Source code for pyodesys.core

# -*- coding: utf-8 -*-
"""
Core functionality for ODESys.

Note that it is possible to use new custom ODE integrators with pyodesys by
providing a module with two functions named ``integrate_adaptive`` and
``integrate_predefined``. See the ``pyodesys.integrators`` module for examples.
"""

from __future__ import absolute_import, division, print_function

import copy
import os
import warnings
from collections import defaultdict

import numpy as np

from .plotting import plot_result, plot_phase_plane
from .results import Result
from .util import _ensure_4args, _default


[docs]class RecoverableError(Exception): pass
[docs]class ODESys(object): """ Object representing an ODE system. ``ODESys`` provides unified interface to: - scipy.integarte.ode - pygslodeiv2 - pyodeint - pycvodes The numerical integration can be performed either in an :meth:`adaptive` or :meth:`predefined` mode. Where locations to report the solution is chosen by the stepper or the user respectively. For convenience in user code one may use :meth:`integrate` which automatically chooses between the two based on the length of ``xout`` provided by the user. Parameters ---------- f : callback first derivatives of dependent variables (y) with respect to dependent variable (x). Signature is any of: - ``rhs(x, y[:]) -> f[:]`` - ``rhs(x, y[:], p[:]) -> f[:]`` - ``rhs(x, y[:], p[:], backend=math) -> f[:]``. jac : callback Jacobian matrix (dfdy). Required for implicit methods. Signature should be either of: - ``jac(x, y[:]) -> J`` - ``jac(x, y[:], p[:]) -J``. If ``nnz < 0``, ``J`` should be a 2D array-like object if ``nnz < 0`` corresponding to a dense or banded jacobian (see also ``band``). If ``nnz >= 0``, ``J`` should be an instance of ``scipy.sparse.csc_matrix``. dfdx : callback Signature ``dfdx(x, y[:], p[:]) -> out[:]`` (used by e.g. GSL) jtimes : callback Jacobian-vector product (Jv). Signature is ```jtimes(x, y[:], v[:]) -> Jv[:]``` This is supported only by ``cvode``. first_step_cb : callback Signature ``step1st(x, y[:], p[:]) -> dx0`` (pass first_step==0 to use). This is available for ``cvode``, ``odeint`` & ``gsl``, but not for ``scipy``. roots_cb : callback Signature ``roots_cb(x, y[:], p[:]=(), backend=math) -> discr[:]``. nroots : int Length of return vector from ``roots_cb``. band : tuple of 2 integers or None (default: None) If jacobian is banded: number of sub- and super-diagonals names : iterable of strings (default : None) Names of variables, used for referencing dependent variables by name and for labels in plots. param_names : iterable of strings (default: None) Names of the parameters, used for referencing parameters by name. indep_name : str Name of the independent variable dep_by_name : bool When ``True`` :meth:`integrate` expects a dictionary as input for y0. par_by_name : bool When ``True`` :meth:`integrate` expects a dictionary as input for params. latex_names : iterable of strings (default : None) Names of variables in LaTeX format (e.g. for labels in plots). latex_param_names : iterable of strings (default : None) Names of parameters in LaTeX format (e.g. for labels in plots). latex_indep_name : str LaTeX formatted name of independent variable. taken_names : iterable of str Names of dependent variables which are calculated in pre_processors pre_processors : iterable of callables (optional) signature: f(x1[:], y1[:], params1[:]) -> x2[:], y2[:], params2[:]. When modifying: insert at beginning. post_processors : iterable of callables (optional) signature: f(x2[:], y2[:, :], params2[:]) -> x1[:], y1[:, :], params1[:] When modifying: insert at end. append_iv : bool See :attr:`append_iv`. autonomous_interface : bool (optional) If given, sets the :attr:`autonomous_interface` to indicate whether the system appears autonomous or not upon call to :meth:`integrate`. autonomous_exprs : bool Describes whether the independent variable appears in the rhs expressions. If