def _integrate_runge_kutta_jit(f, time, ic, time_direction, write_steps, b, c,
a):
n_traj = ic.shape[0]
n_dim = ic.shape[1]
s = len(b)
if write_steps == 0:
n_records = 1
else:
tot = time[::write_steps]
n_records = len(tot)
if tot[-1] != time[-1]:
n_records += 1
recorded_traj = np.zeros((n_traj, n_dim, n_records))
if time_direction == -1:
directed_time = reverse(time)
else:
directed_time = time
for i_traj in range(n_traj):
y = ic[i_traj].copy()
k = np.zeros((s, n_dim))
iw = 0
for ti, (tt, dt) in enumerate(
zip(directed_time[:-1], np.diff(directed_time))):
if write_steps > 0 and np.mod(ti, write_steps) == 0:
recorded_traj[i_traj, :, iw] = y
iw += 1
k.fill(0.)
for i in range(s):
y_s = y + dt * a[i] @ k
k[i] = f(tt + c[i] * dt, y_s)
y_new = y + dt * b @ k
y = y_new
recorded_traj[i_traj, :, -1] = y
return recorded_traj[:, :, ::time_direction]
def _integrate_runge_kutta_tgls_jit(f, fjac, time, ic, tg_ic, time_direction,
write_steps, b, c, a, adjoint, inverse,
boundary):
n_traj = ic.shape[0]
n_dim = ic.shape[1]
s = len(b)
if write_steps == 0:
n_records = 1
else:
tot = time[::write_steps]
n_records = len(tot)
if tot[-1] != time[-1]:
n_records += 1
recorded_traj = np.zeros((n_traj, n_dim, n_records))
recorded_fmatrix = np.zeros(
(n_traj, tg_ic.shape[1], tg_ic.shape[2], n_records))
if time_direction == -1:
directed_time = reverse(time)
else:
directed_time = time
for i_traj in range(n_traj):
y = ic[i_traj].copy()
fm = tg_ic[i_traj].copy()
recorded_traj[i_traj, :, 0] = ic[i_traj]
recorded_fmatrix[i_traj, :, :, 0] = tg_ic[i_traj]
k = np.zeros((s, n_dim))
km = np.zeros((s, tg_ic.shape[1], tg_ic.shape[2]))
iw = 0
for ti, (tt, dt) in enumerate(
zip(directed_time[:-1], np.diff(directed_time))):
if write_steps > 0 and np.mod(ti, write_steps) == 0:
recorded_traj[i_traj, :, iw] = y
recorded_fmatrix[i_traj, :, :, iw] = fm
iw += 1
k.fill(0.)
km.fill(0.)
for i in range(s):
y_s = y + dt * a[i] @ k
k[i] = f(tt + c[i] * dt, y_s)
km_s = fm.copy()
for j in range(len(a[i])):
km_s += dt * a[i, j] * km[j]
hom = inverse * _tangent_linear_system(fjac, tt + c[i] * dt,
y_s, km_s, adjoint)
inhom = boundary(tt + c[i] * dt, y_s)
km[i] = (hom.T + inhom.T).T
y_new = y + dt * b @ k
fm_new = fm.copy()
for j in range(len(b)):
fm_new += dt * b[j] * km[j]
y = y_new
fm = fm_new
recorded_traj[i_traj, :, -1] = y
recorded_fmatrix[i_traj, :, :, -1] = fm
return recorded_traj[:, :, ::
time_direction], recorded_fmatrix[:, :, :, ::
time_direction]
def integrate_runge_kutta(f,
t0,
t,
dt,
ic=None,
forward=True,
write_steps=1,
b=None,
c=None,
a=None):
"""
Integrate the ordinary differential equations (ODEs)
.. math:: \dot{\\boldsymbol{x}} = \\boldsymbol{f}(t, \\boldsymbol{x})
with a specified `Runge-Kutta method`_. The function :math:`\\boldsymbol{f}` should
be a `Numba`_ jitted function. This function must have a signature ``f(t, x)`` where ``x`` is
the state value and ``t`` is the time.
.. _Runge-Kutta method: https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods
.. _Numba: https://numba.pydata.org/
Parameters
----------
f: callable
The `Numba`_-jitted function :math:`\\boldsymbol{f}`.
Should have the signature``f(t, x)`` where ``x`` is the state value and ``t`` is the time.
t0: float
Initial time of the time integration. Corresponds to the initial condition's `ic` time.
