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from __future__ import division, print_function, absolute_import
import itertools
import numpy as np
from numpy.testing import run_module_suite, assert_allclose
from scipy.integrate import ode
def _band_count(a):
"""Returns ml and mu, the lower and upper band sizes of a."""
nrows, ncols = a.shape
ml = 0
for k in range(-nrows+1, 0):
if np.diag(a, k).any():
ml = -k
break
mu = 0
for k in range(nrows-1, 0, -1):
if np.diag(a, k).any():
mu = k
break
return ml, mu
def _linear_func(t, y, a):
"""Linear system dy/dt = a * y"""
return a.dot(y)
def _linear_jac(t, y, a):
"""Jacobian of a * y is a."""
return a
def _linear_banded_jac(t, y, a):
"""Banded Jacobian."""
ml, mu = _band_count(a)
bjac = []
for k in range(mu, 0, -1):
bjac.append(np.r_[[0] * k, np.diag(a, k)])
bjac.append(np.diag(a))
for k in range(-1, -ml-1, -1):
bjac.append(np.r_[np.diag(a, k), [0] * (-k)])
return bjac
def _solve_linear_sys(a, y0, tend=1, dt=0.1,
solver=None, method='bdf', use_jac=True,
with_jacobian=False, banded=False):
"""Use scipy.integrate.ode to solve a linear system of ODEs.
a : square ndarray
Matrix of the linear system to be solved.
y0 : ndarray
Initial condition
tend : float
Stop time.
dt : float
Step size of the output.
solver : str
If not None, this must be "vode", "lsoda" or "zvode".
method : str
Either "bdf" or "adams".
use_jac : bool
Determines if the jacobian function is passed to ode().
with_jacobian : bool
Passed to ode.set_integrator().
banded : bool
Determines whether a banded or full jacobian is used.
If `banded` is True, `lband` and `uband` are determined by the
values in `a`.
"""
if banded:
lband, uband = _band_count(a)
else:
lband = None
uband = None
if use_jac:
if banded:
r = ode(_linear_func, _linear_banded_jac)
else:
r = ode(_linear_func, _linear_jac)
else:
r = ode(_linear_func)
if solver is None:
if np.iscomplexobj(a):
solver = "zvode"
else:
solver = "vode"
r.set_integrator(solver,
with_jacobian=with_jacobian,
method=method,
lband=lband, uband=uband,
rtol=1e-9, atol=1e-10,
)
t0 = 0
r.set_initial_value(y0, t0)
r.set_f_params(a)
r.set_jac_params(a)
t = [t0]
y = [y0]
while r.successful() and r.t < tend:
r.integrate(r.t + dt)
t.append(r.t)
y.append(r.y)
t = np.array(t)
y = np.array(y)
return t, y
def _analytical_solution(a, y0, t):
"""
Analytical solution to the linear differential equations dy/dt = a*y.
The solution is only valid if `a` is diagonalizable.
Returns a 2-d array with shape (len(t), len(y0)).
"""
lam, v = np.linalg.eig(a)
c = np.linalg.solve(v, y0)
e = c * np.exp(lam * t.reshape(-1, 1))
sol = e.dot(v.T)
return sol
def test_banded_ode_solvers():
# Test the "lsoda", "vode" and "zvode" solvers of the `ode` class
# with a system that has a banded Jacobian matrix.
t_exact = np.linspace(0, 1.0, 5)
# --- Real arrays for testing the "lsoda" and "vode" solvers ---
# lband = 2, uband = 1:
a_real = np.array([[-0.6, 0.1, 0.0, 0.0, 0.0],
[0.2, -0.5, 0.9, 0.0, 0.0],
[0.1, 0.1, -0.4, 0.1, 0.0],
[0.0, 0.3, -0.1, -0.9, -0.3],
[0.0, 0.0, 0.1, 0.1, -0.7]])
# lband = 0, uband = 1:
a_real_upper = np.triu(a_real)
# lband = 2, uband = 0:
a_real_lower = np.tril(a_real)
# lband = 0, uband = 0:
a_real_diag = np.triu(a_real_lower)
real_matrices = [a_real, a_real_upper, a_real_lower, a_real_diag]
real_solutions = []
for a in real_matrices:
y0 = np.arange(1, a.shape[0] + 1)
y_exact = _analytical_solution(a, y0, t_exact)
real_solutions.append((y0, t_exact, y_exact))
def check_real(idx, solver, meth, use_jac, with_jac, banded):
a = real_matrices[idx]
y0, t_exact, y_exact = real_solutions[idx]
t, y = _solve_linear_sys(a, y0,
tend=t_exact[-1],
dt=t_exact[1] - t_exact[0],
solver=solver,
method=meth,
use_jac=use_jac,
with_jacobian=with_jac,
banded=banded)
assert_allclose(t, t_exact)
assert_allclose(y, y_exact)
for idx in range(len(real_matrices)):
p = [['vode', 'lsoda'], # solver
['bdf', 'adams'], # method
[False, True], # use_jac
[False, True], # with_jacobian
[False, True]] # banded
for solver, meth, use_jac, with_jac, banded in itertools.product(*p):
yield check_real, idx, solver, meth, use_jac, with_jac, banded
# --- Complex arrays for testing the "zvode" solver ---
# complex, lband = 2, uband = 1:
a_complex = a_real - 0.5j * a_real
# complex, lband = 0, uband = 0:
a_complex_diag = np.diag(np.diag(a_complex))
complex_matrices = [a_complex, a_complex_diag]
complex_solutions = []
for a in complex_matrices:
y0 = np.arange(1, a.shape[0] + 1) + 1j
y_exact = _analytical_solution(a, y0, t_exact)
complex_solutions.append((y0, t_exact, y_exact))
def check_complex(idx, solver, meth, use_jac, with_jac, banded):
a = complex_matrices[idx]
y0, t_exact, y_exact = complex_solutions[idx]
t, y = _solve_linear_sys(a, y0,
tend=t_exact[-1],
dt=t_exact[1] - t_exact[0],
solver=solver,
method=meth,
use_jac=use_jac,
with_jacobian=with_jac,
banded=banded)
assert_allclose(t, t_exact)
assert_allclose(y, y_exact)
for idx in range(len(complex_matrices)):
p = [['bdf', 'adams'], # method
[False, True], # use_jac
[False, True], # with_jacobian
[False, True]] # banded
for meth, use_jac, with_jac, banded in itertools.product(*p):
yield check_complex, idx, "zvode", meth, use_jac, with_jac, banded
if __name__ == "__main__":
run_module_suite()
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