Coverage for /pythoncovmergedfiles/medio/medio/usr/local/lib/python3.8/site-packages/scipy/integrate/_ivp/rk.py: 32%
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« prev ^ index » next coverage.py v7.3.2, created at 2023-12-12 06:31 +0000
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: np.ndarray = NotImplemented
77 A: np.ndarray = NotImplemented
78 B: np.ndarray = NotImplemented
79 E: np.ndarray = NotImplemented
80 P: np.ndarray = NotImplemented
81 order: int = NotImplemented
82 error_estimator_order: int = NotImplemented
83 n_stages: int = 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().__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), while `atol` controls
220 absolute accuracy (number of correct decimal places). To achieve the
221 desired `rtol`, set `atol` to be smaller than the smallest value that
222 can be expected from ``rtol * abs(y)`` so that `rtol` dominates the
223 allowable error. If `atol` is larger than ``rtol * abs(y)`` the
224 number of correct digits is not guaranteed. Conversely, to achieve the
225 desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller
226 than `atol`. If components of y have different scales, it might be
227 beneficial to set different `atol` values for different components by
228 passing array_like with shape (n,) for `atol`. Default values are
229 1e-3 for `rtol` and 1e-6 for `atol`.
230 vectorized : bool, optional
231 Whether `fun` is implemented in a vectorized fashion. Default is False.
233 Attributes
234 ----------
235 n : int
236 Number of equations.
237 status : string
238 Current status of the solver: 'running', 'finished' or 'failed'.
239 t_bound : float
240 Boundary time.
241 direction : float
242 Integration direction: +1 or -1.
243 t : float
244 Current time.
245 y : ndarray
246 Current state.
247 t_old : float
248 Previous time. None if no steps were made yet.
249 step_size : float
250 Size of the last successful step. None if no steps were made yet.
251 nfev : int
252 Number evaluations of the system's right-hand side.
253 njev : int
254 Number of evaluations of the Jacobian. Is always 0 for this solver as it does not use the Jacobian.
255 nlu : int
256 Number of LU decompositions. Is always 0 for this solver.
258 References
259 ----------
260 .. [1] P. Bogacki, L.F. Shampine, "A 3(2) Pair of Runge-Kutta Formulas",
261 Appl. Math. Lett. Vol. 2, No. 4. pp. 321-325, 1989.
262 """
263 order = 3
264 error_estimator_order = 2
265 n_stages = 3
266 C = np.array([0, 1/2, 3/4])
267 A = np.array([
268 [0, 0, 0],
269 [1/2, 0, 0],
270 [0, 3/4, 0]
271 ])
272 B = np.array([2/9, 1/3, 4/9])
273 E = np.array([5/72, -1/12, -1/9, 1/8])
274 P = np.array([[1, -4 / 3, 5 / 9],
275 [0, 1, -2/3],
276 [0, 4/3, -8/9],
277 [0, -1, 1]])
280class RK45(RungeKutta):
281 """Explicit Runge-Kutta method of order 5(4).
283 This uses the Dormand-Prince pair of formulas [1]_. The error is controlled
284 assuming accuracy of the fourth-order method accuracy, but steps are taken
285 using the fifth-order accurate formula (local extrapolation is done).
286 A quartic interpolation polynomial is used for the dense output [2]_.
288 Can be applied in the complex domain.
290 Parameters
291 ----------
292 fun : callable
293 Right-hand side of the system. The calling signature is ``fun(t, y)``.
294 Here ``t`` is a scalar, and there are two options for the ndarray ``y``:
295 It can either have shape (n,); then ``fun`` must return array_like with
296 shape (n,). Alternatively it can have shape (n, k); then ``fun``
297 must return an array_like with shape (n, k), i.e., each column
298 corresponds to a single column in ``y``. The choice between the two
299 options is determined by `vectorized` argument (see below).