Important if the ODEs are non-autonomous.
t: float
Final time of the time integration. Corresponds to the final condition.
Important if the ODEs are non-autonomous.
dt: float
Timestep of the integration.
ic: None or ~numpy.ndarray(float), optional
Initial condition of the system. Can be a 1D or a 2D array:
* 1D: Provide a single initial condition.
Should be of shape (`n_dim`,) where `n_dim` = :math:`\mathrm{dim}(\\boldsymbol{x})`.
* 2D: Provide an ensemble of initial condition.
Should be of shape (`n_traj`, `n_dim`) where `n_dim` = :math:`\mathrm{dim}(\\boldsymbol{x})`,
and where `n_traj` is the number of initial conditions.
If `None`, use a zero initial condition. Default to `None`.
forward: bool, optional
Whether to integrate the ODEs forward or backward in time. In case of backward integration, the
initial condition `ic` becomes a final condition. Default to forward integration.
write_steps: int, optional
Save the state of the integration in memory every `write_steps` steps. The other intermediary
steps are lost. It determine the size of the returned objects. Default is 1.
Set to 0 to return only the final state.
b: None or ~numpy.ndarray, optional
Vector of coefficients :math:`b_i` of the `Runge-Kutta method`_ .
If `None`, use the classic RK4 method coefficients. Default to `None`.
c: None or ~numpy.ndarray, optional
Matrix of coefficients :math:`c_{i,j}` of the `Runge-Kutta method`_ .
If `None`, use the classic RK4 method coefficients. Default to `None`.
a: None or ~numpy.ndarray, optional
Vector of coefficients :math:`a_i` of the `Runge-Kutta method`_ .
If `None`, use the classic RK4 method coefficients. Default to `None`.
Returns
-------
time, traj: ~numpy.ndarray
The result of the integration:
* `time` is the time at which the state of the system was saved. Array of shape (`n_step`,) where
`n_step` is the number of saved states of the integration.
* `traj` are the saved states. 3D array of shape (`n_traj`, `n_dim`, `n_steps`). If `n_traj` = 1,
a 2D array of shape (`n_dim`, `n_steps`) is returned instead.
Examples
--------
>>> from numba import njit
>>> import numpy as np
>>> from qgs.integrators.integrate import integrate_runge_kutta
>>> a = 0.25
>>> F = 16.
>>> G = 3.
>>> b = 6.
>>> # Lorenz 84 example
>>> @njit
... def fL84(t, x):
... xx = -x[1] ** 2 - x[2] ** 2 - a * x[0] + a * F
... yy = x[0] * x[1] - b * x[0] * x[2] - x[1] + G
... zz = b * x[0] * x[1] + x[0] * x[2] - x[2]
... return np.array([xx, yy, zz])
>>> # no ic
>>> # write_steps is 1 by default
>>> tt, traj = integrate_runge_kutta(fL84, t0=0., t=10., dt=0.1) # 101 steps
>>> print(traj.shape)
(3, 101)
>>> # 1 ic
>>> ic = 0.1 * np.random.randn(3)
>>> tt, traj = integrate_runge_kutta(fL84, t0=0., t=10., dt=0.1, ic=ic) # 101 steps
>>> print(ic.shape)
(3,)
>>> print(traj.shape)
(3, 101)
>>> # 4 ic
>>> ic = 0.1 * np.random.randn(4, 3)
>>> tt, traj = integrate_runge_kutta(fL84, t0=0., t=10., dt=0.1, ic=ic) # 101 steps
>>> print(ic.shape)
(4, 3)
>>> print(traj.shape)
(4, 3, 101)
"""
if ic is None:
i = 1
while True:
ic = np.zeros(i)
try:
x = f(0., ic)
except:
i += 1
else:
break
i = len(f(0., ic))
ic = np.zeros(i)
if len(ic.shape) == 1:
ic = ic.reshape((1, -1))
# Default is RK4
if a is None and b is None and c is None:
c = np.array([0., 0.5, 0.5, 1.])
b = np.array([1. / 6, 1. / 3, 1. / 3, 1. / 6])
a = np.zeros((len(c), len(b)))
a[1, 0] = 0.5
a[2, 1] = 0.5
a[3, 2] = 1.
