def odeint(func, y0, t, args=(), Dfun=None, col_deriv=0, full_output=0,
ml=None, mu=None, rtol=None, atol=None, tcrit=None, h0=0.0,
hmax=0.0, hmin=0.0, ixpr=0, mxstep=0, mxhnil=0, mxordn=12,
mxords=5, printmessg=0):
"""
Integrate a system of ordinary differential equations.
Solve a system of ordinary differential equations using lsoda from the
FORTRAN library odepack.
Solves the initial value problem for stiff or non-stiff systems
of first order ode-s::
dy/dt = func(y,t0,...)
where y can be a vector.
Parameters
----------
func : callable(y, t0, ...)
Computes the derivative of y at t0.
y0 : array
Initial condition on y (can be a vector).
t : array
A sequence of time points for which to solve for y. The initial
value point should be the first element of this sequence.
args : tuple
Extra arguments to pass to function.
Dfun : callable(y, t0, ...)
Gradient (Jacobian) of func.
col_deriv : boolean
True if Dfun defines derivatives down columns (faster),
otherwise Dfun should define derivatives across rows.
full_output : boolean
True if to return a dictionary of optional outputs as the second output
printmessg : boolean
Whether to print the convergence message
Returns
-------
y : array, shape (len(y0), len(t))
Array containing the value of y for each desired time in t,
with the initial value y0 in the first row.
infodict : dict, only returned if full_output == True
Dictionary containing additional output information
======= ============================================================
key meaning
======= ============================================================
'hu' vector of step sizes successfully used for each time step.
'tcur' vector with the value of t reached for each time step.
(will always be at least as large as the input times).
'tolsf' vector of tolerance scale factors, greater than 1.0,
computed when a request for too much accuracy was detected.
'tsw' value of t at the time of the last method switch
(given for each time step)
'nst' cumulative number of time steps
'nfe' cumulative number of function evaluations for each time step
'nje' cumulative number of jacobian evaluations for each time step
'nqu' a vector of method orders for each successful step.
'imxer' index of the component of largest magnitude in the
weighted local error vector (e / ewt) on an error return, -1
otherwise.
'lenrw' the length of the double work array required.
'leniw' the length of integer work array required.
'mused' a vector of method indicators for each successful time step:
1: adams (nonstiff), 2: bdf (stiff)
======= ============================================================
Other Parameters
----------------
ml, mu : integer
If either of these are not-None or non-negative, then the
Jacobian is assumed to be banded. These give the number of
lower and upper non-zero diagonals in this banded matrix.
For the banded case, Dfun should return a matrix whose
columns contain the non-zero bands (starting with the
lowest diagonal). Thus, the return matrix from Dfun should
have shape ``len(y0) * (ml + mu + 1) when ml >=0 or mu >=0``
rtol, atol : float
The input parameters rtol and atol determine the error
control performed by the solver. The solver will control the
vector, e, of estimated local errors in y, according to an
inequality of the form ``max-norm of (e / ewt) <= 1``,
where ewt is a vector of positive error weights computed as:
``ewt = rtol * abs(y) + atol``
rtol and atol can be either vectors the same length as y or scalars.
Defaults to 1.49012e-8.
tcrit : array
Vector of critical points (e.g. singularities) where integration
care should be taken.
h0 : float, (0: solver-determined)
The step size to be attempted on the first step.
hmax : float, (0: solver-determined)
The maximum absolute step size allowed.
hmin : float, (0: solver-determined)
The minimum absolute step size allowed.
ixpr : boolean
Whether to generate extra printing at method switches.
mxstep : integer, (0: solver-determined)
Maximum number of (internally defined) steps allowed for each
integration point in t.
mxhnil : integer, (0: solver-determined)
Maximum number of messages printed.
mxordn : integer, (0: solver-determined)
Maximum order to be allowed for the nonstiff (Adams) method.
mxords : integer, (0: solver-determined)
Maximum order to be allowed for the stiff (BDF) method.
See Also
--------
ode : a more object-oriented integrator based on VODE
quad : for finding the area under a curve
"""
if ml is None:
ml = -1 # changed to zero inside function call
if mu is None:
mu = -1 # changed to zero inside function call
t = copy(t)
y0 = copy(y0)
output = _odepack.odeint(func, y0, t, args, Dfun, col_deriv, ml, mu,
full_output, rtol, atol, tcrit, h0, hmax, hmin,
ixpr, mxstep, mxhnil, mxordn, mxords)
if output[-1] < 0:
print _msgs[output[-1]]
print "Run with full_output = 1 to get quantitative information."
else:
if printmessg:
print _msgs[output[-1]]
if full_output:
output[1]['message'] = _msgs[output[-1]]
output = output[:-1]
if len(output) == 1:
return output[0]
else:
return output
def odeint(func,
y0,
t,
args=(),
Dfun=None,
col_deriv=0,
full_output=0,
ml=None,
mu=None,
rtol=None,
atol=None,
tcrit=None,
h0=0.0,
hmax=0.0,
hmin=0.0,
ixpr=0,
mxstep=0,
mxhnil=0,
mxordn=12,
mxords=5,
printmessg=0):
"""
Integrate a system of ordinary differential equations.
Solve a system of ordinary differential equations using lsoda from the
FORTRAN library odepack.
Solves the initial value problem for stiff or non-stiff systems
of first order ode-s::
dy/dt = func(y,t0,...)
where y can be a vector.
Parameters
----------
func : callable(y, t0, ...)
Computes the derivative of y at t0.
y0 : array
Initial condition on y (can be a vector).
t : array
A sequence of time points for which to solve for y. The initial
value point should be the first element of this sequence.
args : tuple
Extra arguments to pass to function.
Dfun : callable(y, t0, ...)
Gradient (Jacobian) of func.
col_deriv : boolean
True if Dfun defines derivatives down columns (faster),
otherwise Dfun should define derivatives across rows.
full_output : boolean
True if to return a dictionary of optional outputs as the second output
printmessg : boolean
Whether to print the convergence message
Returns
-------
y : array, shape (len(y0), len(t))
Array containing the value of y for each desired time in t,
with the initial value y0 in the first row.
infodict : dict, only returned if full_output == True
Dictionary containing additional output information
======= ============================================================
key meaning
======= ============================================================
'hu' vector of step sizes successfully used for each time step.
'tcur' vector with the value of t reached for each time step.
(will always be at least as large as the input times).
'tolsf' vector of tolerance scale factors, greater than 1.0,
computed when a request for too much accuracy was detected.
'tsw' value of t at the time of the last method switch
(given for each time step)
'nst' cumulative number of time steps
'nfe' cumulative number of function evaluations for each time step
'nje' cumulative number of jacobian evaluations for each time step
'nqu' a vector of method orders for each successful step.
'imxer' index of the component of largest magnitude in the
weighted local error vector (e / ewt) on an error return, -1
otherwise.
'lenrw' the length of the double work array required.
'leniw' the length of integer work array required.
'mused' a vector of method indicators for each successful time step:
1: adams (nonstiff), 2: bdf (stiff)
======= ============================================================
Other Parameters
----------------
ml, mu : integer
If either of these are not-None or non-negative, then the
Jacobian is assumed to be banded. These give the number of
lower and upper non-zero diagonals in this banded matrix.
For the banded case, Dfun should return a matrix whose
columns contain the non-zero bands (starting with the
lowest diagonal). Thus, the return matrix from Dfun should
have shape ``len(y0) * (ml + mu + 1) when ml >=0 or mu >=0``
rtol, atol : float
The input parameters rtol and atol determine the error
control performed by the solver. The solver will control the
vector, e, of estimated local errors in y, according to an
inequality of the form ``max-norm of (e / ewt) <= 1``,
where ewt is a vector of positive error weights computed as:
``ewt = rtol * abs(y) + atol``
rtol and atol can be either vectors the same length as y or scalars.
tcrit : array
Vector of critical points (e.g. singularities) where integration
care should be taken.
h0 : float, (0: solver-determined)
The step size to be attempted on the first step.
hmax : float, (0: solver-determined)
The maximum absolute step size allowed.
hmin : float, (0: solver-determined)
The minimum absolute step size allowed.
ixpr : boolean
Whether to generate extra printing at method switches.
mxstep : integer, (0: solver-determined)
Maximum number of (internally defined) steps allowed for each
integration point in t.
mxhnil : integer, (0: solver-determined)
Maximum number of messages printed.
mxordn : integer, (0: solver-determined)
Maximum order to be allowed for the nonstiff (Adams) method.
mxords : integer, (0: solver-determined)
Maximum order to be allowed for the stiff (BDF) method.
