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## Automatically adapted for scipy Oct 21, 2005 by
#!/usr/bin/env python
#Author: Pearu Peterson
#Date: 3 Feb 2002
#$Revision: 2233 $
"""
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
Provides:
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()
Supported integrators:
vode - 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
This integrator accepts the following parameters in
set_integrator() method of the ode class:
atol=float|seq
rtol=float|seq
lband=None|int
rband=None|int
method='adams'|'bdf'
with_jacobian=0|1
nsteps = int
(first|min|max)_step = float
order = int # <=12 for adams, <=5 for bdf
"""
"""
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']
__version__ = "$Id: ode.py 2233 2006-09-24 09:30:41Z rkern $"
from numpy import asarray, array, zeros, sin, int32, isscalar
import re, sys
class ode(object):
"""\
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()
Typical usage:
r = ode(f,jac).set_integrator('vode').set_initial_value(y0,t0)
t1 = <final t>
dt = <step>
while r.successful() and r.t < t1:
r.integrate(r.t+dt)
print r.t, r.y
where f and jac have the following signatures:
def f(t,y[,arg1,..]):
return <f(t,y)>
def jac(t,y[,arg1,..]):
return <df/dy(t,y)>
See also:
odeint - an integrator with a simpler interface based on lsoda from ODEPACK
quad - for finding the area under a curve
"""
def __init__(self,f,jac=None):
"""Define equation y' = f(y,t) where (optional) jac = df/dy.
User-supplied functions must have the following signatures:
def f(t,y,...):
return <f(t,y)>
def jac(t,y,...):
return <jac(t,y)>
where ... means extra parameters that can be set with
set_(f|jac)_params(*args)
methods.
"""
self.stiff = 0
self.f = f
self.jac = jac
self.f_params = ()
self.jac_params = ()
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)
self.y = asarray(y, float)
self.t = t
if not n_prev:
self.set_integrator('') # find first available integrator
self._integrator.reset(len(self.y),self.jac is not None)
return self
def set_integrator(self,name,**integrator_params):
"""Set integrator by name."""
integrator = find_integrator(name)
if integrator is None:
print 'No integrator name match with %s 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], float)
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
#############################################################
#### Nothing interesting for an end-user in what follows ####
#############################################################
def find_integrator(name):
for cl in IntegratorBase.integrator_classes:
if re.match(name,cl.__name__,re.I):
print 'Found integrator',cl.__name__
return cl
return
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 = []
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):
try:
import vode as _vode
except ImportError:
print sys.exc_value
_vode = None
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
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
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
def run(self,*args):
y1,t,istate = self.runner(*(args[:5]+tuple(self.call_args)+args[5:]))
if istate <0:
print '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:
IntegratorBase.integrator_classes.append(vode)
def test1():
def f(t,y):
a = sin(6*t)
return y*y-a+y
ode_runner = ode(f)
ode_runner.set_integrator('vode')
ode_runner.set_initial_value([0.1,0.11,.1]*10)
while ode_runner.successful() and ode_runner.t < 50:
y1 = ode_runner.integrate(ode_runner.t+2)
print ode_runner.t,y1[:3]
def test2():
# Stiff problem. Requires analytic Jacobian.
def f(t,y):
ydot0 = -0.04*y[0] + 1e4*y[1]*y[2]
ydot2 = 3e7*y[1]*y[1]
ydot1 = -ydot0-ydot2
return [ydot0,ydot1,ydot2]
def jac(t,y):
jc = [[-0.04,1e4*y[2] ,1e4*y[1]],
[0.04 ,-1e4*y[2]-6e7*y[1],-1e4*y[1]],
[0.0 ,6e7*y[1] ,0.0]]
return jc
r = ode(f,jac).set_integrator('vode',
rtol=1e-4,
atol=[1e-8,1e-14,1e-6],
method='bdf',
)
r.set_initial_value([1,0,0])
print 'At t=%s y=%s'%(r.t,r.y)
tout = 0.4
for i in range(12):
r.integrate(tout)
print 'At t=%s y=%s'%(r.t,r.y)
tout *= 10
if __name__ == "__main__":
print 'Integrators available:',\
', '.join(map(lambda c:c.__name__,
IntegratorBase.integrator_classes))
test1()
test2()
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