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