def _expm_krylov(alpha, beta, V, v_norm, dt):
# diagonalize Hessenberg matrix
w_hess, u_hess = eigh_tridiagonal(alpha, beta)
xp_w_hess = xp.array(w_hess)
xp_u_hess = xp.array(u_hess)
return V @ xp_u_hess @ (v_norm * xp.exp(dt*xp_w_hess) * xp_u_hess[0])
def to_complex(self, inplace=False):
# `xp.array` always creates new array, so to_complex means copy, which is
# in accordance with NumPy
if inplace:
self.array = xp.array(self.array, dtype=backend.complex_dtype)
return self
else:
return xp.array(self.array, dtype=backend.complex_dtype)
def test_expm(N, imag, block_size):
a1 = np.random.rand(N, N) / N
if imag:
a1 = a1 + np.random.rand(N, N) / N / 1j
a2 = xp.array(a1)
v = np.random.rand(N)
if imag:
v = v + v / 1j
res1 = expm(a1) @ v
res2, _ = expm_krylov(lambda x: a2.dot(x), 1, xp.array(v), block_size)
assert xp.allclose(res1, res2)
def projector(ms: xp.ndarray) -> xp.ndarray:
# projector
proj = xp.tensordot(ms, ms.conj(), axes=(-1, -1))
sz = int(np.prod(ms.shape[:-1]))
Iden = xp.array(xp.diag(xp.ones(sz)), dtype=backend.real_dtype).reshape(proj.shape)
proj = Iden - proj
return proj
class RK45(RungeKutta):
"""Explicit Runge-Kutta method of order 5(4).
This uses the Dormand-Prince pair of formulas [1]_. The error is controlled
assuming accuracy of the fourth-order method accuracy, but steps are taken
using the fifth-order accurate formula (local extrapolation is done).
A quartic interpolation polynomial is used for the dense output [2]_.
Can be applied in the complex domain.
Parameters
----------
fun : callable
Right-hand side of the system. The calling signature is ``fun(t, y)``.
Here ``t`` is a scalar, and there are two options for the ndarray ``y``:
It can either have shape (n,); then ``fun`` must return array_like with
shape (n,). Alternatively it can have shape (n, k); then ``fun``
must return an array_like with shape (n, k), i.e. each column
corresponds to a single column in ``y``. The choice between the two
options is determined by `vectorized` argument (see below).
t0 : float
Initial time.
y0 : array_like, shape (n,)
Initial state.
t_bound : float
Boundary time - the integration won't continue beyond it. It also
determines the direction of the integration.
first_step : float or None, optional
Initial step size. Default is ``None`` which means that the algorithm
should choose.
max_step : float, optional
Maximum allowed step size. Default is xp.inf, i.e. the step size is not
bounded and determined solely by the solver.
rtol, atol : float and array_like, optional
Relative and absolute tolerances. The solver keeps the local error
estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a
relative accuracy (number of correct digits). But if a component of `y`
is approximately below `atol`, the error only needs to fall within
the same `atol` threshold, and the number of correct digits is not
guaranteed. If components of y have different scales, it might be
beneficial to set different `atol` values for different components by
passing array_like with shape (n,) for `atol`. Default values are
1e-3 for `rtol` and 1e-6 for `atol`.
vectorized : bool, optional
Whether `fun` is implemented in a vectorized fashion. Default is False.
Attributes
----------
n : int
Number of equations.
status : string
Current status of the solver: 'running', 'finished' or 'failed'.
t_bound : float
Boundary time.
direction : float
Integration direction: +1 or -1.
t : float
Current time.
y : ndarray
Current state.
t_old : float
Previous time. None if no steps were made yet.
step_size : float
Size of the last successful step. None if no steps were made yet.
nfev : int
Number evaluations of the system's right-hand side.
njev : int
Number of evaluations of the Jacobian. Is always 0 for this solver as it does not use the Jacobian.
nlu : int
Number of LU decompositions. Is always 0 for this solver.
References
----------
.. [1] J. R. Dormand, P. J. Prince, "A family of embedded Runge-Kutta
formulae", Journal of Computational and Applied Mathematics, Vol. 6,
No. 1, pp. 19-26, 1980.