set to ``True`` the underlying solver is allowed to shift the independent variable during integration. nnz : int (default : -1) Maximum number of non-zero entries in sparse jacobian. When non-negative, jacobian is assumed to be dense or banded. Attributes ---------- f_cb : callback For evaluating the vector of derivatives. j_cb : callback or None For evaluating the Jacobian matrix of f. dfdx_cb : callback or None For evaluating the second order derivatives. jtimes_cb : callback or None For evaluating the jacobian-vector product. first_step_cb : callback or None For calculating the first step based on x0, y0 & p. roots_cb : callback nroots : int nnz : int names : tuple of strings param_names : tuple of strings description : str dep_by_name : bool par_by_name : bool latex_names : tuple of str latex_param_names : tuple of str pre_processors : iterable of callbacks post_processors : iterable of callbacks append_iv : bool If ``True`` params[:] passed to :attr:`f_cb`, :attr:`jac_cb` will contain initial values of y. Note that this happens after pre processors have been applied. autonomous_interface : bool or None Indicates whether the system appears autonomous upon call to :meth:`integrate`. ``None`` indicates that it is unknown. Examples -------- >>> odesys = ODESys(lambda x, y, p: p[0]*x + p[1]*y[0]*y[0]) >>> yout, info = odesys.predefined([1], [0, .2, .5], [2, 1]) >>> print(info['success']) True Notes ----- Banded jacobians are supported by "scipy" and "cvode" integrators. """ def __init__(self, f, jac=None, dfdx=None, jtimes=None, first_step_cb=None, roots_cb=None, nroots=None, band=None, names=(), param_names=(), indep_name=None, description=None, dep_by_name=False, par_by_name=False, latex_names=(), latex_param_names=(), latex_indep_name=None, taken_names=None, pre_processors=None, post_processors=None, append_iv=False, autonomous_interface=None, to_arrays_callbacks=None, autonomous_exprs=None, _indep_autonomous_key=None, numpy=None, nnz=-1, **kwargs): self.f_cb = _ensure_4args(f) self.j_cb = _ensure_4args(jac) if jac is not None else None self.jtimes_cb = _ensure_4args(jtimes) if jtimes is not None else None self.dfdx_cb = dfdx self.first_step_cb = first_step_cb self.roots_cb = roots_cb self.nroots = nroots or 0 if band is not None: if not band[0] >= 0 or not band[1] >= 0: raise ValueError("bands needs to be > 0 if provided") self.band = band self.nnz = nnz self.names = tuple(names or ()) self.param_names = tuple(param_names or ()) self.indep_name = indep_name self.description = description self.dep_by_name = dep_by_name self.par_by_name = par_by_name self.latex_names = tuple(latex_names or ()) self.latex_param_names = tuple(latex_param_names or ()) self.latex_indep_name = latex_indep_name self.taken_names = tuple(taken_names or ()) self.pre_processors = pre_processors or [] self.post_processors = post_processors or [] self.append_iv = append_iv self.autonomous_exprs = autonomous_exprs if hasattr(self, 'autonomous_interface'): if autonomous_interface is not None and autonomous_interface != self.autonomous_interface: raise ValueError("Got conflicting autonomous_interface infomation.") else: if (autonomous_interface is None and self.autonomous_exprs and len(self.post_processors) == 0 and len(self.pre_processors) == 0): self.autonomous_interface = True else: self.autonomous_interface = autonomous_interface if self.autonomous_interface not in (True, False, None): raise ValueError("autonomous_interface needs to be a boolean value or None.") self._indep_autonomous_key = _indep_autonomous_key self.to_arrays_callbacks = to_arrays_callbacks self.numpy = numpy or np if len(kwargs) > 0: raise ValueError("Unknown kwargs: %s" % str(kwargs)) @staticmethod def _array_from_dict(d, keys, numpy=np): vals = [d[k] for k in keys] lens = [len(v) for v in vals if hasattr(v, '__len__') and getattr(v, 'ndim', 1) > 0] if len(lens) == 0: return vals, True else: if not all(l == lens[0] for l in lens): raise ValueError("Mixed lenghts in dictionary.") out = numpy.empty((lens[0], len(vals)), dtype=object) for idx, v in enumerate(vals): if getattr(v, 'ndim', -1) == 0: for j in range(lens[0]): out[j, idx] = v else: try: for j in range(lens[0]): out[j, idx] = v[j] except TypeError: out[:, idx] = v return out, False def _conditional_from_dict(self, cont, by_name, names): if isinstance(cont, dict): if not by_name: raise ValueError("not by name, yet a dictionary was passed.") cont, tp = self._array_from_dict(cont, names, numpy=self.numpy) else: tp = False return cont, tp
