Coverage for /pythoncovmergedfiles/medio/medio/usr/local/lib/python3.8/site-packages/scipy/integrate/

1import numpy as np

2from scipy.linalg import lu_factor, lu_solve

3from scipy.sparse import csc_matrix, issparse, eye

4from scipy.sparse.linalg import splu

5from scipy.optimize._numdiff import group_columns

6from .common import (validate_max_step, validate_tol, select_initial_step,

7 norm, num_jac, EPS, warn_extraneous,

8 validate_first_step)

9from .base import OdeSolver, DenseOutput

11S6 = 6 ** 0.5

13# Butcher tableau. A is not used directly, see below.

14C = np.array([(4 - S6) / 10, (4 + S6) / 10, 1])

15E = np.array([-13 - 7 * S6, -13 + 7 * S6, -1]) / 3

17# Eigendecomposition of A is done: A = T L T**-1. There is 1 real eigenvalue

18# and a complex conjugate pair. They are written below.

19MU_REAL = 3 + 3 ** (2 / 3) - 3 ** (1 / 3)

20MU_COMPLEX = (3 + 0.5 * (3 ** (1 / 3) - 3 ** (2 / 3))

21 - 0.5j * (3 ** (5 / 6) + 3 ** (7 / 6)))

23# These are transformation matrices.

24T = np.array([

25 [0.09443876248897524, -0.14125529502095421, 0.03002919410514742],

26 [0.25021312296533332, 0.20412935229379994, -0.38294211275726192],

27 [1, 1, 0]])

28TI = np.array([

29 [4.17871859155190428, 0.32768282076106237, 0.52337644549944951],

30 [-4.17871859155190428, -0.32768282076106237, 0.47662355450055044],

31 [0.50287263494578682, -2.57192694985560522, 0.59603920482822492]])

32# These linear combinations are used in the algorithm.

33TI_REAL = TI[0]

34TI_COMPLEX = TI[1] + 1j * TI[2]

36# Interpolator coefficients.

37P = np.array([

38 [13/3 + 7*S6/3, -23/3 - 22*S6/3, 10/3 + 5 * S6],

39 [13/3 - 7*S6/3, -23/3 + 22*S6/3, 10/3 - 5 * S6],

40 [1/3, -8/3, 10/3]])

43NEWTON_MAXITER = 6 # Maximum number of Newton iterations.

44MIN_FACTOR = 0.2 # Minimum allowed decrease in a step size.

45MAX_FACTOR = 10 # Maximum allowed increase in a step size.

48def solve_collocation_system(fun, t, y, h, Z0, scale, tol,

49 LU_real, LU_complex, solve_lu):

50 """Solve the collocation system.

52 Parameters

53 ----------

54 fun : callable

55 Right-hand side of the system.

56 t : float

57 Current time.

58 y : ndarray, shape (n,)

59 Current state.

60 h : float

61 Step to try.

62 Z0 : ndarray, shape (3, n)

63 Initial guess for the solution. It determines new values of `y` at

64 ``t + h * C`` as ``y + Z0``, where ``C`` is the Radau method constants.

65 scale : ndarray, shape (n)

66 Problem tolerance scale, i.e. ``rtol * abs(y) + atol``.

67 tol : float

68 Tolerance to which solve the system. This value is compared with

69 the normalized by `scale` error.

70 LU_real, LU_complex

71 LU decompositions of the system Jacobians.

72 solve_lu : callable

73 Callable which solves a linear system given a LU decomposition. The

74 signature is ``solve_lu(LU, b)``.