.. [2] L. W. Shampine, "Some Practical Runge-Kutta Formulas", Mathematics
of Computation,, Vol. 46, No. 173, pp. 135-150, 1986.
"""
order = 4
n_stages = 6
C = xp.array([1 / 5, 3 / 10, 4 / 5, 8 / 9, 1], dtype=dtype)
A = [
xp.array([1 / 5], dtype=dtype),
xp.array([3 / 40, 9 / 40], dtype=dtype),
xp.array([44 / 45, -56 / 15, 32 / 9], dtype=dtype),
xp.array([19372 / 6561, -25360 / 2187, 64448 / 6561, -212 / 729],
dtype=dtype),
xp.array(
[9017 / 3168, -355 / 33, 46732 / 5247, 49 / 176, -5103 / 18656],
dtype=dtype),
]
B = xp.array([35 / 384, 0, 500 / 1113, 125 / 192, -2187 / 6784, 11 / 84],
dtype=dtype)
E = xp.array(
[
-71 / 57600, 0, 71 / 16695, -71 / 1920, 17253 / 339200, -22 / 525,
1 / 40
],
dtype=dtype,
)
# Corresponds to the optimum value of c_6 from [2]_.
P = xp.array(
[
[
1,
-8048581381 / 2820520608,
8663915743 / 2820520608,
-12715105075 / 11282082432,
],
[0, 0, 0, 0],
[
0,
131558114200 / 32700410799,
-68118460800 / 10900136933,
87487479700 / 32700410799,
],
[
0,
-1754552775 / 470086768,
14199869525 / 1410260304,
-10690763975 / 1880347072,
],
[
0,
127303824393 / 49829197408,
-318862633887 / 49829197408,
701980252875 / 199316789632,
],
[
0,
-282668133 / 205662961,
2019193451 / 616988883,
-1453857185 / 822651844,
],
[
0, 40617522 / 29380423, -110615467 / 29380423,
69997945 / 29380423
],
],
dtype=dtype,
)
class RK23(RungeKutta):
"""Explicit Runge-Kutta method of order 3(2).
This uses the Bogacki-Shampine pair of formulas [1]_. The error is controlled
assuming accuracy of the second-order method, but steps are taken using the
third-order accurate formula (local extrapolation is done). A cubic Hermite
polynomial is used for the dense output.
Can be applied in the complex domain.
Parameters
----------
fun : callable
Right-hand side of the system. The calling signature is ``fun(t, y)``.
Here ``t`` is a scalar and there are two options for ndarray ``y``.
It can either have shape (n,), then ``fun`` must return array_like with
shape (n,). Or alternatively it can have shape (n, k), then ``fun``
must return array_like with shape (n, k), i.e. each column
corresponds to a single column in ``y``. The choice between the two
options is determined by `vectorized` argument (see below).
t0 : float
Initial time.
y0 : array_like, shape (n,)
Initial state.
t_bound : float
Boundary time - the integration won't continue beyond it. It also
determines the direction of the integration.
first_step : float or None, optional
Initial step size. Default is ``None`` which means that the algorithm
should choose.
max_step : float, optional
Maximum allowed step size. Default is xp.inf, i.e. the step size is not
bounded and determined solely by the solver.
rtol, atol : float and array_like, optional
Relative and absolute tolerances. The solver keeps the local error
estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a
relative accuracy (number of correct digits). But if a component of `y`
is approximately below `atol`, the error only needs to fall within
the same `atol` threshold, and the number of correct digits is not
guaranteed. If components of y have different scales, it might be
beneficial to set different `atol` values for different components by
passing array_like with shape (n,) for `atol`. Default values are
1e-3 for `rtol` and 1e-6 for `atol`.
vectorized : bool, optional
Whether `fun` is implemented in a vectorized fashion. Default is False.
Attributes
----------
n : int
Number of equations.
status : string
Current status of the solver: 'running', 'finished' or 'failed'.
t_bound : float
Boundary time.
direction : float
Integration direction: +1 or -1.
t : float
Current time.
y : ndarray
Current state.
t_old : float
Previous time. None if no steps were made yet.
step_size : float
Size of the last successful step. None if no steps were made yet.
nfev : int
Number evaluations of the system's right-hand side.
njev : int
Number of evaluations of the Jacobian. Is always 0 for this solver as it does not use the Jacobian.
nlu : int
Number of LU decompositions. Is always 0 for this solver.
References
----------
.. [1] P. Bogacki, L.F. Shampine, "A 3(2) Pair of Runge-Kutta Formulas",
Appl. Math. Lett. Vol. 2, No. 4. pp. 321-325, 1989.
"""
order = 2
n_stages = 3
C = xp.array([1 / 2, 3 / 4], dtype=dtype)
A = [xp.array([1 / 2], dtype=dtype), xp.array([0, 3 / 4], dtype=dtype)]
B = xp.array([2 / 9, 1 / 3, 4 / 9], dtype=dtype)
E = xp.array([5 / 72, -1 / 12, -1 / 9, 1 / 8], dtype=dtype)
P = xp.array(
[[1, -4 / 3, 5 / 9], [0, 1, -2 / 3], [0, 4 / 3, -8 / 9], [0, -1, 1]],
dtype=dtype,
)
def _sparse_num_jac(fun, t, y, f, h, factor, y_scale, structure, groups):
n = y.shape[0]
n_groups = xp.max(groups) + 1
h_vecs = xp.empty((n_groups, n))
for group in range(n_groups):
e = xp.equal(group, groups)
h_vecs[group] = h * e
h_vecs = h_vecs.T
f_new = fun(t, y[:, None] + h_vecs)
df = f_new - f[:, None]
i, j, _ = find(structure)
diff = coo_matrix((df[i, groups[j]], (i, j)), shape=(n, n)).tocsc()
max_ind = xp.array(abs(diff).argmax(axis=0)).ravel()
r = xp.arange(n)
max_diff = xp.asarray(xp.abs(diff[max_ind, r])).ravel()
scale = xp.maximum(xp.abs(f[max_ind]), xp.abs(f_new[max_ind, groups[r]]))
diff_too_small = max_diff < NUM_JAC_DIFF_REJECT * scale
if xp.any(diff_too_small):
ind, = xp.nonzero(diff_too_small)
new_factor = NUM_JAC_FACTOR_INCREASE * factor[ind]
h_new = (y[ind] + new_factor * y_scale[ind]) - y[ind]
h_new_all = xp.zeros(n)
h_new_all[ind] = h_new
groups_unique = xp.unique(groups[ind])
groups_map = xp.empty(n_groups, dtype=int)
h_vecs = xp.empty((groups_unique.shape[0], n))
for k, group in enumerate(groups_unique):
e = xp.equal(group, groups)
h_vecs[k] = h_new_all * e
groups_map[group] = k
h_vecs = h_vecs.T
f_new = fun(t, y[:, None] + h_vecs)
df = f_new - f[:, None]
i, j, _ = find(structure[:, ind])
diff_new = coo_matrix((df[i, groups_map[groups[ind[j]]]], (i, j)),
shape=(n, ind.shape[0])).tocsc()
max_ind_new = xp.array(abs(diff_new).argmax(axis=0)).ravel()
r = xp.arange(ind.shape[0])
max_diff_new = xp.asarray(xp.abs(diff_new[max_ind_new, r])).ravel()
scale_new = xp.maximum(
xp.abs(f[max_ind_new]),
xp.abs(f_new[max_ind_new, groups_map[groups[ind]]]))
update = max_diff[ind] * scale_new < max_diff_new * scale[ind]
if xp.any(update):
update, = xp.nonzero(update)
update_ind = ind[update]
factor[update_ind] = new_factor[update]
h[update_ind] = h_new[update]
diff[:, update_ind] = diff_new[:, update]
scale[update_ind] = scale_new[update]
max_diff[update_ind] = max_diff_new[update]
diff.data /= xp.repeat(h, xp.diff(diff.indptr))
factor[max_diff < NUM_JAC_DIFF_SMALL * scale] *= NUM_JAC_FACTOR_INCREASE
factor[max_diff > NUM_JAC_DIFF_BIG * scale] *= NUM_JAC_FACTOR_DECREASE
factor = xp.maximum(factor, NUM_JAC_MIN_FACTOR)
return diff, factor
def solve_ivp(fun,
t_span,
y0,
method="RK45",
t_eval=None,
dense_output=False,
events=None,
vectorized=False,
**options) -> OdeResult:
"""Solve an initial value problem for a system of ODEs.
