step is None:
96 self.h_abs = select_initial_step(
97 self.fun, self.t, self.y, self.f, self.direction,
98 self.error_estimator_order, self.rtol, self.atol)
99 else:
100 self.h_abs = validate_first_step(first_step, t0, t_bound)
101 self.K = np.empty((self.n_stages + 1, self.n), dtype=self.y.dtype)
102 self.error_exponent = -1 / (self.error_estimator_order + 1)
103 self.h_previous = None
105 def _estimate_error(self, K, h):
106 return np.dot(K.T, self.E) * h
108 def _estimate_error_norm(self, K, h, scale):
109 return norm(self._estimate_error(K, h) / scale)
111 def _step_impl(self):
112 t = self.t
113 y = self.y
115 max_step = self.max_step
116 rtol = self.rtol
117 atol = self.atol
119 min_step = 10 * np.abs(np.nextafter(t, self.direction * np.inf) - t)
121 if self.h_abs > max_step:
122 h_abs = max_step
123 elif self.h_abs < min_step:
124 h_abs = min_step
125 else:
126 h_abs = self.h_abs
128 step_accepted = False
129 step_rejected = False
131 while not step_accepted:
132 if h_abs < min_step:
133 return False, self.TOO_SMALL_STEP
135 h = h_abs * self.direction
136 t_new = t + h
138 if self.direction * (t_new - self.t_bound) > 0:
139 t_new = self.t_bound
141 h = t_new - t
142 h_abs = np.abs(h)
144 y_new, f_new = rk_step(self.fun, t, y, self.f, h, self.A,
145 self.B, self.C, self.K)
146 scale = atol + np.maximum(np.abs(y), np.abs(y_new)) * rtol
147 error_norm = self._estimate_error_norm(self.K, h, scale)
149 if error_norm < 1:
150 if error_norm == 0:
151 factor = MAX_FACTOR
152 else:
153 factor = min(MAX_FACTOR,
154 SAFETY * error_norm ** self.error_exponent)
156 if step_rejected:
157 factor = min(1, factor)
159 h_abs *= factor
161 step_accepted = True
162 else:
163 h_abs *= max(MIN_FACTOR,
164 SAFETY * error_norm ** self.error_exponent)
165 step_rejected = True
167 self.h_previous = h
168 self.y_old = y
170 self.t = t_new
171 self.y = y_new
173 self.h_abs = h_abs
174 self.f = f_new
176 return True, None
178 def _dense_output_impl(self):
179 Q = self.K.T.dot(self.P)
180 return RkDenseOutput(self.t_old, self.t, self.y_old, Q)
183class RK23(RungeKutta):
184 """Explicit Runge-Kutta method of order 3(2).
186 This uses the Bogacki-Shampine pair of formulas [1]_. The error is controlled
187 assuming accuracy of the second-order method, but steps are taken using the
188 third-order accurate formula (local extrapolation is done). A cubic Hermite
189 polynomial is used for the dense output.
191 Can be applied in the complex domain.
193 Parameters
194 ----------
195 fun : callable
196 Right-hand side of the system. The calling signature is ``fun(t, y)``.
197 Here ``t`` is a scalar and there are two options for ndarray ``y``.
198 It can either have shape (n,), then ``fun`` must return array_like with
199 shape (n,). Or alternatively it can have shape (n, k), then ``fun``
200 must return array_like with shape (n, k), i.e. each column
201 corresponds to a single column in ``y``. The choice between the two
202 options is determined by `vectorized` argument (see below).
203 t0 : float
204 Initial time.
205 y0 : array_like, shape (n,)
206 Initial state.
207 t_bound : float
208 Boundary time - the integration won't continue beyond it. It also
209 determines the direction of the integration.
210 first_step : float or None, optional
211 Initial step size. Default is ``None`` which means that the algorithm
212 should choose.
213 max_step : float, optional
214 Maximum allowed step size. Default is np.inf, i.e., the step size is not
215 bounded and determined solely by the solver.
216 rtol, atol : float and array_like, optional
217 Relative and absolute tolerances. The solver keeps the local error
218 estimates less than ``atol + rtol * abs(y)``. Here, `rtol` controls a
219 relative accuracy (number of correct digits). But if a component of `y`
220 is approximately below `atol`, the error only needs to fall within
221 the same `atol` threshold, and the number of correct digits is not
222 guaranteed. If components of y have different scales, it might be
223 beneficial to set different `atol` values for different components by
224 passing array_like with shape (n,) for `atol`. Default values are
225 1e-3 for `rtol` and 1e-6 for `atol`.