76 Returns

77 -------

78 converged : bool

79 Whether iterations converged.

80 n_iter : int

81 Number of completed iterations.

82 Z : ndarray, shape (3, n)

83 Found solution.

84 rate : float

85 The rate of convergence.

86 """

87 n = y.shape[0]

88 M_real = MU_REAL / h

89 M_complex = MU_COMPLEX / h

91 W = TI.dot(Z0)

92 Z = Z0

94 F = np.empty((3, n))

95 ch = h * C

97 dW_norm_old = None

98 dW = np.empty_like(W)

99 converged = False

100 rate = None

101 for k in range(NEWTON_MAXITER):

102 for i in range(3):

103 F[i] = fun(t + ch[i], y + Z[i])

104

105 if not np.all(np.isfinite(F)):

106 break

107

108 f_real = F.T.dot(TI_REAL) - M_real * W[0]

109 f_complex = F.T.dot(TI_COMPLEX) - M_complex * (W[1] + 1j * W[2])

110

111 dW_real = solve_lu(LU_real, f_real)

112 dW_complex = solve_lu(LU_complex, f_complex)

113

114 dW[0] = dW_real

115 dW[1] = dW_complex.real

116 dW[2] = dW_complex.imag

117

118 dW_norm = norm(dW / scale)

119 if dW_norm_old is not None:

120 rate = dW_norm / dW_norm_old

121

122 if (rate is not None and (rate >= 1 or

123 rate ** (NEWTON_MAXITER - k) / (1 - rate) * dW_norm > tol)):

124 break

125

126 W += dW

127 Z = T.dot(W)

128

129 if (dW_norm == 0 or

130 rate is not None and rate / (1 - rate) * dW_norm < tol):

131 converged = True

132 break

133

134 dW_norm_old = dW_norm

135

136 return converged, k + 1, Z, rate

137

138

139def predict_factor(h_abs, h_abs_old, error_norm, error_norm_old):

140 """Predict by which factor to increase/decrease the step size.

141

142 The algorithm is described in [1]_.

143

144 Parameters

145 ----------

146 h_abs, h_abs_old : float

147 Current and previous values of the step size, `h_abs_old` can be None

148 (see Notes).

149 error_norm, error_norm_old : float

150 Current and previous values of the error norm, `error_norm_old` can

151 be None (see Notes).

152

153 Returns

154 -------

155 factor : float

156 Predicted factor.

157

158 Notes

159 -----

160 If `h_abs_old` and `error_norm_old` are both not None then a two-step

161 algorithm is used, otherwise a one-step algorithm is used.

162

163 References

164 ----------

165 .. [1] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential

166 Equations II: Stiff and Differential-Algebraic Problems", Sec. IV.8.

167 """

168 if error_norm_old is None or h_abs_old is None or error_norm == 0:

169 multiplier = 1

170 else:

171 multiplier = h_abs / h_abs_old * (error_norm_old / error_norm) ** 0.25

172

173 with np.errstate(divide='ignore'):

174 factor = min(1, multiplier) * error_norm ** -0.25

175

176 return factor

177

178

179class Radau(OdeSolver):

180 """Implicit Runge-Kutta method of Radau IIA family of order 5.

181

182 The implementation follows [1]_. The error is controlled with a

183 third-order accurate embedded formula. A cubic polynomial which satisfies

184 the collocation conditions is used for the dense output.

185

186 Parameters

187 ----------

188 fun : callable

189 Right-hand side of the system. The calling signature is ``fun(t, y)``.

190 Here ``t`` is a scalar, and there are two options for the ndarray ``y``:

191 It can either have shape (n,); then ``fun`` must return array_like with

192 shape (n,). Alternatively it can have shape (n, k); then ``fun``

193 must return an array_like with shape (n, k), i.e., each column

194 corresponds to a single column in ``y``. The choice between the two

195 options is determined by `vectorized` argument (see below). The

196 vectorized implementation allows a faster approximation of the Jacobian

197 by finite differences (required for this solver).

198 t0 : float

199 Initial time.

200 y0 : array_like, shape (n,)

201 Initial state.

202 t_bound : float

203 Boundary time - the integration won't continue beyond it. It also

204 determines the direction of the integration.

205 first_step : float or None, optional

206 Initial step size. Default is ``None`` which means that the algorithm

207 should choose.

208 max_step : float, optional

209 Maximum allowed step size. Default is np.inf, i.e., the step size is not

210 bounded and determined solely by the solver.