[docs] def to_arrays(self, x, y, p, callbacks=None, reshape=True): try: nx = len(x) except TypeError: _x = 0*x, x else: _x = (0*x[0], x[0]) if nx == 0 else x _names = [n for n in self.names if n not in self.taken_names] _y, tp_y = self._conditional_from_dict(y, self.dep_by_name, _names) _p, tp_p = self._conditional_from_dict(p, self.par_by_name, self.param_names) del _names callbacks = callbacks or self.to_arrays_callbacks if callbacks is not None: # e.g. dedimensionalisation if len(callbacks) != 3: raise ValueError("Need 3 callbacks/None values.") _x, _y, _p = [e if cb is None else cb(e) for cb, e in zip(callbacks, [_x, _y, _p])] _y = self.numpy.atleast_1d(_y) if self._indep_autonomous_key: if _y.shape[-1] == self.ny: pass elif _y.shape[-1] == self.ny - 1: _y = self.numpy.concatenate((_y, _x[0]*self.numpy.ones(_y.shape[:-1] + (1,))), axis=-1) else: raise ValueError("y of incorrect shape") arrs = [arr.T if tp else arr for tp, arr in zip([False, tp_y, tp_p], map(self.numpy.atleast_1d, (_x, _y, _p)))] if reshape: extra_shape = None for a in arrs: if a.ndim == 1: continue elif a.ndim == 2: if extra_shape is None: extra_shape = a.shape[0] else: if extra_shape != a.shape[0]: raise ValueError("Size mismatch!") else: raise NotImplementedError("Only 2 dimensions currently supported.") if extra_shape is not None: arrs = [a if a.ndim == 2 else self.numpy.tile(a, (extra_shape, 1)) for a in arrs] return arrs
[docs] def pre_process(self, xout, y0, params=()): """ Transforms input to internal values, used internally. """ for pre_processor in self.pre_processors: xout, y0, params = pre_processor(xout, y0, params) return [self.numpy.atleast_1d(arr) for arr in (xout, y0, params)]
[docs] def post_process(self, xout, yout, params): """ Transforms internal values to output, used internally. """ for post_processor in self.post_processors: xout, yout, params = post_processor(xout, yout, params) return xout, yout, params
[docs] def adaptive(self, y0, x0, xend, params=(), **kwargs): """ Integrate with integrator chosen output. Parameters ---------- integrator : str See :meth:`integrate`. y0 : array_like See :meth:`integrate`. x0 : float Initial value of the independent variable. xend : float Final value of the independent variable. params : array_like See :meth:`integrate`. \\*\\*kwargs : See :meth:`integrate`. Returns ------- Same as :meth:`integrate` """ return self.integrate((x0, xend), y0, params=params, **kwargs)
[docs] def predefined(self, y0, xout, params=(), **kwargs): """ Integrate with user chosen output. Parameters ---------- integrator : str See :meth:`integrate`. y0 : array_like See :meth:`integrate`. xout : array_like params : array_like See :meth:`integrate`. \\*\\*kwargs: See :meth:`integrate` Returns ------- Length 2 tuple : (yout, info) See :meth:`integrate`. """ xout, yout, info = self.integrate(xout, y0, params=params, force_predefined=True, **kwargs) return yout, info
[docs] def integrate(self, x, y0, params=(), atol=1e-8, rtol=1e-8, **kwargs): """ Integrate the system of ordinary differential equations. Solves the initial value problem (IVP). Parameters ---------- x : array_like or pair (start and final time) or float if float: make it a pair: (0, x) if pair or length-2 array: initial and final value of the independent variable if array_like: values of independent variable report at y0 : array_like Initial values at x[0] for the dependent variables. params : array_like (default: tuple()) Value of parameters passed to user-supplied callbacks. integrator : str or None Name of integrator, one of: - 'scipy': :meth:`_integrate_scipy` - 'gsl': :meth:`_integrate_gsl` - 'odeint': :meth:`_integrate_odeint` - 'cvode': :meth:`_integrate_cvode` See respective method for more information. If ``None``: ``os.environ.get('PYODESYS_INTEGRATOR', 'scipy')`` atol : float Absolute tolerance rtol : float Relative tolerance with_jacobian : bool or None (default) Whether to use the jacobian. When ``None`` the choice is done automatically (only used when required). This matters when jacobian is derived at runtime (high computational cost). with_jtimes : bool (default: False) Whether to use the jacobian-vector product. This is only supported by ``cvode`` and only when ``linear_solver`` is one of: gmres', 'gmres_classic', 'bicgstab', 'tfqmr'. See the documentation for ``pycvodes`` for more information. force_predefined : bool (default: False) override behaviour of ``len(x) == 2`` => :meth:`adaptive` \\*\\*kwargs : Additional keyword arguments for ``_integrate_$(integrator)``. Returns ------- Length 3 tuple: (x, yout, info) x : array of values of the independent variable yout : array of the dependent variable(s) for the different values of x. info : dict ('nfev' is guaranteed to be a key) """ arrs = self.to_arrays(x, y0, params) _x, _y, _p = _arrs = self.pre_process(*arrs) ndims = [a.ndim for a in _arrs] if ndims == [1, 1, 1]: twodim = False elif ndims == [2, 2, 2]: twodim = True else: raise ValueError("Pre-processor made ndims inconsistent?") if self.append_iv: _p = self.numpy.concatenate((_p, _y), axis=-1) if hasattr(self, 'ny'): if _y.shape[-1] != self.ny: raise ValueError("Incorrect shape of intern_y0") if isinstance(atol, dict): kwargs['atol'] = [atol[k] for k in self.names] else: kwargs['atol'] = atol kwargs['rtol'] = rtol integrator = kwargs.pop('integrator', None) if integrator is None: integrator = os.environ.get('PYODESYS_INTEGRATOR', 'scipy') args = tuple(map(self.numpy.atleast_2d, (_x, _y, _p))) self._current_integration_kwargs = kwargs if isinstance(integrator, str): nfo = getattr(self, '_integrate_' + integrator)(*args, **kwargs) else: kwargs['with_jacobian'] = getattr(integrator, 'with_jacobian', None) nfo = self._integrate(integrator.integrate_adaptive, integrator.integrate_predefined, *args, **kwargs) if twodim: _xout = [d['internal_xout'] for d in nfo] _yout = [d['internal_yout'] for d in nfo] _params = [d['internal_params'] for d in nfo] res = [Result(*(self.post_process(_xout[i], _yout[i], _params[i]) + (nfo[i], self))) for i in range(len(nfo))] else: _xout = nfo[0]['internal_xout'] _yout = nfo[0]['internal_yout'] self._internal = _xout.copy(), _yout.copy(), _p.copy() nfo = nfo[0] res = Result(*(self.post_process(_xout, _yout, _p) + (nfo, self))) return res
[docs] def chained_parameter_variation(self, *args, **kwargs): """ See :func:`chained_parameter_variation`. """ return chained_parameter_variation(self, *args, **kwargs)
def _integrate_scipy(self, intern_xout, intern_y0, intern_p, atol=1e-8, rtol=1e-8, first_step=None, with_jacobian=None, force_predefined=False, name=None, **kwargs): """ Do not use directly (use ``integrate('scipy', ...)``). Uses `scipy.integrate.ode <http://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.ode.html>`_ Parameters ---------- \\*args : See :meth:`integrate`. name : str (default: 'lsoda'/'dopri5' when jacobian is available/not) What integrator wrapped in scipy.integrate.ode to use. \\*\\*kwargs : Keyword arguments passed onto `set_integrator(...) < http://docs.scipy.org/doc/scipy/reference/generated/ scipy.integrate.ode.set_integrator.html#scipy.integrate.ode.set_integrator>`_ Returns ------- See :meth:`integrate`. """ from scipy.integrate import ode ny = intern_y0.shape[-1] nx = intern_xout.shape[-1] results = [] for _xout, _y0, _p