See Also
--------
ode : a more object-oriented integrator based on VODE
quad : for finding the area under a curve
"""
if ml is None:
ml = -1 # changed to zero inside function call
if mu is None:
mu = -1 # changed to zero inside function call
t = copy(t)
y0 = copy(y0)
output = _odepack.odeint(func, y0, t, args, Dfun, col_deriv, ml, mu,
full_output, rtol, atol, tcrit, h0, hmax, hmin,
ixpr, mxstep, mxhnil, mxordn, mxords)
if output[-1] < 0:
print _msgs[output[-1]]
print "Run with full_output = 1 to get quantitative information."
else:
if printmessg:
print _msgs[output[-1]]
if full_output:
output[1]['message'] = _msgs[output[-1]]
output = output[:-1]
if len(output) == 1:
return output[0]
else:
return output
def odeint(func, y0, t, args=(), Dfun=None, col_deriv=0, full_output=0,
ml=None, mu=None, rtol=None, atol=None, tcrit=None, h0=0.0,
hmax=0.0, hmin=0.0, ixpr=0, mxstep=0, mxhnil=0, mxordn=12,
mxords=5, printmessg=0):
"""Integrate a system of ordinary differential equations.
Description:
Solve a system of ordinary differential equations Using lsoda from the
FORTRAN library odepack.
Solves the initial value problem for stiff or non-stiff systems
of first order ode-s:
dy/dt = func(y,t0,...) where y can be a vector.
Inputs:
func -- func(y,t0,...) computes the derivative of y at t0.
y0 -- initial condition on y (can be a vector).
t -- a sequence of time points for which to solve for y. The intial
value point should be the first element of this sequence.
args -- extra arguments to pass to function.
Dfun -- the gradient (Jacobian) of func (same input signature as func).
col_deriv -- non-zero implies that Dfun defines derivatives down
columns (faster), otherwise Dfun should define derivatives
across rows.
full_output -- non-zero to return a dictionary of optional outputs as
the second output.
printmessg -- print the convergence message.
Outputs: (y, {infodict,})
y -- a rank-2 array containing the value of y in each row for each
desired time in t (with the initial value y0 in the first row).
infodict -- a dictionary of optional outputs:
'hu' : a vector of step sizes successfully used for each time step.
'tcur' : a vector with the value of t reached for each time step.
(will always be at least as large as the input times).
'tolsf' : a vector of tolerance scale factors, greater than 1.0,
computed when a request for too much accuracy was detected.
'tsw' : the value of t at the time of the last method switch
(given for each time step).
'nst' : the cumulative number of time steps.
'nfe' : the cumulative number of function evaluations for eadh
time step.
'nje' : the cumulative number of jacobian evaluations for each
time step.
'nqu' : a vector of method orders for each successful step.
'imxer' : index of the component of largest magnitude in the
weighted local error vector (e / ewt) on an error return.
'lenrw' : the length of the double work array required.
'leniw' : the length of integer work array required.
'mused' : a vector of method indicators for each successful time step:
1 -- adams (nonstiff)
2 -- bdf (stiff)
Additional Inputs:
ml, mu -- If either of these are not-None or non-negative, then the
Jacobian is assumed to be banded. These give the number of
lower and upper non-zero diagonals in this banded matrix.
For the banded case, Dfun should return a matrix whose
columns contain the non-zero bands (starting with the
lowest diagonal). Thus, the return matrix from Dfun should
have shape len(y0) x (ml + mu + 1) when ml >=0 or mu >=0
rtol -- The input parameters rtol and atol determine the error
atol control performed by the solver. The solver will control the
vector, e, of estimated local errors in y, according to an
inequality of the form
max-norm of (e / ewt) <= 1
where ewt is a vector of positive error weights computed as
ewt = rtol * abs(y) + atol
rtol and atol can be either vectors the same length as y or
scalars.
tcrit -- a vector of critical points (e.g. singularities) where
integration care should be taken.
(For the next inputs a zero default means the solver determines it).
h0 -- the step size to be attempted on the first step.
hmax -- the maximum absolute step size allowed.
hmin -- the minimum absolute step size allowed.
ixpr -- non-zero to generate extra printing at method switches.
mxstep -- maximum number of (internally defined) steps allowed
for each integration point in t.
mxhnil -- maximum number of messages printed.
mxordn -- maximum order to be allowed for the nonstiff (Adams) method.
mxords -- maximum order to be allowed for the stiff (BDF) method.
See also:
ode - a more object-oriented integrator based on VODE
quad - for finding the area under a curve
"""
if ml is None:
ml = -1 # changed to zero inside function call
if mu is None:
mu = -1 # changed to zero inside function call
t = copy(t)
y0 = copy(y0)
output = _odepack.odeint(func, y0, t, args, Dfun, col_deriv, ml, mu,
full_output, rtol, atol, tcrit, h0, hmax, hmin,
ixpr, mxstep, mxhnil, mxordn, mxords)
if output[-1] < 0:
print _msgs[output[-1]]
print "Run with full_output = 1 to get quantitative information."
else:
if printmessg:
print _msgs[output[-1]]
if full_output:
output[1]['message'] = _msgs[output[-1]]
output = output[:-1]
if len(output) == 1:
return output[0]
else:
return output