226 vectorized : bool, optional
227 Whether `fun` is implemented in a vectorized fashion. Default is False.
229 Attributes
230 ----------
231 n : int
232 Number of equations.
233 status : string
234 Current status of the solver: 'running', 'finished' or 'failed'.
235 t_bound : float
236 Boundary time.
237 direction : float
238 Integration direction: +1 or -1.
239 t : float
240 Current time.
241 y : ndarray
242 Current state.
243 t_old : float
244 Previous time. None if no steps were made yet.
245 step_size : float
246 Size of the last successful step. None if no steps were made yet.
247 nfev : int
248 Number evaluations of the system's right-hand side.
249 njev : int
250 Number of evaluations of the Jacobian. Is always 0 for this solver as it does not use the Jacobian.
251 nlu : int
252 Number of LU decompositions. Is always 0 for this solver.
254 References
255 ----------
256 .. [1] P. Bogacki, L.F. Shampine, "A 3(2) Pair of Runge-Kutta Formulas",
257 Appl. Math. Lett. Vol. 2, No. 4. pp. 321-325, 1989.
258 """
259 order = 3
260 error_estimator_order = 2
261 n_stages = 3
262 C = np.array([0, 1/2, 3/4])
263 A = np.array([
264 [0, 0, 0],
265 [1/2, 0, 0],
266 [0, 3/4, 0]
267 ])
268 B = np.array([2/9, 1/3, 4/9])
269 E = np.array([5/72, -1/12, -1/9, 1/8])
270 P = np.array([[1, -4 / 3, 5 / 9],
271 [0, 1, -2/3],
272 [0, 4/3, -8/9],
273 [0, -1, 1]])
276class RK45(RungeKutta):
277 """Explicit Runge-Kutta method of order 5(4).
279 This uses the Dormand-Prince pair of formulas [1]_. The error is controlled
280 assuming accuracy of the fourth-order method accuracy, but steps are taken
281 using the fifth-order accurate formula (local extrapolation is done).
282 A quartic interpolation polynomial is used for the dense output [2]_.
284 Can be applied in the complex domain.
286 Parameters
287 ----------
288 fun : callable
289 Right-hand side of the system. The calling signature is ``fun(t, y)``.
290 Here ``t`` is a scalar, and there are two options for the ndarray ``y``:
291 It can either have shape (n,); then ``fun`` must return array_like with
292 shape (n,). Alternatively it can have shape (n, k); then ``fun``
293 must return an array_like with shape (n, k), i.e., each column
294 corresponds to a single column in ``y``. The choice between the two
295 options is determined by `vectorized` argument (see below).
296 t0 : float
297 Initial time.
298 y0 : array_like, shape (n,)
299 Initial state.
300 t_bound : float
301 Boundary time - the integration won't continue beyond it. It also
302 determines the direction of the integration.
303 first_step : float or None, optional
304 Initial step size. Default is ``None`` which means that the algorithm
305 should choose.
306 max_step : float, optional
307 Maximum allowed step size. Default is np.inf, i.e., the step size is not
308 bounded and determined solely by the solver.
309 rtol, atol : float and array_like, optional
310 Relative and absolute tolerances. The solver keeps the local error
311 estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a
312 relative accuracy (number of correct digits). But if a component of `y`
313 is approximately below `atol`, the error only needs to fall within
314 the same `atol` threshold, and the number of correct digits is not
315 guaranteed. If components of y have different scales, it might be
316 beneficial to set different `atol` values for different components by
317 passing array_like with shape (n,) for `atol`. Default values are
318 1e-3 for `rtol` and 1e-6 for `atol`.
319 vectorized : bool, optional
320 Whether `fun` is implemented in a vectorized fashion. Default is False.
322 Attributes
323 ----------
324 n : int
325 Number of equations.
326 status : string
327 Current status of the solver: 'running', 'finished' or 'failed'.
328 t_bound : float
329 Boundary time.
330 direction : float
331 Integration direction: +1 or -1.
332 t : float
333 Current time.
334 y : ndarray
335 Current state.
336 t_old : float
337 Previous time. None if no steps were made yet.
338 step_size : float
339 Size of the last successful step. None if no steps were made yet.
340 nfev : int
341 Number evaluations of the system's right-hand side.
342 njev : int
343 Number of evaluations of the Jacobian. Is always 0 for this solver as it does not use the Jacobian.
344 nlu : int
345 Number of LU decompositions. Is always 0 for this solver.
347 References
348 ----------
349 .. [1] J. R. Dormand, P. J. Prince, "A family of embedded Runge-Kutta
350 formulae", Journal of Computational and Applied Mathematics, Vol. 6,
351 No. 1, pp. 19-26, 1980.
352 .. [2] L. W. Shampine, "Some Practical Runge-Kutta Formulas", Mathematics
353 of Computation,, Vol. 46, No. 173, pp. 135-150, 1986.