This function numerically integrates a system of ordinary differential
equations given an initial value::
dy / dt = f(t, y)
y(t0) = y0
Here t is a one-dimensional independent variable (time), y(t) is an
n-dimensional vector-valued function (state), and an n-dimensional
vector-valued function f(t, y) determines the differential equations.
The goal is to find y(t) approximately satisfying the differential
equations, given an initial value y(t0)=y0.
Some of the solvers support integration in the complex domain, but note that
for stiff ODE solvers, the right-hand side must be complex-differentiable
(satisfy Cauchy-Riemann equations [11]_). To solve a problem in the complex
domain, pass y0 with a complex data type. Another option is always to
rewrite your problem for real and imaginary parts separately.
Parameters
----------
fun : callable
Right-hand side of the system. The calling signature is ``fun(t, y)``.
Here ``t`` is a scalar, and there are two options for the ndarray ``y``:
It can either have shape (n,); then ``fun`` must return array_like with
shape (n,). Alternatively it can have shape (n, k); then ``fun``
must return an array_like with shape (n, k), i.e. each column
corresponds to a single column in ``y``. The choice between the two
options is determined by `vectorized` argument (see below). The
vectorized implementation allows a faster approximation of the Jacobian
by finite differences (required for stiff solvers).
t_span : 2-tuple of floats
Interval of integration (t0, tf). The solver starts with t=t0 and
integrates until it reaches t=tf.
y0 : array_like, shape (n,)
Initial state. For problems in the complex domain, pass `y0` with a
complex data type (even if the initial guess is purely real).
method : string or `OdeSolver`, optional
Integration method to use:
* 'RK45' (default): Explicit Runge-Kutta method of order 5(4) [1]_.
The error is controlled assuming accuracy of the fourth-order
method, but steps are taken using the fifth-order accurate formula
(local extrapolation is done). A quartic interpolation polynomial
is used for the dense output [2]_. Can be applied in the complex domain.
* 'RK23': Explicit Runge-Kutta method of order 3(2) [3]_. The error
is controlled assuming accuracy of the second-order method, but
steps are taken using the third-order accurate formula (local
extrapolation is done). A cubic Hermite polynomial is used for the
dense output. Can be applied in the complex domain.
* 'Radau': Implicit Runge-Kutta method of the Radau IIA family of
order 5 [4]_. The error is controlled with a third-order accurate
embedded formula. A cubic polynomial which satisfies the
collocation conditions is used for the dense output.
* 'BDF': Implicit multi-step variable-order (1 to 5) method based
on a backward differentiation formula for the derivative
approximation [5]_. The implementation follows the one described
in [6]_. A quasi-constant step scheme is used and accuracy is
enhanced using the NDF modification. Can be applied in the complex
domain.
* 'LSODA': Adams/BDF method with automatic stiffness detection and
switching [7]_, [8]_. This is a wrapper of the Fortran solver
from ODEPACK.
You should use the 'RK45' or 'RK23' method for non-stiff problems and
'Radau' or 'BDF' for stiff problems [9]_. If not sure, first try to run
'RK45'. If needs unusually many iterations, diverges, or fails, your
problem is likely to be stiff and you should use 'Radau' or 'BDF'.
'LSODA' can also be a good universal choice, but it might be somewhat
less convenient to work with as it wraps old Fortran code.
You can also pass an arbitrary class derived from `OdeSolver` which
implements the solver.
dense_output : bool, optional
Whether to compute a continuous solution. Default is False.
t_eval : array_like or None, optional
Times at which to store the computed solution, must be sorted and lie
within `t_span`. If None (default), use points selected by the solver.
events : callable, list of callables or None, optional
Types of events to track. Each is defined by a continuous function of
time and state that becomes zero value in case of an event. Each function
must have the signature ``event(t, y)`` and return a float. The solver will
find an accurate value of ``t`` at which ``event(t, y(t)) = 0`` using a
root-finding algorithm. Additionally each ``event`` function might have
the following attributes:
* terminal: bool, whether to terminate integration if this
event occurs. Implicitly False if not assigned.
* direction: float, direction of a zero crossing. If `direction`
is positive, `event` must go from negative to positive, and
vice versa if `direction` is negative. If 0, then either direction
will count. Implicitly 0 if not assigned.
You can assign attributes like ``event.terminal = True`` to any
function in Python. If None (default), events won't be tracked.
vectorized : bool, optional
Whether `fun` is implemented in a vectorized fashion. Default is False.
options
Options passed to a chosen solver. All options available for already
implemented solvers are listed below.
first_step : float or None, optional
Initial step size. Default is ``None`` which means that the algorithm
should choose.
max_step : float, optional
Maximum allowed step size. Default is xp.inf, i.e. the step size is not
bounded and determined solely by the solver.
rtol, atol : float and array_like, optional
Relative and absolute tolerances. The solver keeps the local error
estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a
relative accuracy (number of correct digits). But if a component of `y`
is approximately below `atol`, the error only needs to fall within
the same `atol` threshold, and the number of correct digits is not
guaranteed. If components of y have different scales, it might be
beneficial to set different `atol` values for different components by
passing array_like with shape (n,) for `atol`. Default values are
1e-3 for `rtol` and 1e-6 for `atol`.
jac : {None, array_like, sparse_matrix, callable}, optional
Jacobian matrix of the right-hand side of the system with respect to
y, required by the 'Radau', 'BDF' and 'LSODA' method. The Jacobian matrix
has shape (n, n) and its element (i, j) is equal to ``d f_i / d y_j``.
There are three ways to define the Jacobian:
* If array_like or sparse_matrix, the Jacobian is assumed to
be constant. Not supported by 'LSODA'.
* If callable, the Jacobian is assumed to depend on both
t and y; it will be called as ``jac(t, y)`` as necessary.
For the 'Radau' and 'BDF' methods, the return value might be a
sparse matrix.
* If None (default), the Jacobian will be approximated by
finite differences.
It is generally recommended to provide the Jacobian rather than
relying on a finite-difference approximation.
jac_sparsity : {None, array_like, sparse matrix}, optional
Defines a sparsity structure of the Jacobian matrix for a
finite-difference approximation. Its shape must be (n, n). This argument
is ignored if `jac` is not `None`. If the Jacobian has only few non-zero
elements in *each* row, providing the sparsity structure will greatly
speed up the computations [10]_. A zero entry means that a corresponding
element in the Jacobian is always zero. If None (default), the Jacobian
is assumed to be dense.
Not supported by 'LSODA', see `lband` and `uband` instead.
lband, uband : int or None
Parameters defining the bandwidth of the Jacobian for the 'LSODA' method,
i.e., ``jac[i, j] != 0 only for i - lband <= j <= i + uband``. Setting
these requires your jac routine to return the Jacobian in the packed format:
the returned array must have ``n`` columns and ``uband + lband + 1``
rows in which Jacobian diagonals are written. Specifically
``jac_packed[uband + i - j , j] = jac[i, j]``. The same format is used
in `scipy.linalg.solve_banded` (check for an illustration).
These parameters can be also used with ``jac=None`` to reduce the
number of Jacobian elements estimated by finite differences.
min_step : float, optional
The minimum allowed step size for 'LSODA' method.
By default `min_step` is zero.
Returns
-------
Bunch object with the following fields defined:
t : ndarray, shape (n_points,)
Time points.
y : ndarray, shape (n, n_points)
Values of the solution at `t`.
sol : `OdeSolution` or None
Found solution as `OdeSolution` instance; None if `dense_output` was
set to False.
t_events : list of ndarray or None
Contains for each event type a list of arrays at which an event of
that type event was detected. None if `events` was None.
nfev : int
Number of evaluations of the right-hand side.
njev : int
Number of evaluations of the Jacobian.
nlu : int
Number of LU decompositions.
status : int
Reason for algorithm termination:
* -1: Integration step failed.
* 0: The solver successfully reached the end of `tspan`.
* 1: A termination event occurred.
message : string
Human-readable description of the termination reason.
success : bool
True if the solver reached the interval end or a termination event
occurred (``status >= 0``).
References
----------
.. [1] J. R. Dormand, P. J. Prince, "A family of embedded Runge-Kutta
formulae", Journal of Computational and Applied Mathematics, Vol. 6,
No. 1, pp. 19-26, 1980.
.. [2] L. W. Shampine, "Some Practical Runge-Kutta Formulas", Mathematics
of Computation,, Vol. 46, No. 173, pp. 135-150, 1986.
.. [3] P. Bogacki, L.F. Shampine, "A 3(2) Pair of Runge-Kutta Formulas",
Appl. Math. Lett. Vol. 2, No. 4. pp. 321-325, 1989.
.. [4] E. Hairer, G. Wanner, "Solving Ordinary Differential Equations II:
Stiff and Differential-Algebraic Problems", Sec. IV.8.
.. [5] `Backward Differentiation Formula
<https://en.wikipedia.org/wiki/Backward_differentiation_formula>`_
on Wikipedia.
.. [6] L. F. Shampine, M. W. Reichelt, "THE MATLAB ODE SUITE", SIAM J. SCI.
COMPUTE., Vol. 18, No. 1, pp. 1-22, January 1997.
.. [7] A. C. Hindmarsh, "ODEPACK, A Systematized Collection of ODE
Solvers," IMACS Transactions on Scientific Computation, Vol 1.,
pp. 55-64, 1983.
.. [8] L. Petzold, "Automatic selection of methods for solving stiff and
nonstiff systems of ordinary differential equations", SIAM Journal
on Scientific and Statistical Computing, Vol. 4, No. 1, pp. 136-148,
1983.
.. [9] `Stiff equation <https://en.wikipedia.org/wiki/Stiff_equation>`_ on
Wikipedia.
.. [10] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of
sparse Jacobian matrices", Journal of the Institute of Mathematics
and its Applications, 13, pp. 117-120, 1974.
.. [11] `Cauchy-Riemann equations
<https://en.wikipedia.org/wiki/Cauchy-Riemann_equations>`_ on
Wikipedia.
Examples
--------
Basic exponential decay showing automatically chosen time points.
>>> from scipy.integrate import solve_ivp
>>> def exponential_decay(t, y): return -0.5 * y
>>> sol = solve_ivp(exponential_decay, [0, 10], [2, 4, 8])
>>> print(sol.t)
[ 0. 0.11487653 1.26364188 3.06061781 4.85759374
6.65456967 8.4515456 10. ]
>>> print(sol.y)
[[2. 1.88836035 1.06327177 0.43319312 0.17648948 0.0719045
0.02929499 0.01350938]
[4. 3.7767207 2.12654355 0.86638624 0.35297895 0.143809
0.05858998 0.02701876]
[8. 7.5534414 4.25308709 1.73277247 0.7059579 0.287618
0.11717996 0.05403753]]
Specifying points where the solution is desired.
>>> sol = solve_ivp(exponential_decay, [0, 10], [2, 4, 8],
... t_eval=[0, 1, 2, 4, 10])
>>> print(sol.t)
[ 0 1 2 4 10]
>>> print(sol.y)
[[2. 1.21305369 0.73534021 0.27066736 0.01350938]
[4. 2.42610739 1.47068043 0.54133472 0.02701876]
[8. 4.85221478 2.94136085 1.08266944 0.05403753]]
Cannon fired upward with terminal event upon impact. The ``terminal`` and
``direction`` fields of an event are applied by monkey patching a function.
Here ``y[0]`` is position and ``y[1]`` is velocity. The projectile starts at
position 0 with velocity +10. Note that the integration never reaches t=100
because the event is terminal.
>>> def upward_cannon(t, y): return [y[1], -0.5]
>>> def hit_ground(t, y): return y[1]
>>> hit_ground.terminal = True
>>> hit_ground.direction = -1
>>> sol = solve_ivp(upward_cannon, [0, 100], [0, 10], events=hit_ground)
>>> print(sol.t_events)
[array([ 20.])]
