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# Authors: Nils Wagner, Ed Schofield, Pauli Virtanen, John Travers
"""
Tests for numerical integration.
"""
import numpy
from numpy import arange, zeros, array, dot, sqrt, cos, sin, eye, pi, exp, \
allclose
from numpy.testing import assert_, TestCase, run_module_suite, \
assert_array_almost_equal, assert_raises, assert_allclose
from scipy.integrate import odeint, ode, complex_ode
#------------------------------------------------------------------------------
# Test ODE integrators
#------------------------------------------------------------------------------
class TestOdeint(TestCase):
"""
Check integrate.odeint
"""
def _do_problem(self, problem):
t = arange(0.0, problem.stop_t, 0.05)
z, infodict = odeint(problem.f, problem.z0, t, full_output=True)
assert_(problem.verify(z, t))
def test_odeint(self):
for problem_cls in PROBLEMS:
problem = problem_cls()
if problem.cmplx: continue
self._do_problem(problem)
class TestOde(TestCase):
"""
Check integrate.ode
"""
def _do_problem(self, problem, integrator, method='adams'):
# ode has callback arguments in different order than odeint
f = lambda t, z: problem.f(z, t)
jac = None
if hasattr(problem, 'jac'):
jac = lambda t, z: problem.jac(z, t)
ig = ode(f, jac)
ig.set_integrator(integrator,
atol=problem.atol/10,
rtol=problem.rtol/10,
method=method)
ig.set_initial_value(problem.z0, t=0.0)
z = ig.integrate(problem.stop_t)
assert_(ig.successful(), (problem, method))
assert_(problem.verify(array([z]), problem.stop_t), (problem, method))
def test_vode(self):
"""Check the vode solver"""
for problem_cls in PROBLEMS:
problem = problem_cls()
if problem.cmplx: continue
if not problem.stiff:
self._do_problem(problem, 'vode', 'adams')
self._do_problem(problem, 'vode', 'bdf')
def test_zvode(self):
"""Check the zvode solver"""
for problem_cls in PROBLEMS:
problem = problem_cls()
if not problem.stiff:
self._do_problem(problem, 'zvode', 'adams')
self._do_problem(problem, 'zvode', 'bdf')
def test_dopri5(self):
"""Check the dopri5 solver"""
for problem_cls in PROBLEMS:
problem = problem_cls()
if problem.cmplx: continue
if problem.stiff: continue
if hasattr(problem, 'jac'): continue
self._do_problem(problem, 'dopri5')
def test_dop853(self):
"""Check the dop853 solver"""
for problem_cls in PROBLEMS:
problem = problem_cls()
if problem.cmplx: continue
if problem.stiff: continue
if hasattr(problem, 'jac'): continue
self._do_problem(problem, 'dop853')
def test_concurrent_fail(self):
for sol in ('vode', 'zvode'):
f = lambda t, y: 1.0
r = ode(f).set_integrator(sol)
r.set_initial_value(0, 0)
r2 = ode(f).set_integrator(sol)
r2.set_initial_value(0, 0)
r.integrate(r.t + 0.1)
r2.integrate(r2.t + 0.1)
assert_raises(RuntimeError, r.integrate, r.t + 0.1)
def test_concurrent_ok(self):
f = lambda t, y: 1.0
for k in xrange(3):
for sol in ('vode', 'zvode', 'dopri5', 'dop853'):
r = ode(f).set_integrator(sol)
r.set_initial_value(0, 0)
r2 = ode(f).set_integrator(sol)
r2.set_initial_value(0, 0)
r.integrate(r.t + 0.1)
r2.integrate(r2.t + 0.1)
r2.integrate(r2.t + 0.1)
assert_allclose(r.y, 0.1)
assert_allclose(r2.y, 0.2)
for sol in ('dopri5', 'dop853'):
r = ode(f).set_integrator(sol)
r.set_initial_value(0, 0)
r2 = ode(f).set_integrator(sol)
r2.set_initial_value(0, 0)
r.integrate(r.t + 0.1)
r.integrate(r.t + 0.1)
r2.integrate(r2.t + 0.1)
r.integrate(r.t + 0.1)
r2.integrate(r2.t + 0.1)
assert_allclose(r.y, 0.3)
assert_allclose(r2.y, 0.2)
class TestComplexOde(TestCase):
"""
Check integrate.complex_ode
"""
def _do_problem(self, problem, integrator, method='adams'):
# ode has callback arguments in different order than odeint
f = lambda t, z: problem.f(z, t)
jac = None
if hasattr(problem, 'jac'):
jac = lambda t, z: problem.jac(z, t)
ig = complex_ode(f, jac)
ig.set_integrator(integrator,
atol=problem.atol/10,
rtol=problem.rtol/10,
method=method)
ig.set_initial_value(problem.z0, t=0.0)
z = ig.integrate(problem.stop_t)
assert_(ig.successful(), (problem, method))
assert_(problem.verify(array([z]), problem.stop_t), (problem, method))
def test_vode(self):
"""Check the vode solver"""
for problem_cls in PROBLEMS:
problem = problem_cls()
if not problem.stiff:
self._do_problem(problem, 'vode', 'adams')
else:
self._do_problem(problem, 'vode', 'bdf')
def test_dopri5(self):
"""Check the dopri5 solver"""
for problem_cls in PROBLEMS:
problem = problem_cls()
if problem.stiff: continue
if hasattr(problem, 'jac'): continue
