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1377 lines
47 KiB
Python
1377 lines
47 KiB
Python
# Authors: Pearu Peterson, Pauli Virtanen, John Travers
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"""
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First-order ODE integrators.
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User-friendly interface to various numerical integrators for solving a
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system of first order ODEs with prescribed initial conditions::
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d y(t)[i]
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--------- = f(t,y(t))[i],
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d t
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y(t=0)[i] = y0[i],
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where::
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i = 0, ..., len(y0) - 1
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class ode
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---------
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A generic interface class to numeric integrators. It has the following
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methods::
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integrator = ode(f, jac=None)
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integrator = integrator.set_integrator(name, **params)
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integrator = integrator.set_initial_value(y0, t0=0.0)
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integrator = integrator.set_f_params(*args)
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integrator = integrator.set_jac_params(*args)
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y1 = integrator.integrate(t1, step=False, relax=False)
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flag = integrator.successful()
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class complex_ode
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-----------------
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This class has the same generic interface as ode, except it can handle complex
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f, y and Jacobians by transparently translating them into the equivalent
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real-valued system. It supports the real-valued solvers (i.e., not zvode) and is
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an alternative to ode with the zvode solver, sometimes performing better.
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"""
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# XXX: Integrators must have:
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# ===========================
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# cvode - C version of vode and vodpk with many improvements.
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# Get it from http://www.netlib.org/ode/cvode.tar.gz.
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# To wrap cvode to Python, one must write the extension module by
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# hand. Its interface is too much 'advanced C' that using f2py
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# would be too complicated (or impossible).
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#
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# How to define a new integrator:
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# ===============================
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#
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# class myodeint(IntegratorBase):
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#
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# runner = <odeint function> or None
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#
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# def __init__(self,...): # required
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# <initialize>
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#
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# def reset(self,n,has_jac): # optional
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# # n - the size of the problem (number of equations)
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# # has_jac - whether user has supplied its own routine for Jacobian
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# <allocate memory,initialize further>
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#
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# def run(self,f,jac,y0,t0,t1,f_params,jac_params): # required
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# # this method is called to integrate from t=t0 to t=t1
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# # with initial condition y0. f and jac are user-supplied functions
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# # that define the problem. f_params,jac_params are additional
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# # arguments
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# # to these functions.
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# <calculate y1>
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# if <calculation was unsuccessful>:
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# self.success = 0
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# return t1,y1
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#
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# # In addition, one can define step() and run_relax() methods (they
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# # take the same arguments as run()) if the integrator can support
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# # these features (see IntegratorBase doc strings).
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#
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# if myodeint.runner:
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# IntegratorBase.integrator_classes.append(myodeint)
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__all__ = ['ode', 'complex_ode']
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__version__ = "$Id$"
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__docformat__ = "restructuredtext en"
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import re
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import warnings
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from numpy import asarray, array, zeros, isscalar, real, imag, vstack
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from . import vode as _vode
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from . import _dop
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from . import lsoda as _lsoda
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_dop_int_dtype = _dop.types.intvar.dtype
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_vode_int_dtype = _vode.types.intvar.dtype
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_lsoda_int_dtype = _lsoda.types.intvar.dtype
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# ------------------------------------------------------------------------------
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# User interface
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# ------------------------------------------------------------------------------
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class ode(object):
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"""
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A generic interface class to numeric integrators.
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Solve an equation system :math:`y'(t) = f(t,y)` with (optional) ``jac = df/dy``.
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*Note*: The first two arguments of ``f(t, y, ...)`` are in the
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opposite order of the arguments in the system definition function used
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by `scipy.integrate.odeint`.
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Parameters
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----------
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f : callable ``f(t, y, *f_args)``
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Right-hand side of the differential equation. t is a scalar,
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``y.shape == (n,)``.
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``f_args`` is set by calling ``set_f_params(*args)``.
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`f` should return a scalar, array or list (not a tuple).
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jac : callable ``jac(t, y, *jac_args)``, optional
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Jacobian of the right-hand side, ``jac[i,j] = d f[i] / d y[j]``.
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``jac_args`` is set by calling ``set_jac_params(*args)``.
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Attributes
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----------
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t : float
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Current time.
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y : ndarray
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Current variable values.
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See also
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--------
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odeint : an integrator with a simpler interface based on lsoda from ODEPACK
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quad : for finding the area under a curve
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Notes
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-----
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Available integrators are listed below. They can be selected using
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the `set_integrator` method.
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"vode"
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Real-valued Variable-coefficient Ordinary Differential Equation
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solver, with fixed-leading-coefficient implementation. It provides
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implicit Adams method (for non-stiff problems) and a method based on
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backward differentiation formulas (BDF) (for stiff problems).
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Source: http://www.netlib.org/ode/vode.f
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.. warning::
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This integrator is not re-entrant. You cannot have two `ode`
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instances using the "vode" integrator at the same time.
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This integrator accepts the following parameters in `set_integrator`
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method of the `ode` class:
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- atol : float or sequence
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absolute tolerance for solution
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- rtol : float or sequence
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relative tolerance for solution
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- lband : None or int
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- uband : None or int
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Jacobian band width, jac[i,j] != 0 for i-lband <= j <= i+uband.
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Setting these requires your jac routine to return the jacobian
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in packed format, jac_packed[i-j+uband, j] = jac[i,j]. The
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dimension of the matrix must be (lband+uband+1, len(y)).
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- method: 'adams' or 'bdf'
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Which solver to use, Adams (non-stiff) or BDF (stiff)
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- with_jacobian : bool
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This option is only considered when the user has not supplied a
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Jacobian function and has not indicated (by setting either band)
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that the Jacobian is banded. In this case, `with_jacobian` specifies
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whether the iteration method of the ODE solver's correction step is
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chord iteration with an internally generated full Jacobian or
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functional iteration with no Jacobian.
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- nsteps : int
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Maximum number of (internally defined) steps allowed during one
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call to the solver.
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- first_step : float
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- min_step : float
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- max_step : float
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Limits for the step sizes used by the integrator.
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- order : int
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Maximum order used by the integrator,
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order <= 12 for Adams, <= 5 for BDF.