211 rtol, atol : float and array_like, optional

212 Relative and absolute tolerances. The solver keeps the local error

213 estimates less than ``atol + rtol * abs(y)``. HHere `rtol` controls a

214 relative accuracy (number of correct digits), while `atol` controls

215 absolute accuracy (number of correct decimal places). To achieve the

216 desired `rtol`, set `atol` to be smaller than the smallest value that

217 can be expected from ``rtol * abs(y)`` so that `rtol` dominates the

218 allowable error. If `atol` is larger than ``rtol * abs(y)`` the

219 number of correct digits is not guaranteed. Conversely, to achieve the

220 desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller

221 than `atol`. If components of y have different scales, it might be

222 beneficial to set different `atol` values for different components by

223 passing array_like with shape (n,) for `atol`. Default values are

224 1e-3 for `rtol` and 1e-6 for `atol`.

225 jac : {None, array_like, sparse_matrix, callable}, optional

226 Jacobian matrix of the right-hand side of the system with respect to

227 y, required by this method. The Jacobian matrix has shape (n, n) and

228 its element (i, j) is equal to ``d f_i / d y_j``.

229 There are three ways to define the Jacobian:

230

231 * If array_like or sparse_matrix, the Jacobian is assumed to

232 be constant.

233 * If callable, the Jacobian is assumed to depend on both

234 t and y; it will be called as ``jac(t, y)`` as necessary.

235 For the 'Radau' and 'BDF' methods, the return value might be a

236 sparse matrix.

237 * If None (default), the Jacobian will be approximated by

238 finite differences.

239

240 It is generally recommended to provide the Jacobian rather than

241 relying on a finite-difference approximation.

242 jac_sparsity : {None, array_like, sparse matrix}, optional

243 Defines a sparsity structure of the Jacobian matrix for a

244 finite-difference approximation. Its shape must be (n, n). This argument

245 is ignored if `jac` is not `None`. If the Jacobian has only few non-zero

246 elements in *each* row, providing the sparsity structure will greatly

247 speed up the computations [2]_. A zero entry means that a corresponding

248 element in the Jacobian is always zero. If None (default), the Jacobian

249 is assumed to be dense.

250 vectorized : bool, optional

251 Whether `fun` is implemented in a vectorized fashion. Default is False.

252

253 Attributes

254 ----------

255 n : int

256 Number of equations.

257 status : string

258 Current status of the solver: 'running', 'finished' or 'failed'.

259 t_bound : float

260 Boundary time.

261 direction : float

262 Integration direction: +1 or -1.

263 t : float

264 Current time.

265 y : ndarray

266 Current state.

267 t_old : float

268 Previous time. None if no steps were made yet.

269 step_size : float

270 Size of the last successful step. None if no steps were made yet.

271 nfev : int

272 Number of evaluations of the right-hand side.

273 njev : int

274 Number of evaluations of the Jacobian.

275 nlu : int

276 Number of LU decompositions.

277

278 References

279 ----------

280 .. [1] E. Hairer, G. Wanner, "Solving Ordinary Differential Equations II:

281 Stiff and Differential-Algebraic Problems", Sec. IV.8.

282 .. [2] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of

283 sparse Jacobian matrices", Journal of the Institute of Mathematics

284 and its Applications, 13, pp. 117-120, 1974.

285 """

286 def __init__(self, fun, t0, y0, t_bound, max_step=np.inf,

287 rtol=1e-3, atol=1e-6, jac=None, jac_sparsity=None,

288 vectorized=False, first_step=None, **extraneous):

289 warn_extraneous(extraneous)

290 super().__init__(fun, t0, y0, t_bound, vectorized)

291 self.y_old = None

292 self.max_step = validate_max_step(max_step)

293 self.rtol, self.atol = validate_tol(rtol, atol, self.n)

294 self.f = self.fun(self.t, self.y)

295 # Select initial step assuming the same order which is used to control

296 # the error.