354 """
355 order = 5
356 error_estimator_order = 4
357 n_stages = 6
358 C = np.array([0, 1/5, 3/10, 4/5, 8/9, 1])
359 A = np.array([
360 [0, 0, 0, 0, 0],
361 [1/5, 0, 0, 0, 0],
362 [3/40, 9/40, 0, 0, 0],
363 [44/45, -56/15, 32/9, 0, 0],
364 [19372/6561, -25360/2187, 64448/6561, -212/729, 0],
365 [9017/3168, -355/33, 46732/5247, 49/176, -5103/18656]
366 ])
367 B = np.array([35/384, 0, 500/1113, 125/192, -2187/6784, 11/84])
368 E = np.array([-71/57600, 0, 71/16695, -71/1920, 17253/339200, -22/525,
369 1/40])
370 # Corresponds to the optimum value of c_6 from [2]_.
371 P = np.array([
372 [1, -8048581381/2820520608, 8663915743/2820520608,
373 -12715105075/11282082432],
374 [0, 0, 0, 0],
375 [0, 131558114200/32700410799, -68118460800/10900136933,
376 87487479700/32700410799],
377 [0, -1754552775/470086768, 14199869525/1410260304,
378 -10690763975/1880347072],
379 [0, 127303824393/49829197408, -318862633887/49829197408,
380 701980252875 / 199316789632],
381 [0, -282668133/205662961, 2019193451/616988883, -1453857185/822651844],
382 [0, 40617522/29380423, -110615467/29380423, 69997945/29380423]])
385class DOP853(RungeKutta):
386 """Explicit Runge-Kutta method of order 8.
388 This is a Python implementation of "DOP853" algorithm originally written
389 in Fortran [1]_, [2]_. Note that this is not a literate translation, but
390 the algorithmic core and coefficients are the same.
392 Can be applied in the complex domain.
394 Parameters
395 ----------
396 fun : callable
397 Right-hand side of the system. The calling signature is ``fun(t, y)``.
398 Here, ``t`` is a scalar, and there are two options for the ndarray ``y``:
399 It can either have shape (n,); then ``fun`` must return array_like with
400 shape (n,). Alternatively it can have shape (n, k); then ``fun``
401 must return an array_like with shape (n, k), i.e. each column
402 corresponds to a single column in ``y``. The choice between the two
403 options is determined by `vectorized` argument (see below).
404 t0 : float
405 Initial time.
406 y0 : array_like, shape (n,)
407 Initial state.
408 t_bound : float
409 Boundary time - the integration won't continue beyond it. It also
410 determines the direction of the integration.
411 first_step : float or None, optional
412 Initial step size. Default is ``None`` which means that the algorithm
413 should choose.
414 max_step : float, optional
415 Maximum allowed step size. Default is np.inf, i.e. the step size is not
416 bounded and determined solely by the solver.
417 rtol, atol : float and array_like, optional
418 Relative and absolute tolerances. The solver keeps the local error
419 estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a
420 relative accuracy (number of correct digits). But if a component of `y`
421 is approximately below `atol`, the error only needs to fall within
422 the same `atol` threshold, and the number of correct digits is not
423 guaranteed. If components of y have different scales, it might be
424 beneficial to set different `atol` values for different components by
425 passing array_like with shape (n,) for `atol`. Default values are
426 1e-3 for `rtol` and 1e-6 for `atol`.
427 vectorized : bool, optional
428 Whether `fun` is implemented in a vectorized fashion. Default is False.
430 Attributes
431 ----------
432 n : int
433 Number of equations.
434 status : string
435 Current status of the solver: 'running', 'finished' or 'failed'.
436 t_bound : float
437 Boundary time.
438 direction : float
439 Integration direction: +1 or -1.
440 t : float
441 Current time.
442 y : ndarray
443 Current state.
444 t_old : float
445 Previous time. None if no steps were made yet.
446 step_size : float
447 Size of the last successful step. None if no steps were made yet.
448 nfev : int
449 Number evaluations of the system's right-hand side.
450 njev : int
451 Number of evaluations of the Jacobian. Is always 0 for this solver
452 as it does not use the Jacobian.
453 nlu : int
454 Number of LU decompositions. Is always 0 for this solver.
456 References
457 ----------
458 .. [1] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential
459 Equations I: Nonstiff Problems", Sec. II.
460 .. [2] `Page with original Fortran code of DOP853
461 <http://www.unige.ch/~hairer/software.html>`_.