>>> print(sol.t)
[0.00000000e+00 9.99900010e-05 1.09989001e-03 1.10988901e-02
1.11088891e-01 1.11098890e+00 1.11099890e+01 2.00000000e+01]
"""
if method not in METHODS and not (inspect.isclass(method)
and issubclass(method, OdeSolver)):
raise ValueError(
"`method` must be one of {} or OdeSolver class.".format(METHODS))
t0, tf = float(t_span[0]), float(t_span[1])
if t_eval is not None:
t_eval = xp.asarray(t_eval)
if t_eval.ndim != 1:
raise ValueError("`t_eval` must be 1-dimensional.")
if xp.any(t_eval < min(t0, tf)) or xp.any(t_eval > max(t0, tf)):
raise ValueError("Values in `t_eval` are not within `t_span`.")
d = xp.diff(t_eval)
if tf > t0 and xp.any(d <= 0) or tf < t0 and xp.any(d >= 0):
raise ValueError("Values in `t_eval` are not properly sorted.")
if tf > t0:
t_eval_i = 0
else:
# Make order of t_eval decreasing to use xp.searchsorted.
t_eval = t_eval[::-1]
# This will be an upper bound for slices.
t_eval_i = t_eval.shape[0]
if method in METHODS:
method = METHODS[method]
solver = method(fun, t0, y0, tf, vectorized=vectorized, **options)
if t_eval is None:
ts = [t0]
ys = [y0]
elif t_eval is not None and dense_output:
ts = []
ti = [t0]
ys = []
else:
ts = []
ys = []
interpolants = []
events, is_terminal, event_dir = prepare_events(events)
if events is not None:
g = [event(t0, y0) for event in events]
t_events = [[] for _ in range(len(events))]
else:
t_events = None
status = None
while status is None:
message = solver.step()
if solver.status == "finished":
status = 0
elif solver.status == "failed":
status = -1
break
t_old = solver.t_old
t = solver.t
y = solver.y
if dense_output:
sol = solver.dense_output()
interpolants.append(sol)
else:
sol = None
if events is not None:
g_new = [event(t, y) for event in events]
active_events = find_active_events(g, g_new, event_dir)
if active_events.size > 0:
if sol is None:
sol = solver.dense_output()
root_indices, roots, terminate = handle_events(
sol, events, active_events, is_terminal, t_old, t)
for e, te in zip(root_indices, roots):
t_events[e].append(te)
if terminate:
status = 1
t = roots[-1]
y = sol(t)
g = g_new
if t_eval is None:
ts.append(t)
ys[-1] = y
else:
# The value in t_eval equal to t will be included.
if solver.direction > 0:
t_eval_i_new = xp.searchsorted(t_eval, t, side="right")
t_eval_step = t_eval[t_eval_i:t_eval_i_new]
else:
t_eval_i_new = xp.searchsorted(t_eval, t, side="left")
# It has to be done with two slice operations, because
# you can't slice to 0-th element inclusive using backward
# slicing.
t_eval_step = t_eval[t_eval_i_new:t_eval_i][::-1]
if t_eval_step.size > 0:
if sol is None:
sol = solver.dense_output()
ts.append(t_eval_step)
ys.append(sol(t_eval_step))
t_eval_i = t_eval_i_new
if t_eval is not None and dense_output:
ti.append(t)
message = MESSAGES.get(status, message)
if t_events is not None:
t_events = [xp.asarray(te) for te in t_events]
if t_eval is None:
ts = xp.array(ts)
ys = xp.vstack(ys).T
else:
ts = xp.hstack(ts)
ys = xp.hstack(ys)
if dense_output:
if t_eval is None:
sol = OdeSolution(ts, interpolants)
else:
sol = OdeSolution(ti, interpolants)
else:
sol = None
return OdeResult(
t=ts,
y=ys,
sol=sol,
t_events=t_events,
nfev=solver.nfev,
njev=solver.njev,
nlu=solver.nlu,
status=status,
message=message,
success=status >= 0,
)