in zip(intern_xout, intern_y0, intern_p): if name is None: if self.j_cb is None: name = 'dopri5' else: name = 'lsoda' if with_jacobian is None: if name == 'lsoda': # lsoda might call jacobian with_jacobian = True elif name in ('dop853', 'dopri5'): with_jacobian = False # explicit steppers elif name == 'vode': with_jacobian = kwargs.get('method', 'adams') == 'bdf' def rhs(t, y, p=()): rhs.ncall += 1 return self.f_cb(t, y, p) rhs.ncall = 0 if self.j_cb is not None: def jac(t, y, p=()): jac.ncall += 1 return self.j_cb(t, y, p) jac.ncall = 0 r = ode(rhs, jac=jac if with_jacobian else None) if 'lband' in kwargs or 'uband' in kwargs or 'band' in kwargs: raise ValueError("lband and uband set locally (set `band` at initialization instead)") if self.band is not None: kwargs['lband'], kwargs['uband'] = self.band r.set_integrator(name, atol=atol, rtol=rtol, **kwargs) if len(_p) > 0: r.set_f_params(_p) r.set_jac_params(_p) r.set_initial_value(_y0, _xout[0]) if nx == 2 and not force_predefined: mode = 'adaptive' if name in ('vode', 'lsoda'): warnings.warn("'adaptive' mode with SciPy's integrator (vode/lsoda) may overshoot (itask=2)") warnings.warn("'adaptive' mode with SciPy's integrator is unreliable, consider using e.g. cvode") # vode itask 2 (may overshoot) ysteps = [_y0] xsteps = [_xout[0]] while r.t < _xout[1]: r.integrate(_xout[1], step=True) if not r.successful(): raise RuntimeError("failed") xsteps.append(r.t) ysteps.append(r.y) else: xsteps, ysteps = [], [] def solout(x, y): xsteps.append(x) ysteps.append(y) r.set_solout(solout) r.integrate(_xout[1]) if not r.successful(): raise RuntimeError("failed") _yout = np.array(ysteps) _xout = np.array(xsteps) else: # predefined mode = 'predefined' _yout = np.empty((nx, ny)) _yout[0, :] = _y0 for idx in range(1, nx): r.integrate(_xout[idx]) if not r.successful(): raise RuntimeError("failed") _yout[idx, :] = r.y info = { 'internal_xout': _xout, 'internal_yout': _yout, 'internal_params': _p, 'success': r.successful(), 'nfev': rhs.ncall, 'n_steps': -1, # don't know how to obtain this number 'name': name, 'mode': mode, 'atol': atol, 'rtol': rtol } if self.j_cb is not None: info['njev'] = jac.ncall results.append(info) return results def _integrate(self, adaptive, predefined, intern_xout, intern_y0, intern_p, atol=1e-8, rtol=1e-8, first_step=0.0, with_jacobian=None, force_predefined=False, **kwargs): nx = intern_xout.shape[-1] results = [] for _xout, _y0, _p in zip(intern_xout, intern_y0, intern_p): new_kwargs = dict(dx0=first_step, atol=atol, rtol=rtol, check_indexing=False) new_kwargs.update(kwargs) def _f(x, y, fout): try: if len(_p) > 0: fout[:] = np.asarray(self.f_cb(x, y, _p)) else: fout[:] = np.asarray(self.f_cb(x, y)) except RecoverableError: return 1 # recoverable error if with_jacobian is None: raise Exception("Must pass with_jacobian") elif with_jacobian is True: if self.nnz >= 0: def _j(x, y, data, colptrs, rowvals): if len(_p) > 0: J = self.j_cb(x, y, _p) else: J = self.j_cb(x, y) J = J.asformat("csc") data[:] = J.data colptrs[:] = J.indptr rowvals[:] = J.indices new_kwargs['nnz'] = self.nnz else: def _j(x, y, jout, dfdx_out=None, fy=None): if len(_p) > 0: jout[:, :] = np.asarray(self.j_cb(x, y, _p)) else: jout[:, :] = np.asarray(self.j_cb(x, y)) if dfdx_out is not None: if len(_p) > 0: dfdx_out[:] = np.asarray(self.dfdx_cb(x, y, _p)) else: dfdx_out[:] = np.asarray(self.dfdx_cb(x, y)) else: _j = None with_jtimes = kwargs.pop('with_jtimes', False) if with_jtimes is True: def _jtimes(v, Jv, x, y, fy=None): yv = np.concatenate((y, v)) if len(_p) > 0: Jv[:] = np.asarray(self.jtimes_cb(x, yv, _p)) else: Jv[:] = np.asarray(self.jtimes_cb(x, yv)) new_kwargs['jtimes'] = _jtimes if self.first_step_cb is not None: def _first_step(x, y): if len(_p) > 0: return self.first_step_cb(x, y, _p) else: return self.first_step_cb(x, y) if 'dx0cb' in new_kwargs: raise ValueError("cannot override dx0cb") else: new_kwargs['dx0cb'] = _first_step