
 .. versionadded:: 0.17.0
  If "extrapolate", then points outside the data range will be
 extrapolated. ("nearest" and "linear" kinds only.)

 .. versionadded:: 0.17.0
 assume_sorted : bool, optional
 If False, values of `x` can be in any order and they are sorted first.
 If True, `x` has to be an array of monotonically increasing values.

 Methods
 
 __call__

 See Also
 
 splrep, splev
 Spline interpolation/smoothing based on FITPACK.
 UnivariateSpline : An objectoriented wrapper of the FITPACK routines.
 interp2d : 2D interpolation

 Examples
 
 >>> import matplotlib.pyplot as plt
 >>> from scipy import interpolate
 >>> x = np.arange(0, 10)
 >>> y = np.exp(x/3.0)
 >>> f = interpolate.interp1d(x, y)

 >>> xnew = np.arange(0, 9, 0.1)
 >>> ynew = f(xnew) # use interpolation function returned by `interp1d`
 >>> plt.plot(x, y, 'o', xnew, ynew, '')
 >>> plt.show()

 Method resolution order:
 interp1d
 scipy.interpolate.polyint._Interpolator1D
 builtins.object

 Methods defined here:

 __init__(self, x, y, kind='linear', axis=1, copy=True, bounds_error=None, fill_value=nan, assume_sorted=False)
 Initialize a 1D linear interpolation class.

 
 Data descriptors defined here:

 __dict__
 dictionary for instance variables (if defined)

 __weakref__
 list of weak references to the object (if defined)

 fill_value

 
 Methods inherited from scipy.interpolate.polyint._Interpolator1D:

 __call__(self, x)
 Evaluate the interpolant

 Parameters
 
 x : array_like
 Points to evaluate the interpolant at.

 Returns
 
 y : array_like
 Interpolated values. Shape is determined by replacing
 the interpolation axis in the original array with the shape of x.

 
 Data descriptors inherited from scipy.interpolate.polyint._Interpolator1D:

 dtype
In [5]: runfile('/Users/progprim/Desktop/code/ode/odeintexample1.py', wdir='/Users/progprim/Desktop/code/ode')
In [6]: %reset
Once deleted, variables cannot be recovered. Proceed (y/[n])? y
In [7]: runfile('/Users/progprim/Desktop/code/ode/odeintexample1dHO.py', wdir='/Users/progprim/Desktop/code/ode')
In [8]: ys.shape
Out[8]: (200, 2)
In [9]: ts.shape
Out[9]: (200,)
In [10]: runfile('/Users/progprim/Desktop/code/ode/odeintexample1dHO_energy.py', wdir='/Users/progprim/Desktop/code/ode')
1.12988574323e07
In [11]: odeint?
Signature: 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)
Docstring:
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 nonstiff systems
of first order odes::
dy/dt = func(y, t0, ...)
where y can be a vector.
*Note*: The first two arguments of ``func(y, t0, ...)`` are in the
opposite order of the arguments in the system definition function used
by the `scipy.integrate.ode` class.
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, optional
Extra arguments to pass to function.
Dfun : callable(y, t0, ...)
Gradient (Jacobian) of `func`.
col_deriv : bool, optional
True if `Dfun` defines derivatives down columns (faster),
otherwise `Dfun` should define derivatives across rows.
full_output : bool, optional
True if to return a dictionary of optional outputs as the second output
printmessg : bool, optional
Whether to print the convergence message
Returns

y : array, shape (len(t), len(y0))
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 : int, optional
If either of these are not None or nonnegative, then the
Jacobian is assumed to be banded. These give the number of
lower and upper nonzero diagonals in this banded matrix.
For the banded case, `Dfun` should return a matrix whose
rows contain the nonzero bands (starting with the lowest diagonal).
Thus, the return matrix `jac` from `Dfun` should have shape
``(ml + mu + 1, len(y0))`` when ``ml >=0`` or ``mu >=0``.
The data in `jac` must be stored such that ``jac[i  j + mu, j]``
holds the derivative of the `i`th equation with respect to the `j`th
state variable. If `col_deriv` is True, the transpose of this
`jac` must be returned.
rtol, atol : float, optional
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 ``maxnorm 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 1.49012e8.
tcrit : ndarray, optional
Vector of critical points (e.g. singularities) where integration
care should be taken.
h0 : float, (0: solverdetermined), optional
The step size to be attempted on the first step.
hmax : float, (0: solverdetermined), optional
The maximum absolute step size allowed.
hmin : float, (0: solverdetermined), optional
The minimum absolute step size allowed.
ixpr : bool, optional
Whether to generate extra printing at method switches.
mxstep : int, (0: solverdetermined), optional
Maximum number of (internally defined) steps allowed for each
integration point in t.
mxhnil : int, (0: solverdetermined), optional
Maximum number of messages printed.
mxordn : int, (0: solverdetermined), optional
Maximum order to be allowed for the nonstiff (Adams) method.
mxords : int, (0: solverdetermined), optional
Maximum order to be allowed for the stiff (BDF) method.
See Also