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"zvode"
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Complex-valued Variable-coefficient Ordinary Differential Equation
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solver, with fixed-leading-coefficient implementation. It provides
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implicit Adams method (for non-stiff problems) and a method based on
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backward differentiation formulas (BDF) (for stiff problems).
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Source: http://www.netlib.org/ode/zvode.f
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.. warning::
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This integrator is not re-entrant. You cannot have two `ode`
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instances using the "zvode" integrator at the same time.
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This integrator accepts the same parameters in `set_integrator`
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as the "vode" solver.
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.. note::
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When using ZVODE for a stiff system, it should only be used for
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the case in which the function f is analytic, that is, when each f(i)
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is an analytic function of each y(j). Analyticity means that the
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partial derivative df(i)/dy(j) is a unique complex number, and this
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fact is critical in the way ZVODE solves the dense or banded linear
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systems that arise in the stiff case. For a complex stiff ODE system
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in which f is not analytic, ZVODE is likely to have convergence
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failures, and for this problem one should instead use DVODE on the
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equivalent real system (in the real and imaginary parts of y).
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"lsoda"
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Real-valued Variable-coefficient Ordinary Differential Equation
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solver, with fixed-leading-coefficient implementation. It provides
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automatic method switching between implicit Adams method (for non-stiff
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problems) and a method based on backward differentiation formulas (BDF)
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(for stiff problems).
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Source: http://www.netlib.org/odepack
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.. warning::
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This integrator is not re-entrant. You cannot have two `ode`
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instances using the "lsoda" integrator at the same time.
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This integrator accepts the following parameters in `set_integrator`
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method of the `ode` class:
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- atol : float or sequence
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absolute tolerance for solution
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- rtol : float or sequence
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relative tolerance for solution
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- lband : None or int
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- uband : None or int
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Jacobian band width, jac[i,j] != 0 for i-lband <= j <= i+uband.
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Setting these requires your jac routine to return the jacobian
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in packed format, jac_packed[i-j+uband, j] = jac[i,j].
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- with_jacobian : bool
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*Not used.*
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- nsteps : int
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Maximum number of (internally defined) steps allowed during one
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call to the solver.
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- first_step : float
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- min_step : float
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- max_step : float
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Limits for the step sizes used by the integrator.
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- max_order_ns : int
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Maximum order used in the nonstiff case (default 12).
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- max_order_s : int
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Maximum order used in the stiff case (default 5).
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- max_hnil : int
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Maximum number of messages reporting too small step size (t + h = t)
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(default 0)
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- ixpr : int
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Whether to generate extra printing at method switches (default False).
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"dopri5"
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This is an explicit runge-kutta method of order (4)5 due to Dormand &
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Prince (with stepsize control and dense output).
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Authors:
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E. Hairer and G. Wanner
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Universite de Geneve, Dept. de Mathematiques
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CH-1211 Geneve 24, Switzerland
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e-mail: ernst.hairer@math.unige.ch, gerhard.wanner@math.unige.ch
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This code is described in [HNW93]_.
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This integrator accepts the following parameters in set_integrator()
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method of the ode class:
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- atol : float or sequence
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absolute tolerance for solution
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- rtol : float or sequence
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relative tolerance for solution
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- nsteps : int
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Maximum number of (internally defined) steps allowed during one
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call to the solver.
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- first_step : float
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- max_step : float
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- safety : float
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Safety factor on new step selection (default 0.9)
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- ifactor : float
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- dfactor : float
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Maximum factor to increase/decrease step size by in one step
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- beta : float
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Beta parameter for stabilised step size control.
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- verbosity : int
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Switch for printing messages (< 0 for no messages).
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"dop853"
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This is an explicit runge-kutta method of order 8(5,3) due to Dormand
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& Prince (with stepsize control and dense output).
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Options and references the same as "dopri5".
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Examples
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--------
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A problem to integrate and the corresponding jacobian:
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>>> from scipy.integrate import ode
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>>>
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>>> y0, t0 = [1.0j, 2.0], 0
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>>>
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>>> def f(t, y, arg1):
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... return [1j*arg1*y[0] + y[1], -arg1*y[1]**2]
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>>> def jac(t, y, arg1):
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... return [[1j*arg1, 1], [0, -arg1*2*y[1]]]
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The integration:
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>>> r = ode(f, jac).set_integrator('zvode', method='bdf')
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>>> r.set_initial_value(y0, t0).set_f_params(2.0).set_jac_params(2.0)
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>>> t1 = 10
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>>> dt = 1
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>>> while r.successful() and r.t < t1:
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... print(r.t+dt, r.integrate(r.t+dt))
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1 [-0.71038232+0.23749653j 0.40000271+0.j ]
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2.0 [0.19098503-0.52359246j 0.22222356+0.j ]
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3.0 [0.47153208+0.52701229j 0.15384681+0.j ]
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4.0 [-0.61905937+0.30726255j 0.11764744+0.j ]
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5.0 [0.02340997-0.61418799j 0.09523835+0.j ]
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6.0 [0.58643071+0.339819j 0.08000018+0.j ]
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7.0 [-0.52070105+0.44525141j 0.06896565+0.j ]
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8.0 [-0.15986733-0.61234476j 0.06060616+0.j ]
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9.0 [0.64850462+0.15048982j 0.05405414+0.j ]
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10.0 [-0.38404699+0.56382299j 0.04878055+0.j ]
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References
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----------
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.. [HNW93] E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary
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Differential Equations i. Nonstiff Problems. 2nd edition.
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Springer Series in Computational Mathematics,
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Springer-Verlag (1993)
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"""
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def __init__(self, f, jac=None):
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self.stiff = 0
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self.f = f
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self.jac = jac
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self.f_params = ()
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self.jac_params = ()
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self._y = []
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@property
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def y(self):
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return self._y
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def set_initial_value(self, y, t=0.0):
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"""Set initial conditions y(t) = y."""
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if isscalar(y):
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y = [y]
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n_prev = len(self._y)
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if not n_prev:
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self.set_integrator('') # find first available integrator
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self._y = asarray(y, self._integrator.scalar)
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self.t = t
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self._integrator.reset(len(self._y), self.jac is not None)
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return self
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def set_integrator(self, name, **integrator_params):
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"""
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Set integrator by name.
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Parameters
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----------
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name : str
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Name of the integrator.
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integrator_params
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Additional parameters for the integrator.
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"""
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integrator = find_integrator(name)
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if integrator is None:
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# FIXME: this really should be raise an exception. Will that break
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# any code?
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warnings.warn('No integrator name match with %r or is not '
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'available.' % name)
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else:
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self._integrator = integrator(**integrator_params)
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if not len(self._y):
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self.t = 0.0
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self._y = array([0.0], self._integrator.scalar)
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self._integrator.reset(len(self._y), self.jac is not None)
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return self
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def integrate(self, t, step=False, relax=False):
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"""Find y=y(t), set y as an initial condition, and return y.
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Parameters
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----------
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t : float
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The endpoint of the integration step.
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step : bool
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If True, and if the integrator supports the step method,
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then perform a single integration step and return.
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This parameter is provided in order to expose internals of
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the implementation, and should not be changed from its default
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value in most cases.
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relax : bool
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If True and if the integrator supports the run_relax method,
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then integrate until t_1 >= t and return. ``relax`` is not
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referenced if ``step=True``.
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This parameter is provided in order to expose internals of
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the implementation, and should not be changed from its default
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value in most cases.
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Returns
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-------
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y : float
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The integrated value at t
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"""
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if step and self._integrator.supports_step:
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mth = self._integrator.step
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elif relax and self._integrator.supports_run_relax:
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mth = self._integrator.run_relax
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else:
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mth = self._integrator.run
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try:
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self._y, self.t = mth(self.f, self.jac or (lambda: None),
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self._y, self.t, t,
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self.f_params, self.jac_params)
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except SystemError as e:
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# f2py issue with tuple returns, see ticket 1187.
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raise ValueError(
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'Function to integrate must not return a tuple.'
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) from e
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return self._y
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def successful(self):
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"""Check if integration was successful."""
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try:
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self._integrator
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except AttributeError:
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self.set_integrator('')
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return self._integrator.success == 1
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def get_return_code(self):
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"""Extracts the return code for the integration to enable better control
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if the integration fails.
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In general, a return code > 0 implies success, while a return code < 0
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implies failure.
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Notes
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-----
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This section describes possible return codes and their meaning, for available
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integrators that can be selected by `set_integrator` method.
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"vode"
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=========== =======
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Return Code Message
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=========== =======
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2 Integration successful.
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-1 Excess work done on this call. (Perhaps wrong MF.)
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-2 Excess accuracy requested. (Tolerances too small.)
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-3 Illegal input detected. (See printed message.)
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-4 Repeated error test failures. (Check all input.)
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-5 Repeated convergence failures. (Perhaps bad Jacobian
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supplied or wrong choice of MF or tolerances.)
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-6 Error weight became zero during problem. (Solution
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component i vanished, and ATOL or ATOL(i) = 0.)
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=========== =======
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"zvode"
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=========== =======
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Return Code Message
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=========== =======
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2 Integration successful.
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-1 Excess work done on this call. (Perhaps wrong MF.)
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-2 Excess accuracy requested. (Tolerances too small.)
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-3 Illegal input detected. (See printed message.)
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-4 Repeated error test failures. (Check all input.)
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-5 Repeated convergence failures. (Perhaps bad Jacobian
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supplied or wrong choice of MF or tolerances.)
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-6 Error weight became zero during problem. (Solution
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component i vanished, and ATOL or ATOL(i) = 0.)
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=========== =======
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"dopri5"
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=========== =======
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Return Code Message
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=========== =======
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1 Integration successful.
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2 Integration successful (interrupted by solout).
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-1 Input is not consistent.
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-2 Larger nsteps is needed.
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-3 Step size becomes too small.
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-4 Problem is probably stiff (interrupted).
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=========== =======
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"dop853"
|
|
|
|
=========== =======
|
|
Return Code Message
|
|
=========== =======
|
|
1 Integration successful.
|
|
2 Integration successful (interrupted by solout).
|
|
-1 Input is not consistent.
|
|
-2 Larger nsteps is needed.
|
|
-3 Step size becomes too small.
|
|
-4 Problem is probably stiff (interrupted).
|
|
=========== =======
|
|
|
|
"lsoda"
|
|
|
|
=========== =======
|
|
Return Code Message
|
|
=========== =======
|
|
2 Integration successful.
|
|
-1 Excess work done on this call (perhaps wrong Dfun type).
|
|
-2 Excess accuracy requested (tolerances too small).
|
|
-3 Illegal input detected (internal error).
|
|
-4 Repeated error test failures (internal error).
|
|
-5 Repeated convergence failures (perhaps bad Jacobian or tolerances).
|
|
-6 Error weight became zero during problem.
|
|
-7 Internal workspace insufficient to finish (internal error).
|
|
=========== =======
|
|
"""
|
|
try:
|
|
self._integrator
|
|
except AttributeError:
|
|
self.set_integrator('')
|
|
return self._integrator.istate
|
|
|
|
def set_f_params(self, *args):
|
|
"""Set extra parameters for user-supplied function f."""
|
|
self.f_params = args
|
|
return self
|
|
|
|
def set_jac_params(self, *args):
|
|
"""Set extra parameters for user-supplied function jac."""
|
|
self.jac_params = args
|
|
return self
|
|
|
|
def set_solout(self, solout):
|
|
"""
|
|
Set callable to be called at every successful integration step.
|
|
|
|
Parameters
|
|
----------
|
|
solout : callable
|
|
``solout(t, y)`` is called at each internal integrator step,
|
|
t is a scalar providing the current independent position
|
|
y is the current soloution ``y.shape == (n,)``
|
|
solout should return -1 to stop integration
|
|
otherwise it should return None or 0
|
|
|
|
"""
|
|
if self._integrator.supports_solout:
|
|
self._integrator.set_solout(solout)
|
|
if self._y is not None:
|
|
self._integrator.reset(len(self._y), self.jac is not None)
|
|
else:
|
|
raise ValueError("selected integrator does not support solout,"
|
|
" choose another one")
|
|
|
|
|
|
def _transform_banded_jac(bjac):
|
|
"""
|
|
Convert a real matrix of the form (for example)
|
|
|
|
[0 0 A B] [0 0 0 B]
|
|
[0 0 C D] [0 0 A D]
|
|
[E F G H] to [0 F C H]
|
|
[I J K L] [E J G L]
|
|
[I 0 K 0]
|
|
|
|
That is, every other column is shifted up one.
|
|
"""
|
|
# Shift every other column.
|
|
newjac = zeros((bjac.shape[0] + 1, bjac.shape[1]))
|
|
newjac[1:, ::2] = bjac[:, ::2]
|
|
newjac[:-1, 1::2] = bjac[:, 1::2]
|
|
return newjac
|
|
|
|
|
|
class complex_ode(ode):
|
|
"""
|
|
A wrapper of ode for complex systems.
|
|
|
|
This functions similarly as `ode`, but re-maps a complex-valued
|
|
equation system to a real-valued one before using the integrators.
|
|
|
|
Parameters
|
|
----------
|
|
f : callable ``f(t, y, *f_args)``
|
|
Rhs of the equation. t is a scalar, ``y.shape == (n,)``.
|
|
``f_args`` is set by calling ``set_f_params(*args)``.
|
|
jac : callable ``jac(t, y, *jac_args)``
|
|
Jacobian of the rhs, ``jac[i,j] = d f[i] / d y[j]``.
|
|
``jac_args`` is set by calling ``set_f_params(*args)``.
|
|
|
|
Attributes
|
|
----------
|
|
t : float
|
|
Current time.
|
|
y : ndarray
|
|
Current variable values.
|
|
|
|
Examples
|
|
--------
|
|
For usage examples, see `ode`.
|
|
|
|
"""
|
|
|
|
def __init__(self, f, jac=None):
|
|
self.cf = f
|
|
self.cjac = jac
|
|
if jac is None:
|
|
ode.__init__(self, self._wrap, None)
|
|
else:
|
|
ode.__init__(self, self._wrap, self._wrap_jac)
|
|
|
|
def _wrap(self, t, y, *f_args):
|
|
f = self.cf(*((t, y[::2] + 1j * y[1::2]) + f_args))
|
|
# self.tmp is a real-valued array containing the interleaved
|
|
# real and imaginary parts of f.
|
|
self.tmp[::2] = real(f)
|
|
self.tmp[1::2] = imag(f)
|
|
return self.tmp
|
|
|
|
def _wrap_jac(self, t, y, *jac_args):
|
|
# jac is the complex Jacobian computed by the user-defined function.
|
|
jac = self.cjac(*((t, y[::2] + 1j * y[1::2]) + jac_args))
|
|
|
|
# jac_tmp is the real version of the complex Jacobian. Each complex
|
|
# entry in jac, say 2+3j, becomes a 2x2 block of the form
|
|
# [2 -3]
|
|
# [3 2]
|
|
jac_tmp = zeros((2 * jac.shape[0], 2 * jac.shape[1]))
|
|
jac_tmp[1::2, 1::2] = jac_tmp[::2, ::2] = real(jac)
|
|
jac_tmp[1::2, ::2] = imag(jac)
|
|
jac_tmp[::2, 1::2] = -jac_tmp[1::2, ::2]
|
|
|
|
ml = getattr(self._integrator, 'ml', None)
|
|
mu = getattr(self._integrator, 'mu', None)
|
|
if ml is not None or mu is not None:
|
|
# Jacobian is banded. The user's Jacobian function has computed
|
|
# the complex Jacobian in packed format. The corresponding
|
|
# real-valued version has every other column shifted up.
|
|
jac_tmp = _transform_banded_jac(jac_tmp)
|
|
|
|
return jac_tmp
|
|
|
|
@property
|
|
def y(self):
|
|
return self._y[::2] + 1j * self._y[1::2]
|
|
|
|
def set_integrator(self, name, **integrator_params):
|
|
"""
|
|
Set integrator by name.
|
|
|
|
Parameters
|
|
----------
|
|
name : str
|
|
Name of the integrator
|
|
integrator_params
|
|
Additional parameters for the integrator.
|
|
"""
|
|
if name == 'zvode':
|
|
raise ValueError("zvode must be used with ode, not complex_ode")
|
|
|
|
lband = integrator_params.get('lband')
|
|
uband = integrator_params.get('uband')
|
|
if lband is not None or uband is not None:
|
|
# The Jacobian is banded. Override the user-supplied bandwidths
|
|
# (which are for the complex Jacobian) with the bandwidths of
|
|
# the corresponding real-valued Jacobian wrapper of the complex
|
|
# Jacobian.
|
|
integrator_params['lband'] = 2 * (lband or 0) + 1
|
|
integrator_params['uband'] = 2 * (uband or 0) + 1
|
|
|
|
return ode.set_integrator(self, name, **integrator_params)
|
|
|
|
def set_initial_value(self, y, t=0.0):
|
|
"""Set initial conditions y(t) = y."""
|
|
y = asarray(y)
|
|
self.tmp = zeros(y.size * 2, 'float')
|
|
self.tmp[::2] = real(y)
|
|
self.tmp[1::2] = imag(y)
|
|
return ode.set_initial_value(self, self.tmp, t)
|
|
|
|
def integrate(self, t, step=False, relax=False):
|
|
"""Find y=y(t), set y as an initial condition, and return y.
|
|
|
|
Parameters
|
|
----------
|
|
t : float
|
|
The endpoint of the integration step.
|
|
step : bool
|
|
If True, and if the integrator supports the step method,
|
|
then perform a single integration step and return.
|
|
This parameter is provided in order to expose internals of
|
|
the implementation, and should not be changed from its default
|
|
value in most cases.
|
|
relax : bool
|
|
If True and if the integrator supports the run_relax method,
|
|
then integrate until t_1 >= t and return. ``relax`` is not
|
|
referenced if ``step=True``.
|
|
This parameter is provided in order to expose internals of
|
|
the implementation, and should not be changed from its default
|
|
value in most cases.
|
|
|
|
Returns
|
|
-------
|
|
y : float
|
|
The integrated value at t
|
|
"""
|
|
y = ode.integrate(self, t, step, relax)
|
|
return y[::2] + 1j * y[1::2]
|
|
|
|
def set_solout(self, solout):
|
|
"""
|
|
Set callable to be called at every successful integration step.