if self.roots_cb is not None: def _roots(x, y, out): if len(_p) > 0: out[:] = np.asarray(self.roots_cb(x, y, _p)) else: out[:] = np.asarray(self.roots_cb(x, y)) if 'roots' in new_kwargs: raise ValueError("cannot override roots") else: new_kwargs['roots'] = _roots if 'nroots' in new_kwargs: raise ValueError("cannot override nroots") new_kwargs['nroots'] = self.nroots if nx == 2 and not force_predefined: _xout, yout, info = adaptive(_f, _j, _y0, *_xout, **new_kwargs) info['mode'] = 'adaptive' else: yout, info = predefined(_f, _j, _y0, _xout, **new_kwargs) info['mode'] = 'predefined' info['internal_xout'] = _xout info['internal_yout'] = yout info['internal_params'] = _p results.append(info) return results def _integrate_gsl(self, *args, **kwargs): """ Do not use directly (use ``integrate(..., integrator='gsl')``). Uses `GNU Scientific Library <http://www.gnu.org/software/gsl/>`_ (via `pygslodeiv2 <https://pypi.python.org/pypi/pygslodeiv2>`_) to integrate the ODE system. Parameters ---------- \\*args : see :meth:`integrate` method : str (default: 'bsimp') what stepper to use, see :py:attr:`gslodeiv2.steppers` \\*\\*kwargs : keyword arguments passed onto :py:func:`gslodeiv2.integrate_adaptive`/:py:func:`gslodeiv2.integrate_predefined` Returns ------- See :meth:`integrate` """ import pygslodeiv2 # Python interface GSL's "odeiv2" integrators kwargs['with_jacobian'] = kwargs.get( 'method', 'bsimp') in pygslodeiv2.requires_jac return self._integrate(pygslodeiv2.integrate_adaptive, pygslodeiv2.integrate_predefined, *args, **kwargs) def _integrate_odeint(self, *args, **kwargs): """ Do not use directly (use ``integrate(..., integrator='odeint')``). Uses `Boost.Numeric.Odeint <http://www.odeint.com>`_ (via `pyodeint <https://pypi.python.org/pypi/pyodeint>`_) to integrate the ODE system. """ import pyodeint # Python interface to boost's odeint integrators kwargs['with_jacobian'] = kwargs.get( 'method', 'rosenbrock4') in pyodeint.requires_jac return self._integrate(pyodeint.integrate_adaptive, pyodeint.integrate_predefined, *args, **kwargs) def _integrate_cvode(self, *args, **kwargs): """ Do not use directly (use ``integrate(..., integrator='cvode')``). Uses CVode from CVodes in `SUNDIALS <https://computation.llnl.gov/casc/sundials/>`_ (via `pycvodes <https://pypi.python.org/pypi/pycvodes>`_) to integrate the ODE system. """ import pycvodes # Python interface to SUNDIALS's cvodes integrators kwargs['with_jacobian'] = kwargs.get('method', 'bdf') in pycvodes.requires_jac if 'lband' in kwargs or 'uband' in kwargs or 'band' in kwargs: raise ValueError("lband and uband set locally (set at" " initialization instead)") if self.band is not None: kwargs['lband'], kwargs['uband'] = self.band kwargs['autonomous_exprs'] = self.autonomous_exprs return self._integrate(pycvodes.integrate_adaptive, pycvodes.integrate_predefined, *args, **kwargs) def _plot(self, cb, internal_xout=None, internal_yout=None, internal_params=None, **kwargs): kwargs = kwargs.copy() if 'x' in kwargs or 'y' in kwargs or 'params' in kwargs: raise ValueError("x and y from internal_xout and internal_yout") _internal = getattr(self, '_internal', [None]*3) x, y, p = (_default(internal_xout, _internal[0]), _default(internal_yout, _internal[1]), _default(internal_params, _internal[2])) for post_processor in self.post_processors: x, y, p = post_processor(x, y, p) if 'names' not in kwargs: kwargs['names'] = getattr(self, 'names', None) else: if 'indices' not in kwargs and getattr(self, 'names', None) is not None: kwargs['indices'] = [self.names.index(n) for n in kwargs['names']] kwargs['names'] = self.names return cb(x, y, **kwargs)
[docs] def plot_result(self, **kwargs): """ Plots the integrated dependent variables from last integration. This method will be deprecated. Please use :meth:`Result.plot`. See :func:`pyodesys.plotting.plot_result` """ return self._plot(plot_result, **kwargs)