ode : a more objectoriented integrator based on VODE.
quad : for finding the area under a curve.
Examples

The second order differential equation for the angle `theta` of a
pendulum acted on by gravity with friction can be written::
theta''(t) + b*theta'(t) + c*sin(theta(t)) = 0
where `b` and `c` are positive constants, and a prime (') denotes a
derivative. To solve this equation with `odeint`, we must first convert
it to a system of first order equations. By defining the angular
velocity ``omega(t) = theta'(t)``, we obtain the system::
theta'(t) = omega(t)
omega'(t) = b*omega(t)  c*sin(theta(t))
Let `y` be the vector [`theta`, `omega`]. We implement this system
in python as:
>>> def pend(y, t, b, c):
... theta, omega = y
... dydt = [omega, b*omega  c*np.sin(theta)]
... return dydt
...
We assume the constants are `b` = 0.25 and `c` = 5.0:
>>> b = 0.25
>>> c = 5.0
For initial conditions, we assume the pendulum is nearly vertical
with `theta(0)` = `pi`  0.1, and it initially at rest, so
`omega(0)` = 0. Then the vector of initial conditions is
>>> y0 = [np.pi  0.1, 0.0]
We generate a solution 101 evenly spaced samples in the interval
0 <= `t` <= 10. So our array of times is:
>>> t = np.linspace(0, 10, 101)
Call `odeint` to generate the solution. To pass the parameters
`b` and `c` to `pend`, we give them to `odeint` using the `args`
argument.
>>> from scipy.integrate import odeint
>>> sol = odeint(pend, y0, t, args=(b, c))
The solution is an array with shape (101, 2). The first column
is `theta(t)`, and the second is `omega(t)`. The following code
plots both components.
>>> import matplotlib.pyplot as plt
>>> plt.plot(t, sol[:, 0], 'b', label='theta(t)')
>>> plt.plot(t, sol[:, 1], 'g', label='omega(t)')
>>> plt.legend(loc='best')
>>> plt.xlabel('t')
>>> plt.grid()
>>> plt.show()
File: ~/anaconda/lib/python3.5/sitepackages/scipy/integrate/odepack.py
Type: function
In [12]: help(odeint)
Help on function odeint in module scipy.integrate.odepack:
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 nonstiff systems
of first order odes::
dy/dt = func(y, t0, ...)
where y can be a vector.
*Note*: The first two arguments of ``func(y, t0, ...)`` are in the
opposite order of the arguments in the system definition function used
by the `scipy.integrate.ode` class.
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, optional
Extra arguments to pass to function.
Dfun : callable(y, t0, ...)
Gradient (Jacobian) of `func`.
col_deriv : bool, optional
True if `Dfun` defines derivatives down columns (faster),
otherwise `Dfun` should define derivatives across rows.
full_output : bool, optional
True if to return a dictionary of optional outputs as the second output
printmessg : bool, optional
Whether to print the convergence message
Returns

