Skip to content

HTTPS clone URL

Subversion checkout URL

You can clone with
or
.
Download ZIP
Fetching contributors…

Cannot retrieve contributors at this time

894 lines (754 sloc) 29.112 kB
# Authors: Pearu Peterson, Pauli Virtanen, John Travers
"""
First-order ODE integrators.
User-friendly interface to various numerical integrators for solving a
system of first order ODEs with prescribed initial conditions::
d y(t)[i]
--------- = f(t,y(t))[i],
d t
y(t=0)[i] = y0[i],
where::
i = 0, ..., len(y0) - 1
class ode
---------
A generic interface class to numeric integrators. It has the following
methods::
integrator = ode(f,jac=None)
integrator = integrator.set_integrator(name,**params)
integrator = integrator.set_initial_value(y0,t0=0.0)
integrator = integrator.set_f_params(*args)
integrator = integrator.set_jac_params(*args)
y1 = integrator.integrate(t1,step=0,relax=0)
flag = integrator.successful()
class complex_ode
-----------------
This class has the same generic interface as ode, except it can handle complex
f, y and Jacobians by transparently translating them into the equivalent
real valued system. It supports the real valued solvers (i.e not zvode) and is
an alternative to ode with the zvode solver, sometimes performing better.
"""
# XXX: Integrators must have:
# ===========================
# cvode - C version of vode and vodpk with many improvements.
# Get it from http://www.netlib.org/ode/cvode.tar.gz
# To wrap cvode to Python, one must write extension module by
# hand. Its interface is too much 'advanced C' that using f2py
# would be too complicated (or impossible).
#
# How to define a new integrator:
# ===============================
#
# class myodeint(IntegratorBase):
#
# runner = <odeint function> or None
#
# def __init__(self,...): # required
# <initialize>
#
# def reset(self,n,has_jac): # optional
# # n - the size of the problem (number of equations)
# # has_jac - whether user has supplied its own routine for Jacobian
# <allocate memory,initialize further>
#
# def run(self,f,jac,y0,t0,t1,f_params,jac_params): # required
# # this method is called to integrate from t=t0 to t=t1
# # with initial condition y0. f and jac are user-supplied functions
# # that define the problem. f_params,jac_params are additional
# # arguments
# # to these functions.
# <calculate y1>
# if <calculation was unsuccesful>:
# self.success = 0
# return t1,y1
#
# # In addition, one can define step() and run_relax() methods (they
# # take the same arguments as run()) if the integrator can support
# # these features (see IntegratorBase doc strings).
#
# if myodeint.runner:
# IntegratorBase.integrator_classes.append(myodeint)
__all__ = ['ode', 'complex_ode']
__version__ = "$Id$"
__docformat__ = "restructuredtext en"
import re
import warnings
from numpy import asarray, array, zeros, int32, isscalar, real, imag
import vode as _vode
import _dop
#------------------------------------------------------------------------------
# User interface
#------------------------------------------------------------------------------
class ode(object):
"""
A generic interface class to numeric integrators.
Solve an equation system :math:`y'(t) = f(t,y)` with (optional) ``jac = df/dy``.
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.
See also
--------
odeint : an integrator with a simpler interface based on lsoda from ODEPACK
quad : for finding the area under a curve
Notes
-----
Available integrators are listed below. They can be selected using
the `set_integrator` method.
"vode"
Real-valued Variable-coefficient Ordinary Differential Equation
solver, with fixed-leading-coefficient implementation. It provides
implicit Adams method (for non-stiff problems) and a method based on
backward differentiation formulas (BDF) (for stiff problems).
Source: http://www.netlib.org/ode/vode.f
.. warning::
This integrator is not re-entrant. You cannot have two `ode`
instances using the "vode" integrator at the same time.
This integrator accepts the following parameters in `set_integrator`
method of the `ode` class:
- atol : float or sequence
absolute tolerance for solution
- rtol : float or sequence
relative tolerance for solution
- lband : None or int
- rband : None or int
Jacobian band width, jac[i,j] != 0 for i-lband <= j <= i+rband.
Setting these requires your jac routine to return the jacobian
in packed format, jac_packed[i-j+lband, j] = jac[i,j].
- method: 'adams' or 'bdf'
Which solver to use, Adams (non-stiff) or BDF (stiff)
- with_jacobian : bool
Whether to use the jacobian
- nsteps : int
Maximum number of (internally defined) steps allowed during one
call to the solver.
- first_step : float
- min_step : float
- max_step : float
Limits for the step sizes used by the integrator.
- order : int
Maximum order used by the integrator,
