laywerrobot/lib/python3.6/site-packages/scipy/integrate/_ivp/bdf.py

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2020-08-27 21:55:39 +02:00
from __future__ import division, print_function, absolute_import
import numpy as np
from scipy.linalg import lu_factor, lu_solve
from scipy.sparse import issparse, csc_matrix, eye
from scipy.sparse.linalg import splu
from scipy.optimize._numdiff import group_columns
from .common import (validate_max_step, validate_tol, select_initial_step,
norm, EPS, num_jac, warn_extraneous)
from .base import OdeSolver, DenseOutput
MAX_ORDER = 5
NEWTON_MAXITER = 4
MIN_FACTOR = 0.2
MAX_FACTOR = 10
def compute_R(order, factor):
"""Compute the matrix for changing the differences array."""
I = np.arange(1, order + 1)[:, None]
J = np.arange(1, order + 1)
M = np.zeros((order + 1, order + 1))
M[1:, 1:] = (I - 1 - factor * J) / I
M[0] = 1
return np.cumprod(M, axis=0)
def change_D(D, order, factor):
"""Change differences array in-place when step size is changed."""
R = compute_R(order, factor)
U = compute_R(order, 1)
RU = R.dot(U)
D[:order + 1] = np.dot(RU.T, D[:order + 1])
def solve_bdf_system(fun, t_new, y_predict, c, psi, LU, solve_lu, scale, tol):
"""Solve the algebraic system resulting from BDF method."""
d = 0
y = y_predict.copy()
dy_norm_old = None
converged = False
for k in range(NEWTON_MAXITER):
f = fun(t_new, y)
if not np.all(np.isfinite(f)):
break
dy = solve_lu(LU, c * f - psi - d)
dy_norm = norm(dy / scale)
if dy_norm_old is None:
rate = None
else:
rate = dy_norm / dy_norm_old
if (rate is not None and (rate >= 1 or
rate ** (NEWTON_MAXITER - k) / (1 - rate) * dy_norm > tol)):
break
y += dy
d += dy
if (dy_norm == 0 or
rate is not None and rate / (1 - rate) * dy_norm < tol):
converged = True
break
dy_norm_old = dy_norm
return converged, k + 1, y, d
class BDF(OdeSolver):
"""Implicit method based on backward-differentiation formulas.
This is a variable order method with the order varying automatically from
1 to 5. The general framework of the BDF algorithm is described in [1]_.
This class implements a quasi-constant step size as explained in [2]_.
The error estimation strategy for the constant-step BDF is derived in [3]_.
An accuracy enhancement using modified formulas (NDF) [2]_ is also implemented.
Can be applied in the complex domain.
Parameters
----------
fun : callable
Right-hand side of the system. The calling signature is ``fun(t, y)``.
Here ``t`` is a scalar, and there are two options for the ndarray ``y``:
It can either have shape (n,); then ``fun`` must return array_like with
shape (n,). Alternatively it can have shape (n, k); then ``fun``
must return an array_like with shape (n, k), i.e. each column
corresponds to a single column in ``y``. The choice between the two
options is determined by `vectorized` argument (see below). The
vectorized implementation allows a faster approximation of the Jacobian
by finite differences (required for this solver).
t0 : float
Initial time.
y0 : array_like, shape (n,)
Initial state.
t_bound : float
Boundary time - the integration won't continue beyond it. It also
determines the direction of the integration.
max_step : float, optional
Maximum allowed step size. Default is np.inf, i.e. the step size is not
bounded and determined solely by the solver.
rtol, atol : float and array_like, optional
Relative and absolute tolerances. The solver keeps the local error
estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a
relative accuracy (number of correct digits). But if a component of `y`
is approximately below `atol`, the error only needs to fall within
the same `atol` threshold, and the number of correct digits is not
guaranteed. If components of y have different scales, it might be
beneficial to set different `atol` values for different components by
passing array_like with shape (n,) for `atol`. Default values are
1e-3 for `rtol` and 1e-6 for `atol`.
jac : {None, array_like, sparse_matrix, callable}, optional
Jacobian matrix of the right-hand side of the system with respect to y,
required by this method. The Jacobian matrix has shape (n, n) and its
element (i, j) is equal to ``d f_i / d y_j``.
There are three ways to define the Jacobian:
* If array_like or sparse_matrix, the Jacobian is assumed to
be constant.
* If callable, the Jacobian is assumed to depend on both
t and y; it will be called as ``jac(t, y)`` as necessary.
For the 'Radau' and 'BDF' methods, the return value might be a
sparse matrix.
* If None (default), the Jacobian will be approximated by
finite differences.
It is generally recommended to provide the Jacobian rather than
relying on a finite-difference approximation.
jac_sparsity : {None, array_like, sparse matrix}, optional
Defines a sparsity structure of the Jacobian matrix for a
finite-difference approximation. Its shape must be (n, n). This argument
is ignored if `jac` is not `None`. If the Jacobian has only few non-zero
elements in *each* row, providing the sparsity structure will greatly
speed up the computations [4]_. A zero entry means that a corresponding
element in the Jacobian is always zero. If None (default), the Jacobian
is assumed to be dense.
vectorized : bool, optional
Whether `fun` is implemented in a vectorized fashion. Default is False.
Attributes
----------
n : int
Number of equations.
status : string
Current status of the solver: 'running', 'finished' or 'failed'.
t_bound : float
Boundary time.
direction : float
Integration direction: +1 or -1.
t : float
Current time.
y : ndarray
Current state.
t_old : float
Previous time. None if no steps were made yet.
step_size : float
Size of the last successful step. None if no steps were made yet.
nfev : int
Number of evaluations of the right-hand side.
njev : int
Number of evaluations of the Jacobian.
nlu : int
Number of LU decompositions.
References
----------
.. [1] G. D. Byrne, A. C. Hindmarsh, "A Polyalgorithm for the Numerical
Solution of Ordinary Differential Equations", ACM Transactions on
Mathematical Software, Vol. 1, No. 1, pp. 71-96, March 1975.
.. [2] L. F. Shampine, M. W. Reichelt, "THE MATLAB ODE SUITE", SIAM J. SCI.
COMPUTE., Vol. 18, No. 1, pp. 1-22, January 1997.
.. [3] E. Hairer, G. Wanner, "Solving Ordinary Differential Equations I:
Nonstiff Problems", Sec. III.2.
.. [4] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of
sparse Jacobian matrices", Journal of the Institute of Mathematics
and its Applications, 13, pp. 117-120, 1974.
"""
def __init__(self, fun, t0, y0, t_bound, max_step=np.inf,
rtol=1e-3, atol=1e-6, jac=None, jac_sparsity=None,
vectorized=False, **extraneous):
warn_extraneous(extraneous)
super(BDF, self).__init__(fun, t0, y0, t_bound, vectorized,
support_complex=True)
self.max_step = validate_max_step(max_step)
self.rtol, self.atol = validate_tol(rtol, atol, self.n)
f = self.fun(self.t, self.y)
self.h_abs = select_initial_step(self.fun, self.t, self.y, f,
self.direction, 1,
self.rtol, self.atol)
self.h_abs_old = None
self.error_norm_old = None
self.newton_tol = max(10 * EPS / rtol, min(0.03, rtol ** 0.5))
self.jac_factor = None
self.jac, self.J = self._validate_jac(jac, jac_sparsity)
if issparse(self.J):
def lu(A):
self.nlu += 1
return splu(A)
def solve_lu(LU, b):
return LU.solve(b)
I = eye(self.n, format='csc', dtype=self.y.dtype)
else:
def lu(A):
self.nlu += 1
return lu_factor(A, overwrite_a=True)
def solve_lu(LU, b):
return lu_solve(LU, b, overwrite_b=True)
I = np.identity(self.n, dtype=self.y.dtype)
self.lu = lu
self.solve_lu = solve_lu
self.I = I
kappa = np.array([0, -0.1850, -1/9, -0.0823, -0.0415, 0])
self.gamma = np.hstack((0, np.cumsum(1 / np.arange(1, MAX_ORDER + 1))))
self.alpha = (1 - kappa) * self.gamma
self.error_const = kappa * self.gamma + 1 / np.arange(1, MAX_ORDER + 2)
D = np.empty((MAX_ORDER + 3, self.n), dtype=self.y.dtype)
D[0] = self.y
D[1] = f * self.h_abs * self.direction
self.D = D
self.order = 1
self.n_equal_steps = 0
self.LU = None
def _validate_jac(self, jac, sparsity):
t0 = self.t
y0 = self.y
if jac is None:
if sparsity is not None:
if issparse(sparsity):
sparsity = csc_matrix(sparsity)
groups = group_columns(sparsity)
sparsity = (sparsity, groups)
def jac_wrapped(t, y):
self.njev += 1
f = self.fun_single(t, y)
J, self.jac_factor = num_jac(self.fun_vectorized, t, y, f,
self.atol, self.jac_factor,
sparsity)
return J
J = jac_wrapped(t0, y0)
elif callable(jac):
J = jac(t0, y0)
self.njev += 1
if issparse(J):
J = csc_matrix(J, dtype=y0.dtype)
def jac_wrapped(t, y):
self.njev += 1
return csc_matrix(jac(t, y), dtype=y0.dtype)
else:
J = np.asarray(J, dtype=y0.dtype)
def jac_wrapped(t, y):
self.njev += 1
return np.asarray(jac(t, y), dtype=y0.dtype)
if J.shape != (self.n, self.n):
raise ValueError("`jac` is expected to have shape {}, but "
"actually has {}."
.format((self.n, self.n), J.shape))
else:
if issparse(jac):
J = csc_matrix(jac, dtype=y0.dtype)
else:
J = np.asarray(jac, dtype=y0.dtype)
if J.shape != (self.n, self.n):
raise ValueError("`jac` is expected to have shape {}, but "
"actually has {}."
.format((self.n, self.n), J.shape))
jac_wrapped = None
return jac_wrapped, J
def _step_impl(self):
t = self.t
D = self.D
max_step = self.max_step
min_step = 10 * np.abs(np.nextafter(t, self.direction * np.inf) - t)
if self.h_abs > max_step:
h_abs = max_step
change_D(D, self.order, max_step / self.h_abs)
self.n_equal_steps = 0
elif self.h_abs < min_step:
h_abs = min_step
change_D(D, self.order, min_step / self.h_abs)
self.n_equal_steps = 0
else:
h_abs = self.h_abs
atol = self.atol
rtol = self.rtol
order = self.order
alpha = self.alpha
gamma = self.gamma
error_const = self.error_const
J = self.J
LU = self.LU
current_jac = self.jac is None
step_accepted = False
while not step_accepted:
if h_abs < min_step:
return False, self.TOO_SMALL_STEP
h = h_abs * self.direction
t_new = t + h
if self.direction * (t_new - self.t_bound) > 0:
t_new = self.t_bound
change_D(D, order, np.abs(t_new - t) / h_abs)
self.n_equal_steps = 0
LU = None
h = t_new - t
h_abs = np.abs(h)
y_predict = np.sum(D[:order + 1], axis=0)
scale = atol + rtol * np.abs(y_predict)
psi = np.dot(D[1: order + 1].T, gamma[1: order + 1]) / alpha[order]
converged = False
c = h / alpha[order]
while not converged:
if LU is None:
LU = self.lu(self.I - c * J)
converged, n_iter, y_new, d = solve_bdf_system(
self.fun, t_new, y_predict, c, psi, LU, self.solve_lu,
scale, self.newton_tol)
if not converged:
if current_jac:
break
J = self.jac(t_new, y_predict)
LU = None
current_jac = True
if not converged:
factor = 0.5
h_abs *= factor
change_D(D, order, factor)
self.n_equal_steps = 0
LU = None
continue
safety = 0.9 * (2 * NEWTON_MAXITER + 1) / (2 * NEWTON_MAXITER
+ n_iter)
scale = atol + rtol * np.abs(y_new)
error = error_const[order] * d
error_norm = norm(error / scale)
if error_norm > 1:
factor = max(MIN_FACTOR,
safety * error_norm ** (-1 / (order + 1)))
h_abs *= factor
change_D(D, order, factor)
self.n_equal_steps = 0
# As we didn't have problems with convergence, we don't
# reset LU here.
else:
step_accepted = True
self.n_equal_steps += 1
self.t = t_new
self.y = y_new
self.h_abs = h_abs
self.J = J
self.LU = LU
# Update differences. The principal relation here is
# D^{j + 1} y_n = D^{j} y_n - D^{j} y_{n - 1}. Keep in mind that D
# contained difference for previous interpolating polynomial and
# d = D^{k + 1} y_n. Thus this elegant code follows.
D[order + 2] = d - D[order + 1]
D[order + 1] = d
for i in reversed(range(order + 1)):
D[i] += D[i + 1]
if self.n_equal_steps < order + 1:
return True, None
if order > 1:
error_m = error_const[order - 1] * D[order]
error_m_norm = norm(error_m / scale)
else:
error_m_norm = np.inf
if order < MAX_ORDER:
error_p = error_const[order + 1] * D[order + 2]
error_p_norm = norm(error_p / scale)
else:
error_p_norm = np.inf
error_norms = np.array([error_m_norm, error_norm, error_p_norm])
factors = error_norms ** (-1 / np.arange(order, order + 3))
delta_order = np.argmax(factors) - 1
order += delta_order
self.order = order
factor = min(MAX_FACTOR, safety * np.max(factors))
self.h_abs *= factor
change_D(D, order, factor)
self.n_equal_steps = 0
self.LU = None
return True, None
def _dense_output_impl(self):
return BdfDenseOutput(self.t_old, self.t, self.h_abs * self.direction,
self.order, self.D[:self.order + 1].copy())
class BdfDenseOutput(DenseOutput):
def __init__(self, t_old, t, h, order, D):
super(BdfDenseOutput, self).__init__(t_old, t)
self.order = order
self.t_shift = self.t - h * np.arange(self.order)
self.denom = h * (1 + np.arange(self.order))
self.D = D
def _call_impl(self, t):
if t.ndim == 0:
x = (t - self.t_shift) / self.denom
p = np.cumprod(x)
else:
x = (t - self.t_shift[:, None]) / self.denom[:, None]
p = np.cumprod(x, axis=0)
y = np.dot(self.D[1:].T, p)
if y.ndim == 1:
y += self.D[0]
else:
y += self.D[0, :, None]
return y