|
|
|
|
Parameters
|
|
----------
|
|
solout : callable
|
|
``solout(t, y)`` is called at each internal integrator step,
|
|
t is a scalar providing the current independent position
|
|
y is the current soloution ``y.shape == (n,)``
|
|
solout should return -1 to stop integration
|
|
otherwise it should return None or 0
|
|
|
|
"""
|
|
if self._integrator.supports_solout:
|
|
self._integrator.set_solout(solout, complex=True)
|
|
else:
|
|
raise TypeError("selected integrator does not support solouta,"
|
|
+ "choose another one")
|
|
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# ODE integrators
|
|
# ------------------------------------------------------------------------------
|
|
|
|
def find_integrator(name):
|
|
for cl in IntegratorBase.integrator_classes:
|
|
if re.match(name, cl.__name__, re.I):
|
|
return cl
|
|
return None
|
|
|
|
|
|
class IntegratorConcurrencyError(RuntimeError):
|
|
"""
|
|
Failure due to concurrent usage of an integrator that can be used
|
|
only for a single problem at a time.
|
|
|
|
"""
|
|
|
|
def __init__(self, name):
|
|
msg = ("Integrator `%s` can be used to solve only a single problem "
|
|
"at a time. If you want to integrate multiple problems, "
|
|
"consider using a different integrator "
|
|
"(see `ode.set_integrator`)") % name
|
|
RuntimeError.__init__(self, msg)
|
|
|
|
|
|
class IntegratorBase(object):
|
|
runner = None # runner is None => integrator is not available
|
|
success = None # success==1 if integrator was called successfully
|
|
istate = None # istate > 0 means success, istate < 0 means failure
|
|
supports_run_relax = None
|
|
supports_step = None
|
|
supports_solout = False
|
|
integrator_classes = []
|
|
scalar = float
|
|
|
|
def acquire_new_handle(self):
|
|
# Some of the integrators have internal state (ancient
|
|
# Fortran...), and so only one instance can use them at a time.
|
|
# We keep track of this, and fail when concurrent usage is tried.
|
|
self.__class__.active_global_handle += 1
|
|
self.handle = self.__class__.active_global_handle
|
|
|
|
def check_handle(self):
|
|
if self.handle is not self.__class__.active_global_handle:
|
|
raise IntegratorConcurrencyError(self.__class__.__name__)
|
|
|
|
def reset(self, n, has_jac):
|
|
"""Prepare integrator for call: allocate memory, set flags, etc.
|
|
n - number of equations.
|
|
has_jac - if user has supplied function for evaluating Jacobian.
|
|
"""
|
|
|
|
def run(self, f, jac, y0, t0, t1, f_params, jac_params):
|
|
"""Integrate from t=t0 to t=t1 using y0 as an initial condition.
|
|
Return 2-tuple (y1,t1) where y1 is the result and t=t1
|
|
defines the stoppage coordinate of the result.
|
|
"""
|
|
raise NotImplementedError('all integrators must define '
|
|
'run(f, jac, t0, t1, y0, f_params, jac_params)')
|
|
|
|
def step(self, f, jac, y0, t0, t1, f_params, jac_params):
|
|
"""Make one integration step and return (y1,t1)."""
|
|
raise NotImplementedError('%s does not support step() method' %
|
|
self.__class__.__name__)
|
|
|
|
def run_relax(self, f, jac, y0, t0, t1, f_params, jac_params):
|
|
"""Integrate from t=t0 to t>=t1 and return (y1,t)."""
|
|
raise NotImplementedError('%s does not support run_relax() method' %
|
|
self.__class__.__name__)
|
|
|
|
# XXX: __str__ method for getting visual state of the integrator
|
|
|
|
|
|
def _vode_banded_jac_wrapper(jacfunc, ml, jac_params):
|
|
"""
|
|
Wrap a banded Jacobian function with a function that pads
|
|
the Jacobian with `ml` rows of zeros.
|
|
"""
|
|
|
|
def jac_wrapper(t, y):
|
|
jac = asarray(jacfunc(t, y, *jac_params))
|
|
padded_jac = vstack((jac, zeros((ml, jac.shape[1]))))
|
|
return padded_jac
|
|
|
|
return jac_wrapper
|
|
|
|
|
|
class vode(IntegratorBase):
|
|
runner = getattr(_vode, 'dvode', None)
|
|
|
|
messages = {-1: 'Excess work done on this call. (Perhaps wrong MF.)',
|
|
-2: 'Excess accuracy requested. (Tolerances too small.)',
|
|
-3: 'Illegal input detected. (See printed message.)',
|
|
-4: 'Repeated error test failures. (Check all input.)',
|
|
-5: 'Repeated convergence failures. (Perhaps bad'
|
|
' Jacobian supplied or wrong choice of MF or tolerances.)',
|
|
-6: 'Error weight became zero during problem. (Solution'
|
|
' component i vanished, and ATOL or ATOL(i) = 0.)'
|
|
}
|
|
supports_run_relax = 1
|
|
supports_step = 1
|
|
active_global_handle = 0
|
|
|
|
def __init__(self,
|
|
method='adams',
|
|
with_jacobian=False,
|
|
rtol=1e-6, atol=1e-12,
|
|
lband=None, uband=None,
|
|
order=12,
|
|
nsteps=500,
|
|
max_step=0.0, # corresponds to infinite
|
|
min_step=0.0,
|
|
first_step=0.0, # determined by solver
|
|
):
|
|
|
|
if re.match(method, r'adams', re.I):
|
|
self.meth = 1
|
|
elif re.match(method, r'bdf', re.I):
|
|
self.meth = 2
|
|
else:
|
|
raise ValueError('Unknown integration method %s' % method)
|
|
self.with_jacobian = with_jacobian
|
|
self.rtol = rtol
|
|
self.atol = atol
|
|
self.mu = uband
|
|
self.ml = lband
|
|
|
|
self.order = order
|
|
self.nsteps = nsteps
|
|
self.max_step = max_step
|
|
self.min_step = min_step
|
|
self.first_step = first_step
|
|
self.success = 1
|
|
|
|
self.initialized = False
|
|
|
|
def _determine_mf_and_set_bands(self, has_jac):
|
|
"""
|
|
Determine the `MF` parameter (Method Flag) for the Fortran subroutine `dvode`.
|
|
|
|
In the Fortran code, the legal values of `MF` are:
|
|
10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25,
|
|
-11, -12, -14, -15, -21, -22, -24, -25
|
|
but this Python wrapper does not use negative values.
|
|
|
|
Returns
|
|
|
|
mf = 10*self.meth + miter
|
|
|
|
self.meth is the linear multistep method:
|
|
self.meth == 1: method="adams"
|
|
self.meth == 2: method="bdf"
|
|
|
|
miter is the correction iteration method:
|
|
miter == 0: Functional iteraton; no Jacobian involved.
|
|
miter == 1: Chord iteration with user-supplied full Jacobian.
|
|
miter == 2: Chord iteration with internally computed full Jacobian.
|
|
miter == 3: Chord iteration with internally computed diagonal Jacobian.
|
|
miter == 4: Chord iteration with user-supplied banded Jacobian.
|
|
miter == 5: Chord iteration with internally computed banded Jacobian.
|
|
|
|
Side effects: If either self.mu or self.ml is not None and the other is None,
|
|
then the one that is None is set to 0.
|
|
"""
|
|
|
|
jac_is_banded = self.mu is not None or self.ml is not None
|
|
if jac_is_banded:
|
|
if self.mu is None:
|
|
self.mu = 0
|
|
if self.ml is None:
|
|
self.ml = 0
|
|
|
|
# has_jac is True if the user provided a Jacobian function.
|
|
if has_jac:
|
|
if jac_is_banded:
|
|
miter = 4
|
|
else:
|
|
miter = 1
|
|
else:
|
|
if jac_is_banded:
|
|
if self.ml == self.mu == 0:
|
|
miter = 3 # Chord iteration with internal diagonal Jacobian.
|
|
else:
|
|
miter = 5 # Chord iteration with internal banded Jacobian.
|
|
else:
|
|
# self.with_jacobian is set by the user in the call to ode.set_integrator.
|
|
if self.with_jacobian:
|
|
miter = 2 # Chord iteration with internal full Jacobian.
|
|
else:
|
|
miter = 0 # Functional iteraton; no Jacobian involved.
|
|
|
|
mf = 10 * self.meth + miter
|
|
return mf
|
|
|
|
def reset(self, n, has_jac):
|
|
mf = self._determine_mf_and_set_bands(has_jac)
|
|
|
|
if mf == 10:
|
|
lrw = 20 + 16 * n
|
|
elif mf in [11, 12]:
|
|
lrw = 22 + 16 * n + 2 * n * n
|
|
elif mf == 13:
|
|
lrw = 22 + 17 * n
|
|
elif mf in [14, 15]:
|
|
lrw = 22 + 18 * n + (3 * self.ml + 2 * self.mu) * n
|
|
elif mf == 20:
|
|
lrw = 20 + 9 * n
|
|
elif mf in [21, 22]:
|
|
lrw = 22 + 9 * n + 2 * n * n
|
|
elif mf == 23:
|
|
lrw = 22 + 10 * n
|
|
elif mf in [24, 25]:
|
|
lrw = 22 + 11 * n + (3 * self.ml + 2 * self.mu) * n
|
|
else:
|
|
raise ValueError('Unexpected mf=%s' % mf)
|
|
|
|
if mf % 10 in [0, 3]:
|
|
liw = 30
|
|
else:
|
|
liw = 30 + n
|
|
|
|
rwork = zeros((lrw,), float)
|
|
rwork[4] = self.first_step
|
|
rwork[5] = self.max_step
|
|
rwork[6] = self.min_step
|
|
self.rwork = rwork
|
|
|
|
iwork = zeros((liw,), _vode_int_dtype)
|
|
if self.ml is not None:
|
|
iwork[0] = self.ml
|
|
if self.mu is not None:
|
|
iwork[1] = self.mu
|
|
iwork[4] = self.order
|
|
iwork[5] = self.nsteps
|
|
iwork[6] = 2 # mxhnil
|
|
self.iwork = iwork
|
|
|
|
self.call_args = [self.rtol, self.atol, 1, 1,
|
|
self.rwork, self.iwork, mf]
|
|
self.success = 1
|
|
self.initialized = False
|
|
|
|
def run(self, f, jac, y0, t0, t1, f_params, jac_params):
|
|
if self.initialized:
|
|
self.check_handle()
|
|
else:
|
|
self.initialized = True
|
|
self.acquire_new_handle()
|
|
|
|
if self.ml is not None and self.ml > 0:
|
|
# Banded Jacobian. Wrap the user-provided function with one
|
|
# that pads the Jacobian array with the extra `self.ml` rows
|
|
# required by the f2py-generated wrapper.
|
|
jac = _vode_banded_jac_wrapper(jac, self.ml, jac_params)
|
|
|
|
args = ((f, jac, y0, t0, t1) + tuple(self.call_args) +
|
|
(f_params, jac_params))
|
|
y1, t, istate = self.runner(*args)
|
|
self.istate = istate
|
|
if istate < 0:
|
|
unexpected_istate_msg = 'Unexpected istate={:d}'.format(istate)
|
|
warnings.warn('{:s}: {:s}'.format(self.__class__.__name__,
|
|
self.messages.get(istate, unexpected_istate_msg)))
|
|
self.success = 0
|
|
else:
|
|
self.call_args[3] = 2 # upgrade istate from 1 to 2
|
|
self.istate = 2
|
|
return y1, t
|
|
|
|
def step(self, *args):
|
|
itask = self.call_args[2]
|
|
self.call_args[2] = 2
|
|
r = self.run(*args)
|
|
self.call_args[2] = itask
|
|
return r
|
|
|
|
def run_relax(self, *args):
|
|
itask = self.call_args[2]
|
|
self.call_args[2] = 3
|
|
r = self.run(*args)
|
|
self.call_args[2] = itask
|
|
return r
|
|
|
|
|
|
if vode.runner is not None:
|
|
IntegratorBase.integrator_classes.append(vode)
|
|
|
|
|
|
class zvode(vode):
|
|
runner = getattr(_vode, 'zvode', None)
|
|
|
|
supports_run_relax = 1
|
|
supports_step = 1
|
|
scalar = complex
|
|
active_global_handle = 0
|
|
|
|
def reset(self, n, has_jac):
|
|
mf = self._determine_mf_and_set_bands(has_jac)
|
|
|
|
if mf in (10,):
|
|
lzw = 15 * n
|
|
elif mf in (11, 12):
|
|
lzw = 15 * n + 2 * n ** 2
|
|
elif mf in (-11, -12):
|
|
lzw = 15 * n + n ** 2
|
|
elif mf in (13,):
|
|
lzw = 16 * n
|
|
elif mf in (14, 15):
|
|
lzw = 17 * n + (3 * self.ml + 2 * self.mu) * n
|
|
elif mf in (-14, -15):
|
|
lzw = 16 * n + (2 * self.ml + self.mu) * n
|
|
elif mf in (20,):
|
|
lzw = 8 * n
|
|
elif mf in (21, 22):
|
|
lzw = 8 * n + 2 * n ** 2
|
|
elif mf in (-21, -22):
|
|
lzw = 8 * n + n ** 2
|
|
elif mf in (23,):
|
|
lzw = 9 * n
|
|
elif mf in (24, 25):
|
|
lzw = 10 * n + (3 * self.ml + 2 * self.mu) * n
|
|
elif mf in (-24, -25):
|
|
lzw = 9 * n + (2 * self.ml + self.mu) * n
|
|
|
|
lrw = 20 + n
|
|
|
|
if mf % 10 in (0, 3):
|
|
liw = 30
|
|
else:
|
|
liw = 30 + n
|
|
|
|
zwork = zeros((lzw,), complex)
|
|
self.zwork = zwork
|
|
|
|
rwork = zeros((lrw,), float)
|
|
rwork[4] = self.first_step
|
|
rwork[5] = self.max_step
|
|
rwork[6] = self.min_step
|
|
self.rwork = rwork
|
|
|
|
iwork = zeros((liw,), _vode_int_dtype)
|
|
if self.ml is not None:
|
|
iwork[0] = self.ml
|
|
if self.mu is not None:
|
|
iwork[1] = self.mu
|
|
iwork[4] = self.order
|
|
iwork[5] = self.nsteps
|
|
iwork[6] = 2 # mxhnil
|
|
self.iwork = iwork
|
|
|
|
self.call_args = [self.rtol, self.atol, 1, 1,
|
|
self.zwork, self.rwork, self.iwork, mf]
|
|
self.success = 1
|
|
self.initialized = False
|
|
|
|
|
|
if zvode.runner is not None:
|
|
IntegratorBase.integrator_classes.append(zvode)
|
|
|
|
|
|
class dopri5(IntegratorBase):
|
|
runner = getattr(_dop, 'dopri5', None)
|
|
name = 'dopri5'
|
|
supports_solout = True
|
|
|
|
messages = {1: 'computation successful',
|
|
2: 'computation successful (interrupted by solout)',
|
|
-1: 'input is not consistent',
|
|
-2: 'larger nsteps is needed',
|
|
-3: 'step size becomes too small',
|
|
-4: 'problem is probably stiff (interrupted)',
|
|
}
|
|
|
|
def __init__(self,
|
|
rtol=1e-6, atol=1e-12,
|
|
nsteps=500,
|
|
max_step=0.0,
|
|
first_step=0.0, # determined by solver
|
|
safety=0.9,
|
|
ifactor=10.0,
|
|
dfactor=0.2,
|
|
beta=0.0,
|
|
method=None,
|
|
verbosity=-1, # no messages if negative
|
|
):
|
|
self.rtol = rtol
|
|
self.atol = atol
|
|
self.nsteps = nsteps
|
|
self.max_step = max_step
|
|
self.first_step = first_step
|
|
self.safety = safety
|
|
self.ifactor = ifactor
|
|
self.dfactor = dfactor
|
|
self.beta = beta
|
|
self.verbosity = verbosity
|
|
self.success = 1
|
|
self.set_solout(None)
|
|
|
|
def set_solout(self, solout, complex=False):
|
|
self.solout = solout
|
|
self.solout_cmplx = complex
|
|
if solout is None:
|
|
self.iout = 0
|
|
else:
|
|
self.iout = 1
|
|
|
|
def reset(self, n, has_jac):
|
|
work = zeros((8 * n + 21,), float)
|
|
work[1] = self.safety
|
|
work[2] = self.dfactor
|
|
work[3] = self.ifactor
|
|
work[4] = self.beta
|
|
work[5] = self.max_step
|
|
work[6] = self.first_step
|
|
self.work = work
|
|
iwork = zeros((21,), _dop_int_dtype)
|
|
iwork[0] = self.nsteps
|
|
iwork[2] = self.verbosity
|
|
self.iwork = iwork
|
|
self.call_args = [self.rtol, self.atol, self._solout,
|
|
self.iout, self.work, self.iwork]
|
|
self.success = 1
|
|
|
|
def run(self, f, jac, y0, t0, t1, f_params, jac_params):
|
|
x, y, iwork, istate = self.runner(*((f, t0, y0, t1) +
|
|
tuple(self.call_args) + (f_params,)))
|
|
self.istate = istate
|
|
if istate < 0:
|
|
unexpected_istate_msg = 'Unexpected istate={:d}'.format(istate)
|
|
warnings.warn('{:s}: {:s}'.format(self.__class__.__name__,
|
|
self.messages.get(istate, unexpected_istate_msg)))
|
|
self.success = 0
|
|
return y, x
|
|
|
|
def _solout(self, nr, xold, x, y, nd, icomp, con):
|
|
if self.solout is not None:
|
|
if self.solout_cmplx:
|
|
y = y[::2] + 1j * y[1::2]
|
|
return self.solout(x, y)
|
|
else:
|
|
return 1
|
|
|
|
|
|
if dopri5.runner is not None:
|
|
IntegratorBase.integrator_classes.append(dopri5)
|
|
|
|
|
|
class dop853(dopri5):
|
|
runner = getattr(_dop, 'dop853', None)
|
|
name = 'dop853'
|
|
|
|
def __init__(self,
|
|
rtol=1e-6, atol=1e-12,
|
|
nsteps=500,
|
|
max_step=0.0,
|
|
first_step=0.0, # determined by solver
|
|
safety=0.9,
|
|
ifactor=6.0,
|
|
dfactor=0.3,
|
|
beta=0.0,
|
|
method=None,
|
|
verbosity=-1, # no messages if negative
|
|
):
|
|
super(self.__class__, self).__init__(rtol, atol, nsteps, max_step,
|
|
first_step, safety, ifactor,
|
|
dfactor, beta, method,
|
|
verbosity)
|
|
|
|
def reset(self, n, has_jac):
|
|
work = zeros((11 * n + 21,), float)
|
|
work[1] = self.safety
|
|
work[2] = self.dfactor
|
|
work[3] = self.ifactor
|
|
work[4] = self.beta
|
|
work[5] = self.max_step
|
|
work[6] = self.first_step
|
|
self.work = work
|
|
iwork = zeros((21,), _dop_int_dtype)
|
|
iwork[0] = self.nsteps
|
|
iwork[2] = self.verbosity
|
|
self.iwork = iwork
|
|
self.call_args = [self.rtol, self.atol, self._solout,
|
|
self.iout, self.work, self.iwork]
|
|
self.success = 1
|
|
|
|
|
|
if dop853.runner is not None:
|
|
IntegratorBase.integrator_classes.append(dop853)
|
|
|
|
|
|
class lsoda(IntegratorBase):
|
|
runner = getattr(_lsoda, 'lsoda', None)
|
|
active_global_handle = 0
|
|
|
|
messages = {
|
|
2: "Integration successful.",
|
|
-1: "Excess work done on this call (perhaps wrong Dfun type).",
|
|
-2: "Excess accuracy requested (tolerances too small).",
|
|
-3: "Illegal input detected (internal error).",
|
|
-4: "Repeated error test failures (internal error).",
|
|
-5: "Repeated convergence failures (perhaps bad Jacobian or tolerances).",
|
|
-6: "Error weight became zero during problem.",
|
|
-7: "Internal workspace insufficient to finish (internal error)."
|
|
}
|
|
|
|
def __init__(self,
|
|
with_jacobian=False,
|
|
rtol=1e-6, atol=1e-12,
|
|
lband=None, uband=None,
|
|
nsteps=500,
|
|
max_step=0.0, # corresponds to infinite
|
|
min_step=0.0,
|
|
first_step=0.0, # determined by solver
|
|
ixpr=0,
|
|
max_hnil=0,
|
|
max_order_ns=12,
|
|
max_order_s=5,
|
|
method=None
|
|
):
|
|
|
|
self.with_jacobian = with_jacobian
|
|
self.rtol = rtol
|
|
self.atol = atol
|
|
self.mu = uband
|
|
self.ml = lband
|
|
|
|
self.max_order_ns = max_order_ns
|
|
self.max_order_s = max_order_s
|
|
self.nsteps = nsteps
|
|
self.max_step = max_step
|
|
self.min_step = min_step
|
|
self.first_step = first_step
|
|
self.ixpr = ixpr
|
|
self.max_hnil = max_hnil
|
|
self.success = 1
|
|
|
|
self.initialized = False
|
|
|
|
def reset(self, n, has_jac):
|
|
# Calculate parameters for Fortran subroutine dvode.
|
|
if has_jac:
|
|
if self.mu is None and self.ml is None:
|
|
jt = 1
|
|
else:
|
|
if self.mu is None:
|
|
self.mu = 0
|
|
if self.ml is None:
|
|
self.ml = 0
|
|
jt = 4
|
|
else:
|
|
if self.mu is None and self.ml is None:
|
|
jt = 2
|
|
else:
|
|
if self.mu is None:
|
|
self.mu = 0
|
|
if self.ml is None:
|
|
self.ml = 0
|
|
jt = 5
|
|
lrn = 20 + (self.max_order_ns + 4) * n
|
|
if jt in [1, 2]:
|
|
lrs = 22 + (self.max_order_s + 4) * n + n * n
|
|
elif jt in [4, 5]:
|
|
lrs = 22 + (self.max_order_s + 5 + 2 * self.ml + self.mu) * n
|
|
else:
|
|
raise ValueError('Unexpected jt=%s' % jt)
|
|
lrw = max(lrn, lrs)
|
|
liw = 20 + n
|
|
rwork = zeros((lrw,), float)
|
|
rwork[4] = self.first_step
|
|
rwork[5] = self.max_step
|
|
rwork[6] = self.min_step
|
|
self.rwork = rwork
|
|
iwork = zeros((liw,), _lsoda_int_dtype)
|
|
if self.ml is not None:
|
|
iwork[0] = self.ml
|
|
if self.mu is not None:
|
|
iwork[1] = self.mu
|
|
iwork[4] = self.ixpr
|
|
iwork[5] = self.nsteps
|
|
iwork[6] = self.max_hnil
|
|
iwork[7] = self.max_order_ns
|
|
iwork[8] = self.max_order_s
|
|
self.iwork = iwork
|
|
self.call_args = [self.rtol, self.atol, 1, 1,
|
|
self.rwork, self.iwork, jt]
|
|
self.success = 1
|
|
self.initialized = False
|
|
|
|
def run(self, f, jac, y0, t0, t1, f_params, jac_params):
|
|
if self.initialized:
|
|
self.check_handle()
|
|
else:
|
|
self.initialized = True
|
|
self.acquire_new_handle()
|
|
args = [f, y0, t0, t1] + self.call_args[:-1] + \
|
|
[jac, self.call_args[-1], f_params, 0, jac_params]
|
|
y1, t, istate = self.runner(*args)
|
|
self.istate = istate
|
|
if istate < 0:
|
|
unexpected_istate_msg = 'Unexpected istate={:d}'.format(istate)
|
|
warnings.warn('{:s}: {:s}'.format(self.__class__.__name__,
|
|
self.messages.get(istate, unexpected_istate_msg)))
|
|
self.success = 0
|
|
else:
|
|
self.call_args[3] = 2 # upgrade istate from 1 to 2
|
|
self.istate = 2
|
|
return y1, t
|
|
|
|
def step(self, *args):
|
|
itask = self.call_args[2]
|
|
self.call_args[2] = 2
|
|
r = self.run(*args)
|
|
self.call_args[2] = itask
|
|
return r
|
|
|
|
def run_relax(self, *args):
|
|
itask = self.call_args[2]
|
|
self.call_args[2] = 3
|
|
r = self.run(*args)
|
|
self.call_args[2] = itask
|
|
return r
|
|
|
|
|
|
if lsoda.runner:
|
|
IntegratorBase.integrator_classes.append(lsoda)
|