[docs] def plot_phase_plane(self, indices=None, **kwargs): """ Plots a phase portrait from last integration. This method will be deprecated. Please use :meth:`Result.plot_phase_plane`. See :func:`pyodesys.plotting.plot_phase_plane` """ return self._plot(plot_phase_plane, indices=indices, **kwargs)
def _jac_eigenvals_svd(self, xval, yvals, intern_p): from scipy.linalg import svd J = self.j_cb(xval, yvals, intern_p) return svd(J, compute_uv=False)
[docs] def stiffness(self, xyp=None, eigenvals_cb=None): """ [DEPRECATED] Use :meth:`Result.stiffness`, stiffness ration Running stiffness ratio from last integration. Calculate sittness ratio, i.e. the ratio between the largest and smallest absolute eigenvalue of the jacobian matrix. The user may supply their own routine for calculating the eigenvalues, or they will be calculated from the SVD (singular value decomposition). Note that calculating the SVD for any but the smallest Jacobians may prove to be prohibitively expensive. Parameters ---------- xyp : length 3 tuple (default: None) internal_xout, internal_yout, internal_params, taken from last integration if not specified. eigenvals_cb : callback (optional) Signature (x, y, p) (internal variables), when not provided an internal routine will use ``self.j_cb`` and ``scipy.linalg.svd``. """ if eigenvals_cb is None: if self.band is not None: raise NotImplementedError eigenvals_cb = self._jac_eigenvals_svd if xyp is None: x, y, intern_p = self._internal else: x, y, intern_p = self.pre_process(*xyp) singular_values = [] for xval, yvals in zip(x, y): singular_values.append(eigenvals_cb(xval, yvals, intern_p)) return (np.abs(singular_values).max(axis=-1) / np.abs(singular_values).min(axis=-1))
[docs]class OdeSys(ODESys): """ DEPRECATED, use ODESys instead. """ pass
def _new_x(xout, x, guaranteed_autonomous): if guaranteed_autonomous: return 0, abs(x[-1] - xout[-1]) # rounding else: return xout[-1], x[-1]
[docs]def integrate_auto_switch(odes, kw, x, y0, params=(), **kwargs): """ Auto-switching between formulations of ODE system. In case one has a formulation of a system of ODEs which is preferential in the beginning of the integration, this function allows the user to run the integration with this system where it takes a user-specified maximum number of steps before switching to another formulation (unless final value of the independent variables has been reached). Number of systems used i returned as ``nsys`` in info dict. Parameters ---------- odes : iterable of :class:`OdeSy` instances kw : dict mapping kwarg to iterables of same legnth as ``odes`` x : array_like y0 : array_like params : array_like \\*\\*kwargs: See :meth:`ODESys.integrate` Notes ----- Plays particularly well with :class:`symbolic.TransformedSys`. """ x_arr = np.asarray(x) if x_arr.shape[-1] > 2: raise NotImplementedError("Only adaptive support return_on_error for now") multimode = False if x_arr.ndim < 2 else x_arr.shape[0] nfo_keys = ('nfev', 'njev', 'time_cpu', 'time_wall') next_autonomous = getattr(odes[0], 'autonomous_interface', False) == True # noqa (np.True_) if multimode: tot_x = [np.array([0] if next_autonomous else [x[_][0]]) for _ in range(multimode)] tot_y = [np.asarray([y0[_]]) for _ in range(multimode)] tot_nfo = [defaultdict(int) for _ in range(multimode)] glob_x = [_[0] for _ in x] if next_autonomous else [0.0]*multimode else: tot_x, tot_y, tot_nfo = np.array([0 if next_autonomous else x[0]]), np.asarray([y0]), defaultdict(int) glob_x = x[0] if next_autonomous else 0.0 for oi in range(len(odes)): if oi < len(odes) - 1: next_autonomous = getattr(odes[oi+1], 'autonomous_interface', False) == True # noqa (np.True_) _int_kw = kwargs.copy() for k, v in kw.items(): _int_kw[k] = v[oi] res = odes[oi].integrate(x, y0, params, **_int_kw) if multimode: for idx in range(multimode): tot_x[idx] = np.concatenate((tot_x[idx], res[idx].xout[1:] + glob_x[idx])) tot_y[idx] = np.concatenate((tot_y[idx], res[idx].yout[1:, :])) for k in nfo_keys: if k