if forward:
time_direction = 1
else:
time_direction = -1
time = np.concatenate((np.arange(t0, t, dt), np.full((1, ), t)))
recorded_traj = _integrate_runge_kutta_jit(f, time, ic, time_direction,
write_steps, b, c, a)
if write_steps > 0:
if forward:
if time[::write_steps][-1] == time[-1]:
return time[::write_steps], np.squeeze(recorded_traj)
else:
return np.concatenate((time[::write_steps], np.full(
(1, ), t))), np.squeeze(recorded_traj)
else:
rtime = reverse(time[::-write_steps])
if rtime[0] == time[0]:
return rtime, np.squeeze(recorded_traj)
else:
return np.concatenate((np.full(
(1, ), t0), rtime)), np.squeeze(recorded_traj)
else:
return time[-1], np.squeeze(recorded_traj)
def integrate_runge_kutta_tgls(f,
fjac,
t0,
t,
dt,
ic=None,
tg_ic=None,
forward=True,
adjoint=False,
inverse=False,
boundary=None,
write_steps=1,
b=None,
c=None,
a=None):
"""Integrate simultaneously the ordinary differential equations (ODEs)
.. math:: \dot{\\boldsymbol{x}} = \\boldsymbol{f}(t, \\boldsymbol{x})
and its tangent linear model, i.e. the linearized ODEs
.. math :: \dot{\\boldsymbol{\delta x}} = \\boldsymbol{\mathrm{J}}(t, \\boldsymbol{x}) \cdot \\boldsymbol{\delta x}
where :math:`\\boldsymbol{\mathrm{J}} = \\frac{\partial \\boldsymbol{f}}{\partial \\boldsymbol{x}}` is the
Jacobian matrix of :math:`\\boldsymbol{f}`, with a specified `Runge-Kutta method`_.
To solve this equation, one has to integrate simultaneously both ODEs.
The function :math:`\\boldsymbol{f}` and :math:`\\boldsymbol{J}` should
be `Numba`_ jitted functions. These functions must have a signature ``f(t, x)`` and ``J(t, x)`` where ``x`` is
the state value and ``t`` is the time.
.. _Runge-Kutta method: https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods
.. _Numba: https://numba.pydata.org/
.. _fundamental matrix of solutions: https://en.wikipedia.org/wiki/Fundamental_matrix_(linear_differential_equation)
Parameters
----------
f: callable
The `Numba`_-jitted function :math:`\\boldsymbol{f}`.
Should have the signature``f(t, x)`` where ``x`` is the state value and ``t`` is the time.
fjac: callable
The `Numba`_-jitted Jacobian :math:`\\boldsymbol{J}`.
Should have the signature``J(t, x)`` where ``x`` is the state value and ``t`` is the time.
t0: float
Initial time of the time integration. Corresponds to the initial condition's `ic` time.
Important if the ODEs are non-autonomous.
t: float
Final time of the time integration. Corresponds to the final condition.
Important if the ODEs are non-autonomous.
dt: float
Timestep of the integration.
ic: None or ~numpy.ndarray(float), optional
Initial condition of the ODEs :math:`\dot{\\boldsymbol{x}} = \\boldsymbol{f}(t, \\boldsymbol{x})`.
Can be a 1D or a 2D array:
* 1D: Provide a single initial condition.
Should be of shape (`n_dim`,) where `n_dim` = :math:`\mathrm{dim}(\\boldsymbol{x})`.
* 2D: Provide an ensemble of initial condition.
Should be of shape (`n_traj`, `n_dim`) where `n_dim` = :math:`\mathrm{dim}(\\boldsymbol{x})`,
and where `n_traj` is the number of initial conditions.
If `None`, use a zero initial condition. Default to `None`.
tg_ic: None or ~numpy.ndarray(float), optional
Initial condition of the linear ODEs
:math:`\dot{\\boldsymbol{\delta x}} = \\boldsymbol{\mathrm{J}}(t, \\boldsymbol{x}) \cdot \\boldsymbol{\delta x}`. \n
Can be a 1D, a 2D or a 3D array:
* 1D: Provide a single initial condition. This initial condition of the linear ODEs will be the same used for each
initial condition `ic` of the ODEs :math:`\dot{\\boldsymbol{x}} = \\boldsymbol{f}(t, \\boldsymbol{x})`
Should be of shape (`n_dim`,) where `n_dim` = :math:`\mathrm{dim}(\\boldsymbol{x})`.
* 2D: Two sub-cases:
+ If `tg_ic.shape[0]`=`ic.shape[0]`, assumes that each initial condition `ic[i]` of :math:`\dot{\\boldsymbol{x}} = \\boldsymbol{f}(t, \\boldsymbol{x})`,
correspond to a different initial condition `tg_ic[i]`.
+ Else, assumes and integrate an ensemble of `n_tg_traj` initial condition of the linear ODEs for each
initial condition of :math:`\dot{\\boldsymbol{x}} = \\boldsymbol{f}(t, \\boldsymbol{x})`.
* 3D: An array of shape (`n_traj`, `n_tg_traj`, `n_dim`) which provide an ensemble of `n_tg_ic` initial conditions
specific to each of the `n_traj` initial conditions of :math:`\dot{\\boldsymbol{x}} = \\boldsymbol{f}(t, \\boldsymbol{x})`.
If `None`, use the identity matrix as initial condition, returning the `fundamental matrix of solutions`_ of the
linear ODEs.
Default to `None`.
forward: bool, optional
Whether to integrate the ODEs forward or backward in time. In case of backward integration, the
initial condition `ic` becomes a final condition. Default to forward integration.
adjoint: bool, optional
Wheter to integrate the tangent :math:`\dot{\\boldsymbol{\delta x}} = \\boldsymbol{\mathrm{J}}(t, \\boldsymbol{x}) \cdot \\boldsymbol{\delta x}`
or the adjoint linear model :math:`\dot{\\boldsymbol{\delta x}} = \\boldsymbol{\mathrm{J}}^T(t, \\boldsymbol{x}) \cdot \\boldsymbol{\delta x}`.
Integrate the tangent model by default.
inverse: bool, optional
Wheter or not to invert the Jacobian matrix
:math:`\\boldsymbol{\mathrm{J}}(t, \\boldsymbol{x}) \\rightarrow \\boldsymbol{\mathrm{J}}^{-1}(t, \\boldsymbol{x})`.
`False` by default.
boundary: None or callable, optional
Allow to add a inhomogeneous term to linear ODEs:
:math:`\dot{\\boldsymbol{\delta x}} = \\boldsymbol{\mathrm{J}}(t, \\boldsymbol{x}) \cdot \\boldsymbol{\delta x} + \Psi(t, \\boldsymbol{x})`.
The boundary :math:`\Psi` should have the same signature as :math:`\\boldsymbol{\mathrm{J}}`, i.e. ``func(t, x)``.
If `None`, don't add anything (homogeneous case). `None` by default.
write_steps: int, optional
Save the state of the integration in memory every `write_steps` steps. The other intermediary
steps are lost. It determine the size of the returned objects. Default is 1.
Set to 0 to return only the final state.
b: None or ~numpy.ndarray, optional
Vector of coefficients :math:`b_i` of the `Runge-Kutta method`_ .
If `None`, use the classic RK4 method coefficients. Default to `None`.
c: None or ~numpy.ndarray, optional
Matrix of coefficients :math:`c_{i,j}` of the `Runge-Kutta method`_ .
If `None`, use the classic RK4 method coefficients. Default to `None`.
a: None or ~numpy.ndarray, optional
Vector of coefficients :math:`a_i` of the `Runge-Kutta method`_ .
If `None`, use the classic RK4 method coefficients. Default to `None`.
Returns
-------
time, traj, tg_traj: ~numpy.ndarray
The result of the integration:
* `time` is the time at which the state of the system was saved. Array of shape (`n_step`,) where
`n_step` is the number of saved states of the integration.
* `traj` are the saved states of the ODEs. 3D array of shape (`n_traj`, `n_dim`, `n_steps`). If `n_traj` = 1,
a 2D array of shape (`n_dim`, `n_steps`) is returned instead.
* `tg_traj` are the saved states of the linear ODEs.
Depending on the input initial conditions of both ODEs,
it is at maximum a 4D array of shape (`n_traj`, `n_tg_traj `n_dim`, `n_steps`).
If one of the dimension is 1, it is squeezed.
Examples
--------
>>> from numba import njit
>>> import numpy as np
>>> from qgs.integrators.integrate import integrate_runge_kutta_tgls
>>> a = 0.25
>>> F = 16.
>>> G = 3.
>>> b = 6.