300 t0 : float
301 Initial time.
302 y0 : array_like, shape (n,)
303 Initial state.
304 t_bound : float
305 Boundary time - the integration won't continue beyond it. It also
306 determines the direction of the integration.
307 first_step : float or None, optional
308 Initial step size. Default is ``None`` which means that the algorithm
309 should choose.
310 max_step : float, optional
311 Maximum allowed step size. Default is np.inf, i.e., the step size is not
312 bounded and determined solely by the solver.
313 rtol, atol : float and array_like, optional
314 Relative and absolute tolerances. The solver keeps the local error
315 estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a
316 relative accuracy (number of correct digits), while `atol` controls
317 absolute accuracy (number of correct decimal places). To achieve the
318 desired `rtol`, set `atol` to be smaller than the smallest value that
319 can be expected from ``rtol * abs(y)`` so that `rtol` dominates the
320 allowable error. If `atol` is larger than ``rtol * abs(y)`` the
321 number of correct digits is not guaranteed. Conversely, to achieve the
322 desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller
323 than `atol`. If components of y have different scales, it might be
324 beneficial to set different `atol` values for different components by
325 passing array_like with shape (n,) for `atol`. Default values are
326 1e-3 for `rtol` and 1e-6 for `atol`.
327 vectorized : bool, optional
328 Whether `fun` is implemented in a vectorized fashion. Default is False.
330 Attributes
331 ----------
332 n : int
333 Number of equations.
334 status : string
335 Current status of the solver: 'running', 'finished' or 'failed'.
336 t_bound : float
337 Boundary time.
338 direction : float
339 Integration direction: +1 or -1.
340 t : float
341 Current time.
342 y : ndarray
343 Current state.
344 t_old : float
345 Previous time. None if no steps were made yet.
346 step_size : float
347 Size of the last successful step. None if no steps were made yet.
348 nfev : int
349 Number evaluations of the system's right-hand side.
350 njev : int
351 Number of evaluations of the Jacobian. Is always 0 for this solver as it does not use the Jacobian.
352 nlu : int
353 Number of LU decompositions. Is always 0 for this solver.
355 References
356 ----------
357 .. [1] J. R. Dormand, P. J. Prince, "A family of embedded Runge-Kutta
358 formulae", Journal of Computational and Applied Mathematics, Vol. 6,
359 No. 1, pp. 19-26, 1980.
360 .. [2] L. W. Shampine, "Some Practical Runge-Kutta Formulas", Mathematics
361 of Computation,, Vol. 46, No. 173, pp. 135-150, 1986.
362 """
363 order = 5
364 error_estimator_order = 4
365 n_stages = 6
366 C = np.array([0, 1/5, 3/10, 4/5, 8/9, 1])
367 A = np.array([
368 [0, 0, 0, 0, 0],
369 [1/5, 0, 0, 0, 0],
370 [3/40, 9/40, 0, 0, 0],
371 [44/45, -56/15, 32/9, 0, 0],
372 [19372/6561, -25360/2187, 64448/6561, -212/729, 0],
373 [9017/3168, -355/33, 46732/5247, 49/176, -5103/18656]
374 ])
375 B = np.array([35/384, 0, 500/1113, 125/192, -2187/6784, 11/84])
376 E = np.array([-71/57600, 0, 71/16695, -71/1920, 17253/339200, -22/525,
377 1/40])
378 # Corresponds to the optimum value of c_6 from [2]_.
379 P = np.array([
380 [1, -8048581381/2820520608, 8663915743/2820520608,
381 -12715105075/11282082432],
382 [0, 0, 0, 0],
383 [0, 131558114200/32700410799, -68118460800/10900136933,
384 87487479700/32700410799],
385 [0, -1754552775/470086768, 14199869525/1410260304,
386 -10690763975/1880347072],
387 [0, 127303824393/49829197408, -318862633887/49829197408,
388 701980252875 / 199316789632],
389 [0, -282668133/205662961, 2019193451/616988883, -1453857185/822651844],
390 [0, 40617522/29380423, -110615467/29380423, 69997945/29380423]])
393class DOP853(RungeKutta):
394 """Explicit Runge-Kutta method of order 8.
396 This is a Python implementation of "DOP853" algorithm originally written
397 in Fortran [1]_, [2]_. Note that this is not a literate translation, but
398 the algorithmic core and coefficients are the same.
400 Can be applied in the complex domain.
402 Parameters
403 ----------
404 fun : callable
405 Right-hand side of the system. The calling signature is ``fun(t, y)``.
406 Here, ``t`` is a scalar, and there are two options for the ndarray ``y``:
407 It can either have shape (n,); then ``fun`` must return array_like with
408 shape (n,). Alternatively it can have shape (n, k); then ``fun``
409 must return an array_like with shape (n, k), i.e. each column
410 corresponds to a single column in ``y``. The choice between the two
411 options is determined by `vectorized` argument (see below).
412 t0 : float
413 Initial time.
414 y0 : array_like, shape (n,)
415 Initial state.
416 t_bound : float
417 Boundary time - the integration won't continue beyond it. It also
418 determines the direction of the integration.
419 first_step : float or None, optional
420 Initial step size. Default is ``None`` which means that the algorithm
421 should choose.
422 max_step : float, optional
423 Maximum allowed step size. Default is np.inf, i.e. the step size is not
424 bounded and determined solely by the solver.
425 rtol, atol : float and array_like, optional
426 Relative and absolute tolerances. The solver keeps the local error
427 estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a
428 relative accuracy (number of correct digits), while `atol` controls
429 absolute accuracy (number of correct decimal places). To achieve the
430 desired `rtol`, set `atol` to be smaller than the smallest value that
431 can be expected from ``rtol * abs(y)`` so that `rtol` dominates the
432 allowable error. If `atol` is larger than ``rtol * abs(y)`` the
433 number of correct digits is not guaranteed. Conversely, to achieve the
434 desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller
435 than `atol`. If components of y have different scales, it might be
436 beneficial to set different `atol` values for different components by
437 passing array_like with shape (n,) for `atol`. Default values are
438 1e-3 for `rtol` and 1e-6 for `atol`.
439 vectorized : bool, optional
440 Whether `fun` is implemented in a vectorized fashion. Default is False.
442 Attributes
443 ----------
444 n : int
445 Number of equations.
446 status : string
447 Current status of the solver: 'running', 'finished' or 'failed'.
448 t_bound : float
449 Boundary time.
450 direction : float
451 Integration direction: +1 or -1.
452 t : float
453 Current time.
454 y : ndarray
455 Current state.
456 t_old : float
457 Previous time. None if no steps were made yet.
458 step_size : float
459 Size of the last successful step. None if no steps were made yet.
460 nfev : int
461 Number evaluations of the system's right-hand side.
462 njev : int
463 Number of evaluations of the Jacobian. Is always 0 for this solver
464 as it does not use the Jacobian.
465 nlu : int
466 Number of LU decompositions. Is always 0 for this solver.
468 References
469 ----------
470 .. [1] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential
471 Equations I: Nonstiff Problems", Sec. II.
472 .. [2] `Page with original Fortran code of DOP853
473 <http://www.unige.ch/~hairer/software.html>`_.