in res[idx].info: tot_nfo[idx][k] += res[idx].info[k] tot_nfo[idx]['success'] = res[idx].info['success'] else: tot_x = np.concatenate((tot_x, res.xout[1:] + glob_x)) tot_y = np.concatenate((tot_y, res.yout[1:, :])) for k in nfo_keys: if k in res.info: tot_nfo[k] += res.info[k] tot_nfo['success'] = res.info['success'] if multimode: if all([r.info['success'] for r in res]): break else: if res.info['success']: break if oi < len(odes) - 1: if multimode: _x, y0 = [], [] for idx in range(multimode): _x.append(_new_x(res[idx].xout, x[idx], next_autonomous)) y0.append(res[idx].yout[-1, :]) if next_autonomous: glob_x[idx] += res[idx].xout[-1] x = _x else: x = _new_x(res.xout, x, next_autonomous) y0 = res.yout[-1, :] if next_autonomous: glob_x += res.xout[-1] if multimode: # don't return defaultdict tot_nfo = [dict(nsys=oi+1, **_nfo) for _nfo in tot_nfo] return [Result(tot_x[idx], tot_y[idx], res[idx].params, tot_nfo[idx], odes[0]) for idx in range(len(res))] else: tot_nfo = dict(nsys=oi+1, **tot_nfo) return Result(tot_x, tot_y, res.params, tot_nfo, odes[0])
integrate_chained = integrate_auto_switch # deprecated name
[docs]def chained_parameter_variation(subject, durations, y0, varied_params, default_params=None, integrate_kwargs=None, x0=None, npoints=1, numpy=None): """ Integrate an ODE-system for a serie of durations with some parameters changed in-between Parameters ---------- subject : function or ODESys instance If a function: should have the signature of :meth:`pyodesys.ODESys.integrate` (and resturn a :class:`pyodesys.results.Result` object). If a ODESys instance: the ``integrate`` method will be used. durations : iterable of floats Spans of the independent variable. y0 : dict or array_like varied_params : dict mapping parameter name (or index) to array_like Each array_like need to be of same length as durations. default_params : dict or array_like Default values for the parameters of the ODE system. integrate_kwargs : dict Keyword arguments passed on to ``integrate``. x0 : float-like First value of independent variable. default: 0. npoints : int Number of points per sub-interval. Examples -------- >>> odesys = ODESys(lambda t, y, p: [-p[0]*y[0]]) >>> int_kw = dict(integrator='cvode', method='adams', atol=1e-12, rtol=1e-12) >>> kwargs = dict(default_params=[0], integrate_kwargs=int_kw) >>> res = chained_parameter_variation(odesys, [2, 3], [42], {0: [.7, .1]}, **kwargs) >>> mask1 = res.xout <= 2 >>> import numpy as np >>> np.allclose(res.yout[mask1, 0], 42*np.exp(-.7*res.xout[mask1])) True >>> mask2 = 2 <= res.xout >>> np.allclose(res.yout[mask2, 0], res.yout[mask2, 0][0]*np.exp(-.1*(res.xout[mask2] - res.xout[mask2][0]))) True """ assert len(durations) > 0, 'need at least 1 duration (preferably many)' assert npoints > 0, 'need at least 1 point per duration' for k, v in varied_params.items(): if len(v) != len(durations): raise ValueError("Mismathced lengths of durations and varied_params") if isinstance(subject, ODESys): integrate = subject.integrate numpy = numpy or subject.numpy else: integrate = subject numpy = numpy or np default_params = default_params or {} integrate_kwargs = integrate_kwargs or {} def _get_idx(cont, idx): if isinstance(cont, dict): return {k: (v[idx] if hasattr(v, '__len__') and getattr(v, 'ndim', 1) > 0 else v) for k, v in cont.items()} else: return cont[idx] durations = numpy.cumsum(durations) for idx_dur in range(len(durations)): params = copy.copy(default_params) for k, v in varied_params.items(): params[k] = v[idx_dur] if idx_dur == 0: if x0 is None: x0 = durations[0]*0 out = integrate(numpy.linspace(x0, durations[0], npoints + 1), y0, params, **integrate_kwargs) else: if isinstance(out, Result): out.extend_by_integration(durations[idx_dur], params, npoints=npoints, **integrate_kwargs) else: for idx_res, r in enumerate(out): r.extend_by_integration(durations[idx_dur], _get_idx(params, idx_res), npoints=npoints, **integrate_kwargs) return out