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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(f"`{x}`" for x in extraneous)),
stacklevel=3)
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})`.",
stacklevel=3)
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]``.
alt_segment : boolean
Requests the alternative interpolant segment selection scheme. At each
solver integration point, two interpolant segments are available. The
default (False) and alternative (True) behaviours select the segment
for which the requested time corresponded to ``t`` and ``t_old``,
respectively. This functionality is only relevant for testing the
interpolants' accuracy: different integrators use different
construction strategies.
Attributes
----------
t_min, t_max : float
Time range of the interpolation.
"""
def __init__(self, ts, interpolants, alt_segment=False):
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.side = "right" if alt_segment else "left"
self.ts_sorted = ts
else:
self.t_min = ts[-1]
self.t_max = ts[0]
self.ascending = False
self.side = "left" if alt_segment else "right"
self.ts_sorted = ts[::-1]
def _call_single(self, t):
# Here we preserve a certain symmetry that when t is in self.ts,
# if alt_segment=False, then we prioritize a segment with a lower
# index.
ind = np.searchsorted(self.ts_sorted, t, side=self.side)
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.
segments = np.searchsorted(self.ts_sorted, t_sorted, side=self.side)
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