462 """
463 n_stages = dop853_coefficients.N_STAGES
464 order = 8
465 error_estimator_order = 7
466 A = dop853_coefficients.A[:n_stages, :n_stages]
467 B = dop853_coefficients.B
468 C = dop853_coefficients.C[:n_stages]
469 E3 = dop853_coefficients.E3
470 E5 = dop853_coefficients.E5
471 D = dop853_coefficients.D
473 A_EXTRA = dop853_coefficients.A[n_stages + 1:]
474 C_EXTRA = dop853_coefficients.C[n_stages + 1:]
476 def __init__(self, fun, t0, y0, t_bound, max_step=np.inf,
477 rtol=1e-3, atol=1e-6, vectorized=False,
478 first_step=None, **extraneous):
479 super(DOP853, self).__init__(fun, t0, y0, t_bound, max_step,
480 rtol, atol, vectorized, first_step,
481 **extraneous)
482 self.K_extended = np.empty((dop853_coefficients.N_STAGES_EXTENDED,
483 self.n), dtype=self.y.dtype)
484 self.K = self.K_extended[:self.n_stages + 1]
486 def _estimate_error(self, K, h): # Left for testing purposes.
487 err5 = np.dot(K.T, self.E5)
488 err3 = np.dot(K.T, self.E3)
489 denom = np.hypot(np.abs(err5), 0.1 * np.abs(err3))
490 correction_factor = np.ones_like(err5)
491 mask = denom > 0
492 correction_factor[mask] = np.abs(err5[mask]) / denom[mask]
493 return h * err5 * correction_factor
495 def _estimate_error_norm(self, K, h, scale):
496 err5 = np.dot(K.T, self.E5) / scale
497 err3 = np.dot(K.T, self.E3) / scale
499 err5_norm_2 = np.sum(err5**2)
500 err3_norm_2 = np.sum(err3**2)
501 denom = err5_norm_2 + 0.01 * err3_norm_2
502 return np.abs(h) * err5_norm_2 / np.sqrt(denom * len(scale))
504 def _dense_output_impl(self):
505 K = self.K_extended
506 h = self.h_previous
507 for s, (a, c) in enumerate(zip(self.A_EXTRA, self.C_EXTRA),
508 start=self.n_stages + 1):
509 dy = np.dot(K[:s].T, a[:s]) * h
510 K[s] = self.fun(self.t_old + c * h, self.y_old + dy)
512 F = np.empty((dop853_coefficients.INTERPOLATOR_POWER, self.n),
513 dtype=self.y_old.dtype)
515 f_old = K[0]
516 delta_y = self.y - self.y_old
518 F[0] = delta_y
519 F[1] = h * f_old - delta_y
520 F[2] = 2 * delta_y - h * (self.f + f_old)
521 F[3:] = h * np.dot(self.D, K)
523 return Dop853DenseOutput(self.t_old, self.t, self.y_old, F)
526class RkDenseOutput(DenseOutput):
527 def __init__(self, t_old, t, y_old, Q):
528 super(RkDenseOutput, self).__init__(t_old, t)
529 self.h = t - t_old
530 self.Q = Q
531 self.order = Q.shape[1] - 1
532 self.y_old = y_old
534 def _call_impl(self, t):
535 x = (t - self.t_old) / self.h
536 if t.ndim == 0:
537 p = np.tile(x, self.order + 1)
538 p = np.cumprod(p)
539 else:
540 p = np.tile(x, (self.order + 1, 1))
541 p = np.cumprod(p, axis=0)
542 y = self.h * np.dot(self.Q, p)
543 if y.ndim == 2:
544 y += self.y_old[:, None]
545 else:
546 y += self.y_old
548 return y
551class Dop853DenseOutput(DenseOutput):
552 def __init__(self, t_old, t, y_old, F):
553 super(Dop853DenseOutput, self).__init__(t_old, t)
554 self.h = t - t_old
555 self.F = F
556 self.y_old = y_old
558 def _call_impl(self, t):
559 x = (t - self.t_old) / self.h
561 if t.ndim == 0:
562 y = np.zeros_like(self.y_old)
563 else:
564 x = x[:, None]
565 y = np.zeros((len(x), len(self.y_old)), dtype=self.y_old.dtype)
567 for i, f in enumerate(reversed(self.F)):
568 y += f
569 if i % 2 == 0:
570 y *= x
571 else:
572 y *= 1 - x
573 y += self.y_old
575 return y.T