291 lines
10 KiB
Python
291 lines
10 KiB
Python
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import numpy as np
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def check_arguments(fun, y0, support_complex):
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"""Helper function for checking arguments common to all solvers."""
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y0 = np.asarray(y0)
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if np.issubdtype(y0.dtype, np.complexfloating):
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if not support_complex:
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raise ValueError("`y0` is complex, but the chosen solver does "
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"not support integration in a complex domain.")
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dtype = complex
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else:
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dtype = float
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y0 = y0.astype(dtype, copy=False)
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if y0.ndim != 1:
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raise ValueError("`y0` must be 1-dimensional.")
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if not np.isfinite(y0).all():
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raise ValueError("All components of the initial state `y0` must be finite.")
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def fun_wrapped(t, y):
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return np.asarray(fun(t, y), dtype=dtype)
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return fun_wrapped, y0
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class OdeSolver:
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"""Base class for ODE solvers.
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In order to implement a new solver you need to follow the guidelines:
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1. A constructor must accept parameters presented in the base class
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(listed below) along with any other parameters specific to a solver.
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2. A constructor must accept arbitrary extraneous arguments
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``**extraneous``, but warn that these arguments are irrelevant
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using `common.warn_extraneous` function. Do not pass these
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arguments to the base class.
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3. A solver must implement a private method `_step_impl(self)` which
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propagates a solver one step further. It must return tuple
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``(success, message)``, where ``success`` is a boolean indicating
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whether a step was successful, and ``message`` is a string
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containing description of a failure if a step failed or None
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otherwise.
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4. A solver must implement a private method `_dense_output_impl(self)`,
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which returns a `DenseOutput` object covering the last successful
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step.
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5. A solver must have attributes listed below in Attributes section.
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Note that ``t_old`` and ``step_size`` are updated automatically.
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6. Use `fun(self, t, y)` method for the system rhs evaluation, this
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way the number of function evaluations (`nfev`) will be tracked
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automatically.
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7. For convenience, a base class provides `fun_single(self, t, y)` and
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`fun_vectorized(self, t, y)` for evaluating the rhs in
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non-vectorized and vectorized fashions respectively (regardless of
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how `fun` from the constructor is implemented). These calls don't
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increment `nfev`.
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8. If a solver uses a Jacobian matrix and LU decompositions, it should
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track the number of Jacobian evaluations (`njev`) and the number of
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LU decompositions (`nlu`).
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9. By convention, the function evaluations used to compute a finite
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difference approximation of the Jacobian should not be counted in
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`nfev`, thus use `fun_single(self, t, y)` or
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`fun_vectorized(self, t, y)` when computing a finite difference
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approximation of the 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: the time derivative of the state ``y``
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at time ``t``. The calling signature is ``fun(t, y)``, where ``t`` is a
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scalar and ``y`` is an ndarray with ``len(y) = len(y0)``. ``fun`` must
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return an array of the same shape as ``y``. See `vectorized` for more
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information.
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t0 : float
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Initial time.
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y0 : array_like, shape (n,)
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Initial state.
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t_bound : float
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Boundary time --- the integration won't continue beyond it. It also
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determines the direction of the integration.
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vectorized : bool
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Whether `fun` can be called in a vectorized fashion. Default is False.
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If ``vectorized`` is False, `fun` will always be called with ``y`` of
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shape ``(n,)``, where ``n = len(y0)``.
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If ``vectorized`` is True, `fun` may be called with ``y`` of shape
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``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave
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such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of
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the returned array is the time derivative of the state corresponding
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with a column of ``y``).
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Setting ``vectorized=True`` allows for faster finite difference
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approximation of the Jacobian by methods 'Radau' and 'BDF', but
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will result in slower execution for other methods. It can also
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result in slower overall execution for 'Radau' and 'BDF' in some
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circumstances (e.g. small ``len(y0)``).
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support_complex : bool, optional
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Whether integration in a complex domain should be supported.
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Generally determined by a derived solver class capabilities.
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Default is False.
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Attributes
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----------
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n : int
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Number of equations.
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status : string
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Current status of the solver: 'running', 'finished' or 'failed'.
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t_bound : float
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Boundary time.
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direction : float
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Integration direction: +1 or -1.