y : array, shape (len(t), len(y0))
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 : int, optional
If either of these are not None or nonnegative, then the
Jacobian is assumed to be banded. These give the number of
lower and upper nonzero diagonals in this banded matrix.
For the banded case, `Dfun` should return a matrix whose
rows contain the nonzero bands (starting with the lowest diagonal).
Thus, the return matrix `jac` from `Dfun` should have shape
``(ml + mu + 1, len(y0))`` when ``ml >=0`` or ``mu >=0``.
The data in `jac` must be stored such that ``jac[i  j + mu, j]``
holds the derivative of the `i`th equation with respect to the `j`th
state variable. If `col_deriv` is True, the transpose of this
`jac` must be returned.
rtol, atol : float, optional
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 ``maxnorm 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 1.49012e8.
tcrit : ndarray, optional
Vector of critical points (e.g. singularities) where integration
care should be taken.
h0 : float, (0: solverdetermined), optional
The step size to be attempted on the first step.
hmax : float, (0: solverdetermined), optional
The maximum absolute step size allowed.
hmin : float, (0: solverdetermined), optional
The minimum absolute step size allowed.
ixpr : bool, optional
Whether to generate extra printing at method switches.
mxstep : int, (0: solverdetermined), optional
Maximum number of (internally defined) steps allowed for each
integration point in t.
mxhnil : int, (0: solverdetermined), optional
Maximum number of messages printed.
mxordn : int, (0: solverdetermined), optional
Maximum order to be allowed for the nonstiff (Adams) method.
mxords : int, (0: solverdetermined), optional
Maximum order to be allowed for the stiff (BDF) method.
See Also

ode : a more objectoriented integrator based on VODE.
quad : for finding the area under a curve.
Examples

The second order differential equation for the angle `theta` of a
pendulum acted on by gravity with friction can be written::
theta''(t) + b*theta'(t) + c*sin(theta(t)) = 0
where `b` and `c` are positive constants, and a prime (') denotes a
derivative. To solve this equation with `odeint`, we must first convert
it to a system of first order equations. By defining the angular
velocity ``omega(t) = theta'(t)``, we obtain the system::
theta'(t) = omega(t)
omega'(t) = b*omega(t)  c*sin(theta(t))
Let `y` be the vector [`theta`, `omega`]. We implement this system
in python as:
>>> def pend(y, t, b, c):
... theta, omega = y
... dydt = [omega, b*omega  c*np.sin(theta)]
... return dydt
...
We assume the constants are `b` = 0.25 and `c` = 5.0:
>>> b = 0.25
>>> c = 5.0
For initial conditions, we assume the pendulum is nearly vertical
with `theta(0)` = `pi`  0.1, and it initially at rest, so
`omega(0)` = 0. Then the vector of initial conditions is
>>> y0 = [np.pi  0.1, 0.0]
We generate a solution 101 evenly spaced samples in the interval
0 <= `t` <= 10. So our array of times is:
>>> t = np.linspace(0, 10, 101)
Call `odeint` to generate the solution. To pass the parameters
`b` and `c` to `pend`, we give them to `odeint` using the `args`
argument.
>>> from scipy.integrate import odeint
>>> sol = odeint(pend, y0, t, args=(b, c))
The solution is an array with shape (101, 2). The first column
is `theta(t)`, and the second is `omega(t)`. The following code
plots both components.
>>> import matplotlib.pyplot as plt
>>> plt.plot(t, sol[:, 0], 'b', label='theta(t)')
>>> plt.plot(t, sol[:, 1], 'g', label='omega(t)')
>>> plt.legend(loc='best')
>>> plt.xlabel('t')
>>> plt.grid()
>>> plt.show()
In [13]: runfile('/Users/progprim/Desktop/code/ode/odeintexample1dHO.py', wdir='/Users/progprim/Desktop/code/ode')
In [14]: runfile('/Users/progprim/Desktop/code/ode/odeintexample1dHO.py', wdir='/Users/progprim/Desktop/code/ode')
In [15]: runfile('/Users/progprim/Desktop/code/ode/odeintexample1dHO.py', wdir='/Users/progprim/Desktop/code/ode')
In [16]: runfile('/Users/progprim/Desktop/code/ode/odeintexample1dHO.py', wdir='/Users/progprim/Desktop/code/ode')
In [17]: runfile('/Users/progprim/Desktop/code/ode/odeintexample1dHO.py', wdir='/Users/progprim/Desktop/code/ode')
In [18]: runfile('/Users/progprim/Desktop/code/ode/odeintexample1dHO.py', wdir='/Users/progprim/Desktop/code/ode')
In [19]: runfile('/Users/progprim/Desktop/code/ode/odeintexample1dHO.py', wdir='/Users/progprim/Desktop/code/ode')
In [20]: runfile('/Users/progprim/Desktop/code/ode/odeintexample1dHO.py', wdir='/Users/progprim/Desktop/code/ode')
In [21]: runfile('/Users/progprim/Desktop/code/ode/odeintexample1dHO.py', wdir='/Users/progprim/Desktop/code/ode')
In [22]: