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# 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=False, relax=False) 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 unsuccessful>: # 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)
# ------------------------------------------------------------------------------ # User interface # ------------------------------------------------------------------------------
""" A generic interface class to numeric integrators.
Solve an equation system :math:`y'(t) = f(t,y)` with (optional) ``jac = df/dy``.
*Note*: The first two arguments of ``f(t, y, ...)`` are in the opposite order of the arguments in the system definition function used by `scipy.integrate.odeint`.
Parameters ---------- f : callable ``f(t, y, *f_args)`` Right-hand side of the differential equation. t is a scalar, ``y.shape == (n,)``. ``f_args`` is set by calling ``set_f_params(*args)``. `f` should return a scalar, array or list (not a tuple). jac : callable ``jac(t, y, *jac_args)``, optional Jacobian of the right-hand side, ``jac[i,j] = d f[i] / d y[j]``. ``jac_args`` is set by calling ``set_jac_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 - uband : None or int Jacobian band width, jac[i,j] != 0 for i-lband <= j <= i+uband. Setting these requires your jac routine to return the jacobian in packed format, jac_packed[i-j+uband, j] = jac[i,j]. The dimension of the matrix must be (lband+uband+1, len(y)). - method: 'adams' or 'bdf' Which solver to use, Adams (non-stiff) or BDF (stiff) - with_jacobian : bool This option is only considered when the user has not supplied a Jacobian function and has not indicated (by setting either band) that the Jacobian is banded. In this case, `with_jacobian` specifies whether the iteration method of the ODE solver's correction step is chord iteration with an internally generated full Jacobian or functional iteration with no 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).
"lsoda"
Real-valued Variable-coefficient Ordinary Differential Equation solver, with fixed-leading-coefficient implementation. It provides automatic method switching between implicit Adams method (for non-stiff problems) and a method based on backward differentiation formulas (BDF) (for stiff problems).
Source: http://www.netlib.org/odepack
.. warning::
This integrator is not re-entrant. You cannot have two `ode` instances using the "lsoda" 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 - uband : None or int Jacobian band width, jac[i,j] != 0 for i-lband <= j <= i+uband. Setting these requires your jac routine to return the jacobian in packed format, jac_packed[i-j+uband, j] = jac[i,j]. - with_jacobian : bool *Not used.* - 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. - max_order_ns : int Maximum order used in the nonstiff case (default 12). - max_order_s : int Maximum order used in the stiff case (default 5). - max_hnil : int Maximum number of messages reporting too small step size (t + h = t) (default 0) - ixpr : int Whether to generate extra printing at method switches (default False).
"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. - verbosity : int Switch for printing messages (< 0 for no messages).
"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') >>> 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: ... print(r.t+dt, r.integrate(r.t+dt)) 1 [-0.71038232+0.23749653j 0.40000271+0.j ] 2.0 [0.19098503-0.52359246j 0.22222356+0.j ] 3.0 [0.47153208+0.52701229j 0.15384681+0.j ] 4.0 [-0.61905937+0.30726255j 0.11764744+0.j ] 5.0 [0.02340997-0.61418799j 0.09523835+0.j ] 6.0 [0.58643071+0.339819j 0.08000018+0.j ] 7.0 [-0.52070105+0.44525141j 0.06896565+0.j ] 8.0 [-0.15986733-0.61234476j 0.06060616+0.j ] 9.0 [0.64850462+0.15048982j 0.05405414+0.j ] 10.0 [-0.38404699+0.56382299j 0.04878055+0.j ]
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)
"""
self.stiff = 0 self.f = f self.jac = jac self.f_params = () self.jac_params = () self._y = []
def y(self): return self._y
"""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
""" 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
"""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 """ 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
try: self._y, self.t = mth(self.f, self.jac or (lambda: None), self._y, self.t, t, self.f_params, self.jac_params) except SystemError: # f2py issue with tuple returns, see ticket 1187. raise ValueError('Function to integrate must not return a tuple.')
return self._y
"""Check if integration was successful.""" try: self._integrator except AttributeError: self.set_integrator('') return self._integrator.success == 1
"""Extracts the return code for the integration to enable better control if the integration fails.
In general, a return code > 0 implies success while a return code < 0 implies failure.
Notes ----- This section describes possible return codes and their meaning, for available integrators that can be selected by `set_integrator` method.
"vode"
=========== ======= Return Code Message =========== ======= 2 Integration successful. -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.) =========== =======
"zvode"
=========== ======= Return Code Message =========== ======= 2 Integration successful. -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.) =========== =======
"dopri5"
=========== ======= 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). =========== =======
"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
"""Set extra parameters for user-supplied function f.""" self.f_params = args return self
"""Set extra parameters for user-supplied function jac.""" self.jac_params = args return self
""" 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")
""" 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
""" 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`.
"""
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)
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
# 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
def y(self): return self._y[::2] + 1j * self._y[1::2]
""" 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)
"""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)
"""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]
""" 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 # ------------------------------------------------------------------------------
for cl in IntegratorBase.integrator_classes: if re.match(name, cl.__name__, re.I): return cl return None
""" Failure due to concurrent usage of an integrator that can be used only for a single problem at a time.
"""
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)
# 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
if self.handle is not self.__class__.active_global_handle: raise IntegratorConcurrencyError(self.__class__.__name__)
"""Prepare integrator for call: allocate memory, set flags, etc. n - number of equations. has_jac - if user has supplied function for evaluating Jacobian. """
"""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)')
"""Make one integration step and return (y1,t1).""" raise NotImplementedError('%s does not support step() method' % self.__class__.__name__)
"""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
""" 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
-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.)' }
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
""" 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
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,), 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
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
itask = self.call_args[2] self.call_args[2] = 2 r = self.run(*args) self.call_args[2] = itask return r
itask = self.call_args[2] self.call_args[2] = 3 r = self.run(*args) self.call_args[2] = itask return r
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,), 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
2: 'comput. 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)', }
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)
self.solout = solout self.solout_cmplx = complex if solout is None: self.iout = 0 else: self.iout = 1
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 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
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
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
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)
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 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
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)." }
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
# 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,), int32) 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
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
itask = self.call_args[2] self.call_args[2] = 2 r = self.run(*args) self.call_args[2] = itask return r
itask = self.call_args[2] self.call_args[2] = 3 r = self.run(*args) self.call_args[2] = itask return r
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