297 if first_step is None:

298 self.h_abs = select_initial_step(

299 self.fun, self.t, self.y, self.f, self.direction,

300 3, self.rtol, self.atol)

301 else:

302 self.h_abs = validate_first_step(first_step, t0, t_bound)

303 self.h_abs_old = None

304 self.error_norm_old = None

305

306 self.newton_tol = max(10 * EPS / rtol, min(0.03, rtol ** 0.5))

307 self.sol = None

308

309 self.jac_factor = None

310 self.jac, self.J = self._validate_jac(jac, jac_sparsity)

311 if issparse(self.J):

312 def lu(A):

313 self.nlu += 1

314 return splu(A)

315

316 def solve_lu(LU, b):

317 return LU.solve(b)

318

319 I = eye(self.n, format='csc')

320 else:

321 def lu(A):

322 self.nlu += 1

323 return lu_factor(A, overwrite_a=True)

324

325 def solve_lu(LU, b):

326 return lu_solve(LU, b, overwrite_b=True)

327

328 I = np.identity(self.n)

329

330 self.lu = lu

331 self.solve_lu = solve_lu

332 self.I = I

333

334 self.current_jac = True

335 self.LU_real = None

336 self.LU_complex = None

337 self.Z = None

338

339 def _validate_jac(self, jac, sparsity):

340 t0 = self.t

341 y0 = self.y

342

343 if jac is None:

344 if sparsity is not None:

345 if issparse(sparsity):

346 sparsity = csc_matrix(sparsity)

347 groups = group_columns(sparsity)

348 sparsity = (sparsity, groups)

349

350 def jac_wrapped(t, y, f):

351 self.njev += 1

352 J, self.jac_factor = num_jac(self.fun_vectorized, t, y, f,

353 self.atol, self.jac_factor,

354 sparsity)

355 return J

356 J = jac_wrapped(t0, y0, self.f)

357 elif callable(jac):

358 J = jac(t0, y0)

359 self.njev = 1

360 if issparse(J):

361 J = csc_matrix(J)

362

363 def jac_wrapped(t, y, _=None):

364 self.njev += 1

365 return csc_matrix(jac(t, y), dtype=float)

366

367 else:

368 J = np.asarray(J, dtype=float)

369

370 def jac_wrapped(t, y, _=None):

371 self.njev += 1

372 return np.asarray(jac(t, y), dtype=float)

373

374 if J.shape != (self.n, self.n):

375 raise ValueError("`jac` is expected to have shape {}, but "

376 "actually has {}."

377 .format((self.n, self.n), J.shape))

378 else:

379 if issparse(jac):

380 J = csc_matrix(jac)

381 else:

382 J = np.asarray(jac, dtype=float)

383

384 if J.shape != (self.n, self.n):

385 raise ValueError("`jac` is expected to have shape {}, but "

386 "actually has {}."

387 .format((self.n, self.n), J.shape))

388 jac_wrapped = None

389

390 return jac_wrapped, J

391

392 def _step_impl(self):

393 t = self.t

394 y = self.y

395 f = self.f

396

397 max_step = self.max_step

398 atol = self.atol

399 rtol = self.rtol

400

401 min_step = 10 * np.abs(np.nextafter(t, self.direction * np.inf) - t)

402 if self.h_abs > max_step:

403 h_abs = max_step

404 h_abs_old = None

405 error_norm_old = None

406 elif self.h_abs < min_step:

407 h_abs = min_step

408 h_abs_old = None

409 error_norm_old = None

410 else:

411 h_abs = self.h_abs

412 h_abs_old = self.h_abs_old

413 error_norm_old = self.error_norm_old

414

415 J = self.J

416 LU_real = self.LU_real

417 LU_complex = self.LU_complex

418

419 current_jac = self.current_jac

420 jac = self.jac

421

422 rejected = False

423 step_accepted = False

424 message = None

425 while not step_accepted:

426 if h_abs < min_step:

427 return False, self.TOO_SMALL_STEP

428

429 h = h_abs * self.direction

430 t_new = t + h

431

432 if self.direction * (t_new - self.t_bound) > 0:

433 t_new = self.t_bound

434

435 h = t_new - t

436 h_abs = np.abs(h)

437

438 if self.sol is None:

439 Z0 = np.zeros((3, y.shape[0]))

440 else:

441 Z0 = self.sol(t + h * C).T - y

442

443 scale = atol + np.abs(y) * rtol

444

445 converged = False

446 while not converged:

447 if LU_real is None or LU_complex is None:

448 LU_real = self.lu(MU_REAL / h * self.I - J)

449 LU_complex = self.lu(MU_COMPLEX / h * self.I - J)

450

451 converged, n_iter, Z, rate = solve_collocation_system(

452 self.fun, t, y, h, Z0, scale, self.newton_tol,

453 LU_real, LU_complex, self.solve_lu)

454

455 if not converged:

456 if current_jac:

457 break

458

459 J = self.jac(t, y, f)

460 current_jac = True

461 LU_real = None

462 LU_complex = None

463

464 if not converged:

465 h_abs *= 0.5

466 LU_real = None

467 LU_complex = None

468 continue

469

470 y_new = y + Z[-1]

471 ZE = Z.T.dot(E) / h

472 error = self.solve_lu(LU_real, f + ZE)

473 scale = atol + np.maximum(np.abs(y), np.abs(y_new)) * rtol

474 error_norm = norm(error / scale)

475 safety = 0.9 * (2 * NEWTON_MAXITER + 1) / (2 * NEWTON_MAXITER

476 + n_iter)

477

478 if rejected and error_norm > 1:

479 error = self.solve_lu(LU_real, self.fun(t, y + error) + ZE)

480 error_norm = norm(error / scale)

481

482 if error_norm > 1:

483 factor = predict_factor(h_abs, h_abs_old,

484 error_norm, error_norm_old)

485 h_abs *= max(MIN_FACTOR, safety * factor)

486

487 LU_real = None

488 LU_complex = None

489 rejected = True

490 else:

491 step_accepted = True

492

493 recompute_jac = jac is not None and n_iter > 2 and rate > 1e-3

494

495 factor = predict_factor(h_abs, h_abs_old, error_norm, error_norm_old)

496 factor = min(MAX_FACTOR, safety * factor)

497

498 if not recompute_jac and factor < 1.2:

499 factor = 1

500 else:

501 LU_real = None

502 LU_complex = None

503

504 f_new = self.fun(t_new, y_new)

505 if recompute_jac:

506 J = jac(t_new, y_new, f_new)

507 current_jac = True

508 elif jac is not None:

509 current_jac = False

510

511 self.h_abs_old = self.h_abs

512 self.error_norm_old = error_norm

513

514 self.h_abs = h_abs * factor

515

516 self.y_old = y

517

518 self.t = t_new

519 self.y = y_new

520 self.f = f_new

521

522 self.Z = Z

523

524 self.LU_real = LU_real

525 self.LU_complex = LU_complex

526 self.current_jac = current_jac

527 self.J = J

528

529 self.t_old = t

530 self.sol = self._compute_dense_output()

531

532 return step_accepted, message

533

534 def _compute_dense_output(self):

535 Q = np.dot(self.Z.T, P)

536 return RadauDenseOutput(self.t_old, self.t, self.y_old, Q)

537

538 def _dense_output_impl(self):

539 return self.sol

540

541

542class RadauDenseOutput(DenseOutput):

543 def __init__(self, t_old, t, y_old, Q):

544 super().__init__(t_old, t)

545 self.h = t - t_old

546 self.Q = Q

547 self.order = Q.shape[1] - 1

548 self.y_old = y_old

549

550 def _call_impl(self, t):

551 x = (t - self.t_old) / self.h

552 if t.ndim == 0:

553 p = np.tile(x, self.order + 1)

554 p = np.cumprod(p)

555 else:

556 p = np.tile(x, (self.order + 1, 1))

557 p = np.cumprod(p, axis=0)

558 # Here we don't multiply by h, not a mistake.

559 y = np.dot(self.Q, p)

560 if y.ndim == 2:

561 y += self.y_old[:, None]

562 else:

563 y += self.y_old

564

565 return y

Coverage for /pythoncovmergedfiles/medio/medio/usr/local/lib/python3.8/site-packages/scipy/integrate/_ivp/radau.py: 12%

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