order <= 12 for Adams, <= 5 for BDF.
"zvode"
Complex-valued Variable-coefficient Ordinary Differential Equation
solver, with fixed-leading-coefficient implementation. It provides
implicit Adams method (for non-stiff problems) and a method based on
backward differentiation formulas (BDF) (for stiff problems).
Source: http://www.netlib.org/ode/zvode.f
.. warning::
This integrator is not re-entrant. You cannot have two `ode`
instances using the "zvode" integrator at the same time.
This integrator accepts the same parameters in `set_integrator`
as the "vode" solver.
.. note::
When using ZVODE for a stiff system, it should only be used for
the case in which the function f is analytic, that is, when each f(i)
is an analytic function of each y(j). Analyticity means that the
partial derivative df(i)/dy(j) is a unique complex number, and this
fact is critical in the way ZVODE solves the dense or banded linear
systems that arise in the stiff case. For a complex stiff ODE system
in which f is not analytic, ZVODE is likely to have convergence
failures, and for this problem one should instead use DVODE on the
equivalent real system (in the real and imaginary parts of y).
"dopri5"
This is an explicit runge-kutta method of order (4)5 due to Dormand &
Prince (with stepsize control and dense output).
Authors:
E. Hairer and G. Wanner
Universite de Geneve, Dept. de Mathematiques
CH-1211 Geneve 24, Switzerland
e-mail: ernst.hairer@math.unige.ch, gerhard.wanner@math.unige.ch
This code is described in [HNW93]_.
This integrator accepts the following parameters in set_integrator()
method of the ode class:
- atol : float or sequence
absolute tolerance for solution
- rtol : float or sequence
relative tolerance for solution
- nsteps : int
Maximum number of (internally defined) steps allowed during one
call to the solver.
- first_step : float
- max_step : float
- safety : float
Safety factor on new step selection (default 0.9)
- ifactor : float
- dfactor : float
Maximum factor to increase/decrease step size by in one step
- beta : float
Beta parameter for stabilised step size control.
"dop853"
This is an explicit runge-kutta method of order 8(5,3) due to Dormand
& Prince (with stepsize control and dense output).
Options and references the same as "dopri5".
Examples
--------
A problem to integrate and the corresponding jacobian:
>>> from scipy.integrate import ode
>>>
>>> y0, t0 = [1.0j, 2.0], 0
>>>
>>> def f(t, y, arg1):
>>> return [1j*arg1*y[0] + y[1], -arg1*y[1]**2]
>>> def jac(t, y, arg1):
>>> return [[1j*arg1, 1], [0, -arg1*2*y[1]]]
The integration:
>>> r = ode(f, jac).set_integrator('zvode', method='bdf', with_jacobian=True)
>>> r.set_initial_value(y0, t0).set_f_params(2.0).set_jac_params(2.0)
>>> t1 = 10
>>> dt = 1
>>> while r.successful() and r.t < t1:
>>> r.integrate(r.t+dt)
>>> print r.t, r.y
References
----------
.. [HNW93] E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary
Differential Equations i. Nonstiff Problems. 2nd edition.
Springer Series in Computational Mathematics,
Springer-Verlag (1993)
"""
def __init__(self, f, jac=None):
self.stiff = 0
self.f = f
self.jac = jac
self.f_params = ()
self.jac_params = ()
self._y = []
@property
def y(self):
return self._y
def set_initial_value(self, y, t=0.0):
"""Set initial conditions y(t) = y."""
if isscalar(y):
y = [y]
n_prev = len(self._y)
if not n_prev:
self.set_integrator('') # find first available integrator
self._y = asarray(y, self._integrator.scalar)
self.t = t
self._integrator.reset(len(self._y), self.jac is not None)
return self
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.
"""
integrator = find_integrator(name)
if integrator is None:
# FIXME: this really should be raise an exception. Will that break
# any code?
warnings.warn('No integrator name match with %r or is not '
'available.' % name)
else:
self._integrator = integrator(**integrator_params)
if not len(self._y):
self.t = 0.0
self._y = array([0.0], self._integrator.scalar)
self._integrator.reset(len(self._y), self.jac is not None)
return self
def integrate(self, t, step=0, relax=0):
"""Find y=y(t), set y as an initial condition, and return y."""
if step and self._integrator.supports_step:
mth = self._integrator.step
elif relax and self._integrator.supports_run_relax:
mth = self._integrator.run_relax
else:
mth = self._integrator.run
self._y, self.t = mth(self.f, self.jac or (lambda: None),
self._y, self.t, t,
self.f_params, self.jac_params)
return self._y
def successful(self):
"""Check if integration was successful."""
try:
self._integrator
except AttributeError:
self.set_integrator('')
return self._integrator.success == 1
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
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 not None:
ode.__init__(self, self._wrap, self._wrap_jac)
else:
ode.__init__(self, self._wrap, None)
def _wrap(self, t, y, *f_args):
f = self.cf(*((t, y[::2] + 1j * y[1::2]) + f_args))
self.tmp[::2] = real(f)
self.tmp[1::2] = imag(f)
return self.tmp
def _wrap_jac(self, t, y, *jac_args):
jac = self.cjac(*((t, y[::2] + 1j * y[1::2]) + jac_args))
self.jac_tmp[1::2, 1::2] = self.jac_tmp[::2, ::2] = real(jac)
self.jac_tmp[1::2, ::2] = imag(jac)
self.jac_tmp[::2, 1::2] = -self.jac_tmp[1::2, ::2]
return self.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 should be used with ode, not zode")
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)
if self.cjac is not None:
self.jac_tmp = zeros((y.size * 2, y.size * 2), 'float')
return ode.set_initial_value(self, self.tmp, t)
def integrate(self, t, step=0, relax=0):
"""Find y=y(t), set y as an initial condition, and return y."""
y = ode.integrate(self, t, step, relax)
return y[::2] + 1j * y[1::2]
#------------------------------------------------------------------------------
# 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
supports_run_relax = None
supports_step = None
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
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=0,
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 reset(self, n, has_jac):
# Calculate parameters for Fortran subroutine dvode.
if has_jac:
if self.mu is None and self.ml is None:
miter = 1
else:
if self.mu is None:
self.mu = 0
if self.ml is None:
self.ml = 0
miter = 4
else:
if self.mu is None and self.ml is None:
if self.with_jacobian:
miter = 2
else:
miter = 0
else:
if self.mu is None:
self.mu = 0
if self.ml is None:
self.ml = 0
if self.ml == self.mu == 0:
miter = 3
else:
miter = 5
mf = 10 * self.meth + miter
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 miter 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,), int32)
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, *args):
if self.initialized:
self.check_handle()
else:
self.initialized = True
self.acquire_new_handle()
y1, t, istate = self.runner(*(args[:5] + tuple(self.call_args) +
args[5:]))
if istate < 0:
warnings.warn('vode: ' +
self.messages.get(istate,
'Unexpected istate=%s' % istate))
self.success = 0
else:
self.call_args[3] = 2 # upgrade istate from 1 to 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):
# Calculate parameters for Fortran subroutine dvode.
if has_jac:
if self.mu is None and self.ml is None:
miter = 1
else:
if self.mu is None:
self.mu = 0
if self.ml is None:
self.ml = 0
miter = 4
else:
if self.mu is None and self.ml is None:
if self.with_jacobian:
miter = 2
else:
miter = 0
else:
if self.mu is None:
self.mu = 0
if self.ml is None:
self.ml = 0
if self.ml == self.mu == 0:
miter = 3
else:
miter = 5
mf = 10 * self.meth + miter
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 miter 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,), int32)
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
def run(self, *args):
if self.initialized:
self.check_handle()
else:
self.initialized = True
self.acquire_new_handle()
y1, t, istate = self.runner(*(args[:5] + tuple(self.call_args) +
args[5:]))
if istate < 0:
warnings.warn('zvode: ' +
self.messages.get(istate, 'Unexpected istate=%s' % istate))
self.success = 0
else:
self.call_args[3] = 2 # upgrade istate from 1 to 2
return y1, t
if zvode.runner is not None:
IntegratorBase.integrator_classes.append(zvode)
class dopri5(IntegratorBase):
runner = getattr(_dop, 'dopri5', None)
name = 'dopri5'
messages = {1: 'computation successful',
2: 'comput. successful (interrupted by solout)',
-1: 'input is not consistent',
-2: 'larger nmax 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
):
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.success = 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,), int32)
iwork[0] = self.nsteps
self.iwork = iwork
self.call_args = [self.rtol, self.atol, self._solout,
self.work, self.iwork]
self.success = 1
def run(self, f, jac, y0, t0, t1, f_params, jac_params):
x, y, iwork, idid = self.runner(*((f, t0, y0, t1) +
tuple(self.call_args) + (f_params,)))
if idid < 0:
warnings.warn(self.name + ': ' +
self.messages.get(idid, 'Unexpected idid=%s' % idid))
self.success = 0
return y, x
def _solout(self, *args):
# dummy solout function
pass
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
):
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.success = 1
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,), int32)
iwork[0] = self.nsteps
self.iwork = iwork
self.call_args = [self.rtol, self.atol, self._solout,
self.work, self.iwork]
self.success = 1
if dop853.runner is not None:
IntegratorBase.integrator_classes.append(dop853)
Jump to Line
Something went wrong with that request. Please try again.