>>> # Lorenz 84 example
>>> @njit
... def fL84(t, x):
... xx = -x[1] ** 2 - x[2] ** 2 - a * x[0] + a * F
... yy = x[0] * x[1] - b * x[0] * x[2] - x[1] + G
... zz = b * x[0] * x[1] + x[0] * x[2] - x[2]
... return np.array([xx, yy, zz])
>>> @njit
... def DfL84(t, x):
... return np.array([[ -a , -2. * x[1], -2. * x[2]],
... [x[1] - b * x[2], -1. + x[0], -b * x[0]],
... [b * x[1] + x[2], b * x[0], -1. + x[0]]])
>>> # 4 ic, no tg_ic (fundamental matrix computation of an ensemble of ic)
>>> ic = 0.1 * np.random.randn(4, 3)
>>> tt, traj, dtraj = integrate_runge_kutta_tgls(fL84, DfL84, t0=0., t=10., dt=0.1,
... ic=ic, write_steps=1) # 101 steps
>>> print(ic.shape)
(4, 3)
>>> print(traj.shape)
(4, 3, 101)
>>> print(dtraj.shape)
(4, 3, 3, 101)
>>> # 1 ic, 1 tg_ic (one ic and its tg_ic)
>>> ic = 0.1 * np.random.randn(3)
>>> tg_ic = 0.001 * np.random.randn(3)
>>> tt, traj, dtraj = integrate_runge_kutta_tgls(fL84, DfL84, t0=0., t=10., dt=0.1,
... ic=ic, tg_ic=tg_ic) # 101 steps
>>> print(ic.shape)
(3,)
>>> print(tg_ic.shape)
(3,)
>>> print(traj.shape)
(3, 101)
>>> print(dtraj.shape)
(3, 101)
>>> # 4 ic, 1 same tg_ic (an ensemble of ic with the same tg_ic)
>>> ic = 0.1 * np.random.randn(4, 3)
>>> tt, traj, dtraj = integrate_runge_kutta_tgls(fL84, DfL84, t0=0., t=10., dt=0.1,
... ic=ic, tg_ic=tg_ic) # 101 steps
>>> print(ic.shape)
(4, 3)
>>> print(tg_ic.shape)
(3,)
>>> print(traj.shape)
(4, 3, 101)
>>> print(dtraj.shape)
(4, 3, 101)
>>> # 1 ic, 4 tg_ic (an ic with an ensemble of tg_ic in its tangent space)
>>> ic = 0.1 * np.random.randn(3)
>>> tg_ic = 0.001 * np.random.randn(4, 3)
>>> tt, traj, dtraj = integrate_runge_kutta_tgls(fL84, DfL84, t0=0., t=10., dt=0.1,
... ic=ic, tg_ic=tg_ic) # 101 steps
>>> print(ic.shape)
(3,)
>>> print(tg_ic.shape)
(4, 3)
>>> print(traj.shape)
(3, 101)
>>> print(dtraj.shape)
(4, 3, 101)
>>> # 2 ic, same 4 tg_ic (an ensemble of 2 ic, both with the same ensemble
>>> # of tg_ic in their tangent space)
>>> ic = 0.1 * np.random.randn(2, 3)
>>> tg_ic = 0.001 * np.random.randn(4, 3)
>>> tt, traj, dtraj = integrate_runge_kutta_tgls(fL84, DfL84, t0=0., t=10., dt=0.1,
... ic=ic, tg_ic=tg_ic) # 101 steps
>>> print(ic.shape)
(2, 3)
>>> print(tg_ic.shape)
(4, 3)
>>> print(traj.shape)
(2, 3, 101)
>>> print(dtraj.shape)
(2, 4, 3, 101)
>>> # 2 ic, 4 different tg_ic (an ensemble of 2 ic, with different ensemble
>>> # of tg_ic in their tangent space)
>>> ic = 0.1 * np.random.randn(2, 3)
>>> tg_ic = 0.001 * np.random.randn(2, 4, 3)
>>> tt, traj, dtraj = integrate_runge_kutta_tgls(fL84, DfL84, t0=0., t=10., dt=0.1,
... ic=ic, tg_ic=tg_ic) # 101 steps
>>> print(ic.shape)
(2, 3)
>>> print(tg_ic.shape)
(2, 4, 3)
>>> print(traj.shape)
(2, 3, 101)
>>> print(dtraj.shape)
(2, 4, 3, 101)
"""
if ic is None:
i = 1
while True:
ic = np.zeros(i)
try:
x = f(0., ic)
except:
i += 1
else:
break
i = len(f(0., ic))
ic = np.zeros(i)
if len(ic.shape) == 1:
ic = ic.reshape((1, -1))
n_traj = ic.shape[0]
if tg_ic is None:
tg_ic = np.eye(ic.shape[1])
tg_ic_sav = tg_ic.copy()
if len(tg_ic.shape) == 1:
tg_ic = tg_ic.reshape((1, -1, 1))
ict = tg_ic.copy()
for i in range(n_traj - 1):
ict = np.concatenate((ict, tg_ic))
tg_ic = ict
elif len(tg_ic.shape) == 2:
if tg_ic.shape[0] == n_traj:
tg_ic = tg_ic[..., np.newaxis]
else:
tg_ic = tg_ic[np.newaxis, ...]
