|

| - If "extrapolate", then points outside the data range will be

| extrapolated. ("nearest" and "linear" kinds only.)

|

| 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__

|

| --------

| splrep, splev

| Spline interpolation/smoothing based on FITPACK.

| UnivariateSpline : An object-oriented wrapper of the FITPACK routines.

| interp2d : 2-D 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')

1.12988574323e-07

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 non-stiff systems

of first order ode-s::

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, ...)

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

======= ============================================================

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 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

rows contain the non-zero 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 ``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 : ndarray, optional

Vector of critical points (e.g. singularities) where integration

care should be taken.

h0 : float, (0: solver-determined), optional

The step size to be attempted on the first step.

hmax : float, (0: solver-determined), optional

The maximum absolute step size allowed.

hmin : float, (0: solver-determined), optional

The minimum absolute step size allowed.

ixpr : bool, optional

Whether to generate extra printing at method switches.

mxstep : int, (0: solver-determined), optional

Maximum number of (internally defined) steps allowed for each

integration point in t.

mxhnil : int, (0: solver-determined), optional

Maximum number of messages printed.

mxordn : int, (0: solver-determined), optional

Maximum order to be allowed for the non-stiff (Adams) method.

mxords : int, (0: solver-determined), optional

Maximum order to be allowed for the stiff (BDF) method.

--------

ode : a more object-oriented 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/site-packages/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 non-stiff systems

of first order ode-s::

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, ...)

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

======= ============================================================

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 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

rows contain the non-zero 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 ``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 : ndarray, optional

Vector of critical points (e.g. singularities) where integration

care should be taken.

h0 : float, (0: solver-determined), optional

The step size to be attempted on the first step.

hmax : float, (0: solver-determined), optional

The maximum absolute step size allowed.

hmin : float, (0: solver-determined), optional

The minimum absolute step size allowed.

ixpr : bool, optional

Whether to generate extra printing at method switches.

mxstep : int, (0: solver-determined), optional

Maximum number of (internally defined) steps allowed for each

integration point in t.

mxhnil : int, (0: solver-determined), optional

Maximum number of messages printed.

mxordn : int, (0: solver-determined), optional

Maximum order to be allowed for the non-stiff (Adams) method.

mxords : int, (0: solver-determined), optional

Maximum order to be allowed for the stiff (BDF) method.

--------

ode : a more object-oriented 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]: