434 lines
14 KiB
Python
434 lines
14 KiB
Python
from itertools import groupby
|
|
from warnings import warn
|
|
import numpy as np
|
|
from scipy.sparse import find, coo_matrix
|
|
|
|
|
|
EPS = np.finfo(float).eps
|
|
|
|
|
|
def validate_first_step(first_step, t0, t_bound):
|
|
"""Assert that first_step is valid and return it."""
|
|
if first_step <= 0:
|
|
raise ValueError("`first_step` must be positive.")
|
|
if first_step > np.abs(t_bound - t0):
|
|
raise ValueError("`first_step` exceeds bounds.")
|
|
return first_step
|
|
|
|
|
|
def validate_max_step(max_step):
|
|
"""Assert that max_Step is valid and return it."""
|
|
if max_step <= 0:
|
|
raise ValueError("`max_step` must be positive.")
|
|
return max_step
|
|
|
|
|
|
def warn_extraneous(extraneous):
|
|
"""Display a warning for extraneous keyword arguments.
|
|
|
|
The initializer of each solver class is expected to collect keyword
|
|
arguments that it doesn't understand and warn about them. This function
|
|
prints a warning for each key in the supplied dictionary.
|
|
|
|
Parameters
|
|
----------
|
|
extraneous : dict
|
|
Extraneous keyword arguments
|
|
"""
|
|
if extraneous:
|
|
warn("The following arguments have no effect for a chosen solver: {}."
|
|
.format(", ".join("`{}`".format(x) for x in extraneous)))
|
|
|
|
|
|
def validate_tol(rtol, atol, n):
|
|
"""Validate tolerance values."""
|
|
|
|
if np.any(rtol < 100 * EPS):
|
|
warn("At least one element of `rtol` is too small. "
|
|
f"Setting `rtol = np.maximum(rtol, {100 * EPS})`.")
|
|
rtol = np.maximum(rtol, 100 * EPS)
|
|
|
|
atol = np.asarray(atol)
|
|
if atol.ndim > 0 and atol.shape != (n,):
|
|
raise ValueError("`atol` has wrong shape.")
|
|
|
|
if np.any(atol < 0):
|
|
raise ValueError("`atol` must be positive.")
|
|
|
|
return rtol, atol
|
|
|
|
|
|
def norm(x):
|
|
"""Compute RMS norm."""
|
|
return np.linalg.norm(x) / x.size ** 0.5
|
|
|
|
|
|
def select_initial_step(fun, t0, y0, f0, direction, order, rtol, atol):
|
|
"""Empirically select a good initial step.
|
|
|
|
The algorithm is described in [1]_.
|
|
|
|
Parameters
|
|
----------
|
|
fun : callable
|
|
Right-hand side of the system.
|
|
t0 : float
|
|
Initial value of the independent variable.
|
|
y0 : ndarray, shape (n,)
|
|
Initial value of the dependent variable.
|
|
f0 : ndarray, shape (n,)
|
|
Initial value of the derivative, i.e., ``fun(t0, y0)``.
|
|
direction : float
|
|
Integration direction.
|
|
order : float
|
|
Error estimator order. It means that the error controlled by the
|
|
algorithm is proportional to ``step_size ** (order + 1)`.
|
|
rtol : float
|
|
Desired relative tolerance.
|
|
atol : float
|
|
Desired absolute tolerance.
|
|
|
|
Returns
|
|
-------
|
|
h_abs : float
|
|
Absolute value of the suggested initial step.
|
|
|
|
References
|
|
----------
|
|
.. [1] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential
|
|
Equations I: Nonstiff Problems", Sec. II.4.
|
|
"""
|
|
if y0.size == 0:
|
|
return np.inf
|
|
|
|
scale = atol + np.abs(y0) * rtol
|
|
d0 = norm(y0 / scale)
|
|
d1 = norm(f0 / scale)
|
|
if d0 < 1e-5 or d1 < 1e-5:
|
|
h0 = 1e-6
|
|
else:
|
|
h0 = 0.01 * d0 / d1
|
|
|
|
y1 = y0 + h0 * direction * f0
|
|
f1 = fun(t0 + h0 * direction, y1)
|
|
d2 = norm((f1 - f0) / scale) / h0
|
|
|
|
if d1 <= 1e-15 and d2 <= 1e-15:
|
|
h1 = max(1e-6, h0 * 1e-3)
|
|
else:
|
|
h1 = (0.01 / max(d1, d2)) ** (1 / (order + 1))
|
|
|
|
return min(100 * h0, h1)
|
|
|
|
|
|
class OdeSolution:
|
|
"""Continuous ODE solution.
|
|
|
|
It is organized as a collection of `DenseOutput` objects which represent
|
|
local interpolants. It provides an algorithm to select a right interpolant
|
|
for each given point.
|
|
|
|
The interpolants cover the range between `t_min` and `t_max` (see
|
|
Attributes below). Evaluation outside this interval is not forbidden, but
|
|
the accuracy is not guaranteed.
|
|
|
|
When evaluating at a breakpoint (one of the values in `ts`) a segment with
|
|
the lower index is selected.
|
|
|
|
Parameters
|
|
----------
|
|
ts : array_like, shape (n_segments + 1,)
|
|
Time instants between which local interpolants are defined. Must
|
|
be strictly increasing or decreasing (zero segment with two points is
|
|
also allowed).
|
|
interpolants : list of DenseOutput with n_segments elements
|
|
Local interpolants. An i-th interpolant is assumed to be defined
|
|
between ``ts[i]`` and ``ts[i + 1]``.
|
|
|
|
Attributes
|
|
----------
|
|
t_min, t_max : float
|
|
Time range of the interpolation.
|
|
"""
|
|
def __init__(self, ts, interpolants):
|
|
ts = np.asarray(ts)
|
|
d = np.diff(ts)
|
|
# The first case covers integration on zero segment.
|
|
if not ((ts.size == 2 and ts[0] == ts[-1])
|
|
or np.all(d > 0) or np.all(d < 0)):
|
|
raise ValueError("`ts` must be strictly increasing or decreasing.")
|
|
|
|
self.n_segments = len(interpolants)
|
|
if ts.shape != (self.n_segments + 1,):
|
|
raise ValueError("Numbers of time stamps and interpolants "
|
|
"don't match.")
|
|
|
|
self.ts = ts
|
|
self.interpolants = interpolants
|
|
if ts[-1] >= ts[0]:
|
|
self.t_min = ts[0]
|
|
self.t_max = ts[-1]
|
|
self.ascending = True
|
|
self.ts_sorted = ts
|
|
else:
|
|
self.t_min = ts[-1]
|
|
self.t_max = ts[0]
|
|
self.ascending = False
|
|
self.ts_sorted = ts[::-1]
|
|
|
|
def _call_single(self, t):
|
|
# Here we preserve a certain symmetry that when t is in self.ts,
|
|
# then we prioritize a segment with a lower index.
|
|
if self.ascending:
|
|
ind = np.searchsorted(self.ts_sorted, t, side='left')
|
|
else:
|
|
ind = np.searchsorted(self.ts_sorted, t, side='right')
|
|
|
|
segment = min(max(ind - 1, 0), self.n_segments - 1)
|
|
if not self.ascending:
|
|
segment = self.n_segments - 1 - segment
|
|
|
|
return self.interpolants[segment](t)
|
|
|
|
def __call__(self, t):
|
|
"""Evaluate the solution.
|
|
|
|
Parameters
|
|
----------
|
|
t : float or array_like with shape (n_points,)
|
|
Points to evaluate at.
|
|
|
|
Returns
|
|
-------
|
|
y : ndarray, shape (n_states,) or (n_states, n_points)
|
|
Computed values. Shape depends on whether `t` is a scalar or a
|
|
1-D array.
|
|
"""
|
|
t = np.asarray(t)
|
|
|
|
if t.ndim == 0:
|
|
return self._call_single(t)
|
|
|
|
order = np.argsort(t)
|
|
reverse = np.empty_like(order)
|
|
reverse[order] = np.arange(order.shape[0])
|
|
t_sorted = t[order]
|
|
|
|
# See comment in self._call_single.
|
|
if self.ascending:
|
|
segments = np.searchsorted(self.ts_sorted, t_sorted, side='left')
|
|
else:
|
|
segments = np.searchsorted(self.ts_sorted, t_sorted, side='right')
|
|
segments -= 1
|
|
segments[segments < 0] = 0
|
|
segments[segments > self.n_segments - 1] = self.n_segments - 1
|
|
if not self.ascending:
|
|
segments = self.n_segments - 1 - segments
|
|
|
|
ys = []
|
|
group_start = 0
|
|
for segment, group in groupby(segments):
|
|
group_end = group_start + len(list(group))
|
|
y = self.interpolants[segment](t_sorted[group_start:group_end])
|
|
ys.append(y)
|
|
group_start = group_end
|
|
|
|
ys = np.hstack(ys)
|
|
ys = ys[:, reverse]
|
|
|
|
return ys
|
|
|
|
|
|
NUM_JAC_DIFF_REJECT = EPS ** 0.875
|
|
NUM_JAC_DIFF_SMALL = EPS ** 0.75
|
|
NUM_JAC_DIFF_BIG = EPS ** 0.25
|
|
NUM_JAC_MIN_FACTOR = 1e3 * EPS
|
|
NUM_JAC_FACTOR_INCREASE = 10
|
|
NUM_JAC_FACTOR_DECREASE = 0.1
|
|
|
|
|
|
def num_jac(fun, t, y, f, threshold, factor, sparsity=None):
|
|
"""Finite differences Jacobian approximation tailored for ODE solvers.
|
|
|
|
This function computes finite difference approximation to the Jacobian
|
|
matrix of `fun` with respect to `y` using forward differences.
|
|
The Jacobian matrix has shape (n, n) and its element (i, j) is equal to
|
|
``d f_i / d y_j``.
|
|
|
|
A special feature of this function is the ability to correct the step
|
|
size from iteration to iteration. The main idea is to keep the finite
|
|
difference significantly separated from its round-off error which
|
|
approximately equals ``EPS * np.abs(f)``. It reduces a possibility of a
|
|
huge error and assures that the estimated derivative are reasonably close
|
|
to the true values (i.e., the finite difference approximation is at least
|
|
qualitatively reflects the structure of the true Jacobian).
|
|
|
|
Parameters
|
|
----------
|
|
fun : callable
|
|
Right-hand side of the system implemented in a vectorized fashion.
|
|
t : float
|
|
Current time.
|
|
y : ndarray, shape (n,)
|
|
Current state.
|
|
f : ndarray, shape (n,)
|
|
Value of the right hand side at (t, y).
|
|
threshold : float
|
|
Threshold for `y` value used for computing the step size as
|
|
``factor * np.maximum(np.abs(y), threshold)``. Typically, the value of
|
|
absolute tolerance (atol) for a solver should be passed as `threshold`.
|
|
factor : ndarray with shape (n,) or None
|
|
Factor to use for computing the step size. Pass None for the very
|
|
evaluation, then use the value returned from this function.
|
|
sparsity : tuple (structure, groups) or None
|
|
Sparsity structure of the Jacobian, `structure` must be csc_matrix.
|
|
|
|
Returns
|
|
-------
|
|
J : ndarray or csc_matrix, shape (n, n)
|
|
Jacobian matrix.
|
|
factor : ndarray, shape (n,)
|
|
Suggested `factor` for the next evaluation.
|
|
"""
|
|
y = np.asarray(y)
|
|
n = y.shape[0]
|
|
if n == 0:
|
|
return np.empty((0, 0)), factor
|
|
|
|
if factor is None:
|
|
factor = np.full(n, EPS ** 0.5)
|
|
else:
|
|
factor = factor.copy()
|
|
|
|
# Direct the step as ODE dictates, hoping that such a step won't lead to
|
|
# a problematic region. For complex ODEs it makes sense to use the real
|
|
# part of f as we use steps along real axis.
|
|
f_sign = 2 * (np.real(f) >= 0).astype(float) - 1
|
|
y_scale = f_sign * np.maximum(threshold, np.abs(y))
|
|
h = (y + factor * y_scale) - y
|
|
|
|
# Make sure that the step is not 0 to start with. Not likely it will be
|
|
# executed often.
|
|
for i in np.nonzero(h == 0)[0]:
|
|
while h[i] == 0:
|
|
factor[i] *= 10
|
|
h[i] = (y[i] + factor[i] * y_scale[i]) - y[i]
|
|
|
|
if sparsity is None:
|
|
return _dense_num_jac(fun, t, y, f, h, factor, y_scale)
|
|
else:
|
|
structure, groups = sparsity
|
|
return _sparse_num_jac(fun, t, y, f, h, factor, y_scale,
|
|
structure, groups)
|
|
|
|
|
|
def _dense_num_jac(fun, t, y, f, h, factor, y_scale):
|
|
n = y.shape[0]
|
|
h_vecs = np.diag(h)
|
|
f_new = fun(t, y[:, None] + h_vecs)
|
|
diff = f_new - f[:, None]
|
|
max_ind = np.argmax(np.abs(diff), axis=0)
|
|
r = np.arange(n)
|
|
max_diff = np.abs(diff[max_ind, r])
|
|
scale = np.maximum(np.abs(f[max_ind]), np.abs(f_new[max_ind, r]))
|
|
|
|
diff_too_small = max_diff < NUM_JAC_DIFF_REJECT * scale
|
|
if np.any(diff_too_small):
|
|
ind, = np.nonzero(diff_too_small)
|
|
new_factor = NUM_JAC_FACTOR_INCREASE * factor[ind]
|
|
h_new = (y[ind] + new_factor * y_scale[ind]) - y[ind]
|
|
h_vecs[ind, ind] = h_new