self._do_problem(problem, 'dopri5')
def test_dop853(self):
"""Check the dop853 solver"""
for problem_cls in PROBLEMS:
problem = problem_cls()
if problem.stiff: continue
if hasattr(problem, 'jac'): continue
self._do_problem(problem, 'dop853')
#------------------------------------------------------------------------------
# Test problems
#------------------------------------------------------------------------------
class ODE:
"""
ODE problem
"""
stiff = False
cmplx = False
stop_t = 1
z0 = []
atol = 1e-6
rtol = 1e-5
class SimpleOscillator(ODE):
r"""
Free vibration of a simple oscillator::
m \ddot{u} + k u = 0, u(0) = u_0 \dot{u}(0) \dot{u}_0
Solution::
u(t) = u_0*cos(sqrt(k/m)*t)+\dot{u}_0*sin(sqrt(k/m)*t)/sqrt(k/m)
"""
stop_t = 1 + 0.09
z0 = array([1.0, 0.1], float)
k = 4.0
m = 1.0
def f(self, z, t):
tmp = zeros((2,2), float)
tmp[0,1] = 1.0
tmp[1,0] = -self.k / self.m
return dot(tmp, z)
def verify(self, zs, t):
omega = sqrt(self.k / self.m)
u = self.z0[0]*cos(omega*t)+self.z0[1]*sin(omega*t)/omega
return allclose(u, zs[:,0], atol=self.atol, rtol=self.rtol)
class ComplexExp(ODE):
r"""The equation :lm:`\dot u = i u`"""
stop_t = 1.23*pi
z0 = exp([1j,2j,3j,4j,5j])
cmplx = True
def f(self, z, t):
return 1j*z
def jac(self, z, t):
return 1j*eye(5)
def verify(self, zs, t):
u = self.z0 * exp(1j*t)
return allclose(u, zs, atol=self.atol, rtol=self.rtol)
class Pi(ODE):
r"""Integrate 1/(t + 1j) from t=-10 to t=10"""
stop_t = 20
z0 = [0]
cmplx = True
def f(self, z, t):
return array([1./(t - 10 + 1j)])
def verify(self, zs, t):
u = -2j*numpy.arctan(10)
return allclose(u, zs[-1,:], atol=self.atol, rtol=self.rtol)
PROBLEMS = [SimpleOscillator, ComplexExp, Pi]
#------------------------------------------------------------------------------
def f(t, x):
dxdt = [x[1], -x[0]]
return dxdt
def jac(t, x):
j = array([[ 0.0, 1.0],
[-1.0, 0.0]])
return j
def f1(t, x, omega):
dxdt = [omega*x[1], -omega*x[0]]
return dxdt
def jac1(t, x, omega):
j = array([[ 0.0, omega],
[-omega, 0.0]])
return j
def f2(t, x, omega1, omega2):
dxdt = [omega1*x[1], -omega2*x[0]]
return dxdt
def jac2(t, x, omega1, omega2):
j = array([[ 0.0, omega1],
[-omega2, 0.0]])
return j
def fv(t, x, omega):
dxdt = [omega[0]*x[1], -omega[1]*x[0]]
return dxdt
def jacv(t, x, omega):
j = array([[ 0.0, omega[0]],
[-omega[1], 0.0]])
return j
class ODECheckParameterUse(object):
"""Call an ode-class solver with several cases of parameter use."""
# This class is intentionally not a TestCase subclass.
# solver_name must be set before tests can be run with this class.
# Set these in subclasses.
solver_name = ''
solver_uses_jac = False
def _get_solver(self, f, jac):
solver = ode(f, jac)
if self.solver_uses_jac:
solver.set_integrator(self.solver_name, atol=1e-9, rtol=1e-7,
with_jacobian=self.solver_uses_jac)
else:
# XXX Shouldn't set_integrator *always* accept the keyword arg
# 'with_jacobian', and perhaps raise an exception if it is set
# to True if the solver can't actually use it?
solver.set_integrator(self.solver_name, atol=1e-9, rtol=1e-7)
return solver
def _check_solver(self, solver):
ic = [1.0, 0.0]
solver.set_initial_value(ic, 0.0)
solver.integrate(pi)
assert_array_almost_equal(solver.y, [-1.0, 0.0])
def test_no_params(self):
solver = self._get_solver(f, jac)
self._check_solver(solver)
def test_one_scalar_param(self):
solver = self._get_solver(f1, jac1)
omega = 1.0
solver.set_f_params(omega)
if self.solver_uses_jac:
solver.set_jac_params(omega)
self._check_solver(solver)
def test_two_scalar_params(self):
solver = self._get_solver(f2, jac2)
omega1 = 1.0
omega2 = 1.0
solver.set_f_params(omega1, omega2)
if self.solver_uses_jac:
solver.set_jac_params(omega1, omega2)
self._check_solver(solver)
def test_vector_param(self):
solver = self._get_solver(fv, jacv)
omega = [1.0, 1.0]
solver.set_f_params(omega)
if self.solver_uses_jac:
solver.set_jac_params(omega)
self._check_solver(solver)
class DOPRI5CheckParameterUse(ODECheckParameterUse, TestCase):
solver_name = 'dopri5'
solver_uses_jac = False
class DOP853CheckParameterUse(ODECheckParameterUse, TestCase):
solver_name = 'dop853'
solver_uses_jac = False
class VODECheckParameterUse(ODECheckParameterUse, TestCase):
solver_name = 'vode'
solver_uses_jac = True
class ZVODECheckParameterUse(ODECheckParameterUse, TestCase):
solver_name = 'zvode'
solver_uses_jac = True
if __name__ == "__main__":
run_module_suite()
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