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t : float
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Current time.
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y : ndarray
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Current state.
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t_old : float
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Previous time. None if no steps were made yet.
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step_size : float
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Size of the last successful step. None if no steps were made yet.
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nfev : int
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Number of the system's rhs evaluations.
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njev : int
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Number of the Jacobian evaluations.
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nlu : int
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Number of LU decompositions.
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"""
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TOO_SMALL_STEP = "Required step size is less than spacing between numbers."
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def __init__(self, fun, t0, y0, t_bound, vectorized,
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support_complex=False):
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self.t_old = None
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self.t = t0
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self._fun, self.y = check_arguments(fun, y0, support_complex)
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self.t_bound = t_bound
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self.vectorized = vectorized
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if vectorized:
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def fun_single(t, y):
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return self._fun(t, y[:, None]).ravel()
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fun_vectorized = self._fun
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else:
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fun_single = self._fun
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def fun_vectorized(t, y):
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f = np.empty_like(y)
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for i, yi in enumerate(y.T):
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f[:, i] = self._fun(t, yi)
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return f
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def fun(t, y):
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self.nfev += 1
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return self.fun_single(t, y)
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self.fun = fun
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self.fun_single = fun_single
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self.fun_vectorized = fun_vectorized
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self.direction = np.sign(t_bound - t0) if t_bound != t0 else 1
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self.n = self.y.size
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self.status = 'running'
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self.nfev = 0
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self.njev = 0
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self.nlu = 0
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@property
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def step_size(self):
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if self.t_old is None:
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return None
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else:
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return np.abs(self.t - self.t_old)
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def step(self):
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"""Perform one integration step.
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Returns
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-------
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message : string or None
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Report from the solver. Typically a reason for a failure if
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`self.status` is 'failed' after the step was taken or None
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otherwise.
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"""
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if self.status != 'running':
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raise RuntimeError("Attempt to step on a failed or finished "
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"solver.")
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if self.n == 0 or self.t == self.t_bound:
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# Handle corner cases of empty solver or no integration.
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self.t_old = self.t
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self.t = self.t_bound
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message = None
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self.status = 'finished'
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else:
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t = self.t
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success, message = self._step_impl()
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if not success:
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self.status = 'failed'
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else:
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self.t_old = t
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if self.direction * (self.t - self.t_bound) >= 0:
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self.status = 'finished'
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return message
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def dense_output(self):
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"""Compute a local interpolant over the last successful step.
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Returns
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-------
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sol : `DenseOutput`
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Local interpolant over the last successful step.
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"""
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if self.t_old is None:
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raise RuntimeError("Dense output is available after a successful "
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"step was made.")
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if self.n == 0 or self.t == self.t_old:
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# Handle corner cases of empty solver and no integration.
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return ConstantDenseOutput(self.t_old, self.t, self.y)
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else:
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return self._dense_output_impl()
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def _step_impl(self):
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raise NotImplementedError
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def _dense_output_impl(self):
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raise NotImplementedError
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class DenseOutput:
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"""Base class for local interpolant over step made by an ODE solver.
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It interpolates between `t_min` and `t_max` (see Attributes below).
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Evaluation outside this interval is not forbidden, but the accuracy is not
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guaranteed.
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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, t_old, t):
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self.t_old = t_old
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self.t = t
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self.t_min = min(t, t_old)
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self.t_max = max(t, t_old)
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def __call__(self, t):
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"""Evaluate the interpolant.
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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 the solution at.
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Returns
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-------
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y : ndarray, shape (n,) or (n, n_points)
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Computed values. Shape depends on whether `t` was 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 > 1:
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raise ValueError("`t` must be a float or a 1-D array.")
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return self._call_impl(t)
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def _call_impl(self, t):
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raise NotImplementedError
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class ConstantDenseOutput(DenseOutput):
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"""Constant value interpolator.
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This class used for degenerate integration cases: equal integration limits
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or a system with 0 equations.
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"""
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def __init__(self, t_old, t, value):
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super().__init__(t_old, t)
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self.value = value
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def _call_impl(self, t):
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if t.ndim == 0:
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return self.value
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else:
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ret = np.empty((self.value.shape[0], t.shape[0]))
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ret[:] = self.value[:, None]
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return ret
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