474 """
475 n_stages = dop853_coefficients.N_STAGES
476 order = 8
477 error_estimator_order = 7
478 A = dop853_coefficients.A[:n_stages, :n_stages]
479 B = dop853_coefficients.B
480 C = dop853_coefficients.C[:n_stages]
481 E3 = dop853_coefficients.E3
482 E5 = dop853_coefficients.E5
483 D = dop853_coefficients.D
485 A_EXTRA = dop853_coefficients.A[n_stages + 1:]
486 C_EXTRA = dop853_coefficients.C[n_stages + 1:]
488 def __init__(self, fun, t0, y0, t_bound, max_step=np.inf,
489 rtol=1e-3, atol=1e-6, vectorized=False,
490 first_step=None, **extraneous):
491 super().__init__(fun, t0, y0, t_bound, max_step, rtol, atol,
492 vectorized, first_step, **extraneous)
493 self.K_extended = np.empty((dop853_coefficients.N_STAGES_EXTENDED,
494 self.n), dtype=self.y.dtype)
495 self.K = self.K_extended[:self.n_stages + 1]
497 def _estimate_error(self, K, h): # Left for testing purposes.
498 err5 = np.dot(K.T, self.E5)
499 err3 = np.dot(K.T, self.E3)
500 denom = np.hypot(np.abs(err5), 0.1 * np.abs(err3))
501 correction_factor = np.ones_like(err5)
502 mask = denom > 0
503 correction_factor[mask] = np.abs(err5[mask]) / denom[mask]
504 return h * err5 * correction_factor
506 def _estimate_error_norm(self, K, h, scale):
507 err5 = np.dot(K.T, self.E5) / scale
508 err3 = np.dot(K.T, self.E3) / scale
509 err5_norm_2 = np.linalg.norm(err5)**2
510 err3_norm_2 = np.linalg.norm(err3)**2
511 if err5_norm_2 == 0 and err3_norm_2 == 0:
512 return 0.0
513 denom = err5_norm_2 + 0.01 * err3_norm_2
514 return np.abs(h) * err5_norm_2 / np.sqrt(denom * len(scale))
516 def _dense_output_impl(self):
517 K = self.K_extended
518 h = self.h_previous
519 for s, (a, c) in enumerate(zip(self.A_EXTRA, self.C_EXTRA),
520 start=self.n_stages + 1):
521 dy = np.dot(K[:s].T, a[:s]) * h
522 K[s] = self.fun(self.t_old + c * h, self.y_old + dy)
524 F = np.empty((dop853_coefficients.INTERPOLATOR_POWER, self.n),
525 dtype=self.y_old.dtype)
527 f_old = K[0]
528 delta_y = self.y - self.y_old
530 F[0] = delta_y
531 F[1] = h * f_old - delta_y
532 F[2] = 2 * delta_y - h * (self.f + f_old)
533 F[3:] = h * np.dot(self.D, K)
535 return Dop853DenseOutput(self.t_old, self.t, self.y_old, F)
538class RkDenseOutput(DenseOutput):
539 def __init__(self, t_old, t, y_old, Q):
540 super().__init__(t_old, t)
541 self.h = t - t_old
542 self.Q = Q
543 self.order = Q.shape[1] - 1
544 self.y_old = y_old
546 def _call_impl(self, t):
547 x = (t - self.t_old) / self.h
548 if t.ndim == 0:
549 p = np.tile(x, self.order + 1)
550 p = np.cumprod(p)
551 else:
552 p = np.tile(x, (self.order + 1, 1))
553 p = np.cumprod(p, axis=0)
554 y = self.h * np.dot(self.Q, p)
555 if y.ndim == 2:
556 y += self.y_old[:, None]
557 else:
558 y += self.y_old
560 return y
563class Dop853DenseOutput(DenseOutput):
564 def __init__(self, t_old, t, y_old, F):
565 super().__init__(t_old, t)
566 self.h = t - t_old
567 self.F = F
568 self.y_old = y_old
570 def _call_impl(self, t):
571 x = (t - self.t_old) / self.h
573 if t.ndim == 0:
574 y = np.zeros_like(self.y_old)
575 else:
576 x = x[:, None]
577 y = np.zeros((len(x), len(self.y_old)), dtype=self.y_old.dtype)
579 for i, f in enumerate(reversed(self.F)):
580 y += f
581 if i % 2 == 0:
582 y *= x
583 else:
584 y *= 1 - x
585 y += self.y_old
587 return y.T