tg_ic = np.swapaxes(tg_ic, 1, 2)
ict = tg_ic.copy()
for i in range(n_traj - 1):
ict = np.concatenate((ict, tg_ic))
tg_ic = ict
elif len(tg_ic.shape) == 3:
if tg_ic.shape[1] != ic.shape[1]:
tg_ic = np.swapaxes(tg_ic, 1, 2)
# Default is RK4
if a is None and b is None and c is None:
c = np.array([0., 0.5, 0.5, 1.])
b = np.array([1. / 6, 1. / 3, 1. / 3, 1. / 6])
a = np.zeros((len(c), len(b)))
a[1, 0] = 0.5
a[2, 1] = 0.5
a[3, 2] = 1.
if forward:
time_direction = 1
else:
time_direction = -1
time = np.concatenate((np.arange(t0, t, dt), np.full((1, ), t)))
if boundary is None:
boundary = _zeros_func
inv = 1.
if inverse:
inv *= -1.
recorded_traj, recorded_fmatrix = _integrate_runge_kutta_tgls_jit(
f, fjac, time, ic, tg_ic, time_direction, write_steps, b, c, a,
adjoint, inv, boundary)
if len(tg_ic_sav.shape) == 2:
if recorded_fmatrix.shape[1:3] != tg_ic_sav.shape:
recorded_fmatrix = np.swapaxes(recorded_fmatrix, 1, 2)
elif len(tg_ic_sav.shape) == 3:
if tg_ic_sav.shape[1] != ic.shape[1]:
if recorded_fmatrix.shape[:3] != tg_ic_sav.shape:
recorded_fmatrix = np.swapaxes(recorded_fmatrix, 1, 2)
if write_steps > 0:
if forward:
if time[::write_steps][-1] == time[-1]:
return time[::write_steps], np.squeeze(
recorded_traj), np.squeeze(recorded_fmatrix)
else:
return np.concatenate((time[::write_steps], np.full((1,), t))), np.squeeze(recorded_traj),\
np.squeeze(recorded_fmatrix)
else:
rtime = reverse(time[::-write_steps])
if rtime[0] == time[0]:
return rtime, np.squeeze(recorded_traj), np.squeeze(
recorded_fmatrix)
else:
return np.concatenate((np.full((1,), t0), rtime)), np.squeeze(recorded_traj),\
np.squeeze(recorded_fmatrix)
else:
return time[-1], np.squeeze(recorded_traj), np.squeeze(
recorded_fmatrix)
c = p.get_color()
plt.plot(t, rl[i][0], color=c, ls='--')
for j in range(1, ic.shape[1]):
p, = plt.plot(time, r[i, j], color=c)
plt.plot(t, rl[i][j], color=c, ls='--')
timet, rt = integrate_runge_kutta(f,
0.,
10.,
0.01,
ic=ic,
forward=False,
write_steps=3)
rlt = list()
for i in range(ic.shape[0]):
rlt.append(odeint(fr, ic[i], reverse(t)).T)
plt.figure()
for i in range(ic.shape[0]):
p, = plt.plot(timet, rt[i, 0])
c = p.get_color()
plt.plot(t, reverse(rlt[i][0]), color=c, ls='--')
for j in range(1, ic.shape[1]):
p, = plt.plot(timet, rt[i, j], color=c)
plt.plot(t, reverse(rlt[i][j]), color=c, ls='--')
tt, re = integrate_runge_kutta(f, 0., 10., 0.01, ic=ic, write_steps=0)
print(tt)
print(r[0, :, -1], re[0])
plt.show(block=False)