|
|
f_new = fun(t, y[:, None] + h_vecs[:, ind])
|
|
diff_new = f_new - f[:, None]
|
|
max_ind = np.argmax(np.abs(diff_new), axis=0)
|
|
r = np.arange(ind.shape[0])
|
|
max_diff_new = np.abs(diff_new[max_ind, r])
|
|
scale_new = np.maximum(np.abs(f[max_ind]), np.abs(f_new[max_ind, r]))
|
|
|
|
update = max_diff[ind] * scale_new < max_diff_new * scale[ind]
|
|
if np.any(update):
|
|
update, = np.nonzero(update)
|
|
update_ind = ind[update]
|
|
factor[update_ind] = new_factor[update]
|
|
h[update_ind] = h_new[update]
|
|
diff[:, update_ind] = diff_new[:, update]
|
|
scale[update_ind] = scale_new[update]
|
|
max_diff[update_ind] = max_diff_new[update]
|
|
|
|
diff /= h
|
|
|
|
factor[max_diff < NUM_JAC_DIFF_SMALL * scale] *= NUM_JAC_FACTOR_INCREASE
|
|
factor[max_diff > NUM_JAC_DIFF_BIG * scale] *= NUM_JAC_FACTOR_DECREASE
|
|
factor = np.maximum(factor, NUM_JAC_MIN_FACTOR)
|
|
|
|
return diff, factor
|
|
|
|
|
|
def _sparse_num_jac(fun, t, y, f, h, factor, y_scale, structure, groups):
|
|
n = y.shape[0]
|
|
n_groups = np.max(groups) + 1
|
|
h_vecs = np.empty((n_groups, n))
|
|
for group in range(n_groups):
|
|
e = np.equal(group, groups)
|
|
h_vecs[group] = h * e
|
|
h_vecs = h_vecs.T
|
|
|
|
f_new = fun(t, y[:, None] + h_vecs)
|
|
df = f_new - f[:, None]
|
|
|
|
i, j, _ = find(structure)
|
|
diff = coo_matrix((df[i, groups[j]], (i, j)), shape=(n, n)).tocsc()
|
|
max_ind = np.array(abs(diff).argmax(axis=0)).ravel()
|
|
r = np.arange(n)
|
|
max_diff = np.asarray(np.abs(diff[max_ind, r])).ravel()
|
|
scale = np.maximum(np.abs(f[max_ind]),
|
|
np.abs(f_new[max_ind, groups[r]]))
|
|
|
|
diff_too_small = max_diff < NUM_JAC_DIFF_REJECT * scale
|
|
if np.any(diff_too_small):
|
|
ind, = np.nonzero(diff_too_small)
|
|
new_factor = NUM_JAC_FACTOR_INCREASE * factor[ind]
|
|
h_new = (y[ind] + new_factor * y_scale[ind]) - y[ind]
|
|
h_new_all = np.zeros(n)
|
|
h_new_all[ind] = h_new
|
|
|
|
groups_unique = np.unique(groups[ind])
|
|
groups_map = np.empty(n_groups, dtype=int)
|
|
h_vecs = np.empty((groups_unique.shape[0], n))
|
|
for k, group in enumerate(groups_unique):
|
|
e = np.equal(group, groups)
|
|
h_vecs[k] = h_new_all * e
|
|
groups_map[group] = k
|
|
h_vecs = h_vecs.T
|
|
|
|
f_new = fun(t, y[:, None] + h_vecs)
|
|
df = f_new - f[:, None]
|
|
i, j, _ = find(structure[:, ind])
|
|
diff_new = coo_matrix((df[i, groups_map[groups[ind[j]]]],
|
|
(i, j)), shape=(n, ind.shape[0])).tocsc()
|
|
|
|
max_ind_new = np.array(abs(diff_new).argmax(axis=0)).ravel()
|
|
r = np.arange(ind.shape[0])
|
|
max_diff_new = np.asarray(np.abs(diff_new[max_ind_new, r])).ravel()
|
|
scale_new = np.maximum(
|
|
np.abs(f[max_ind_new]),
|
|
np.abs(f_new[max_ind_new, groups_map[groups[ind]]]))
|
|
|
|
update = max_diff[ind] * scale_new < max_diff_new * scale[ind]
|
|
if np.any(update):
|
|
update, = np.nonzero(update)
|
|
update_ind = ind[update]
|
|
factor[update_ind] = new_factor[update]
|
|
h[update_ind] = h_new[update]
|
|
diff[:, update_ind] = diff_new[:, update]
|
|
scale[update_ind] = scale_new[update]
|
|
max_diff[update_ind] = max_diff_new[update]
|
|
|
|
diff.data /= np.repeat(h, np.diff(diff.indptr))
|
|
|
|
factor[max_diff < NUM_JAC_DIFF_SMALL * scale] *= NUM_JAC_FACTOR_INCREASE
|
|
factor[max_diff > NUM_JAC_DIFF_BIG * scale] *= NUM_JAC_FACTOR_DECREASE
|
|
factor = np.maximum(factor, NUM_JAC_MIN_FACTOR)
|
|
|
|
return diff, factor
|