835 lines
24 KiB
Python
835 lines
24 KiB
Python
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# Authors: Nils Wagner, Ed Schofield, Pauli Virtanen, John Travers
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"""
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Tests for numerical integration.
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"""
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import numpy as np
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from numpy import (arange, zeros, array, dot, sqrt, cos, sin, eye, pi, exp,
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allclose)
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from numpy.testing import (
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assert_, assert_array_almost_equal,
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assert_allclose, assert_array_equal, assert_equal, assert_warns)
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from pytest import raises as assert_raises
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from scipy.integrate import odeint, ode, complex_ode
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#------------------------------------------------------------------------------
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# Test ODE integrators
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#------------------------------------------------------------------------------
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class TestOdeint:
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# Check integrate.odeint
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def _do_problem(self, problem):
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t = arange(0.0, problem.stop_t, 0.05)
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# Basic case
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z, infodict = odeint(problem.f, problem.z0, t, full_output=True)
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assert_(problem.verify(z, t))
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# Use tfirst=True
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z, infodict = odeint(lambda t, y: problem.f(y, t), problem.z0, t,
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full_output=True, tfirst=True)
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assert_(problem.verify(z, t))
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if hasattr(problem, 'jac'):
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# Use Dfun
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z, infodict = odeint(problem.f, problem.z0, t, Dfun=problem.jac,
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full_output=True)
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assert_(problem.verify(z, t))
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# Use Dfun and tfirst=True
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z, infodict = odeint(lambda t, y: problem.f(y, t), problem.z0, t,
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Dfun=lambda t, y: problem.jac(y, t),
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full_output=True, tfirst=True)
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assert_(problem.verify(z, t))
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def test_odeint(self):
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for problem_cls in PROBLEMS:
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problem = problem_cls()
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if problem.cmplx:
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continue
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self._do_problem(problem)
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class TestODEClass:
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ode_class = None # Set in subclass.
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def _do_problem(self, problem, integrator, method='adams'):
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# ode has callback arguments in different order than odeint
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def f(t, z):
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return problem.f(z, t)
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jac = None
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if hasattr(problem, 'jac'):
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def jac(t, z):
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return problem.jac(z, t)
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integrator_params = {}
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if problem.lband is not None or problem.uband is not None:
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integrator_params['uband'] = problem.uband
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integrator_params['lband'] = problem.lband
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ig = self.ode_class(f, jac)
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ig.set_integrator(integrator,
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atol=problem.atol/10,
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rtol=problem.rtol/10,
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method=method,
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**integrator_params)
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ig.set_initial_value(problem.z0, t=0.0)
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z = ig.integrate(problem.stop_t)
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assert_array_equal(z, ig.y)
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assert_(ig.successful(), (problem, method))
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assert_(ig.get_return_code() > 0, (problem, method))
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assert_(problem.verify(array([z]), problem.stop_t), (problem, method))
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class TestOde(TestODEClass):
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ode_class = ode
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def test_vode(self):
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# Check the vode solver
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for problem_cls in PROBLEMS:
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problem = problem_cls()
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if problem.cmplx:
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continue
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if not problem.stiff:
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self._do_problem(problem, 'vode', 'adams')
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self._do_problem(problem, 'vode', 'bdf')
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def test_zvode(self):
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# Check the zvode solver
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for problem_cls in PROBLEMS:
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problem = problem_cls()
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if not problem.stiff:
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self._do_problem(problem, 'zvode', 'adams')
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self._do_problem(problem, 'zvode', 'bdf')
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def test_lsoda(self):
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# Check the lsoda solver
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for problem_cls in PROBLEMS:
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problem = problem_cls()
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if problem.cmplx:
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continue
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self._do_problem(problem, 'lsoda')
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def test_dopri5(self):
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# Check the dopri5 solver
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for problem_cls in PROBLEMS:
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problem = problem_cls()
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if problem.cmplx:
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continue
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if problem.stiff:
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continue
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if hasattr(problem, 'jac'):
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continue
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self._do_problem(problem, 'dopri5')
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def test_dop853(self):
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# Check the dop853 solver
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for problem_cls in PROBLEMS:
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problem = problem_cls()
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if problem.cmplx:
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continue
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if problem.stiff:
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continue
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if hasattr(problem, 'jac'):
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continue
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self._do_problem(problem, 'dop853')
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def test_concurrent_fail(self):
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for sol in ('vode', 'zvode', 'lsoda'):
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def f(t, y):
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return 1.0
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r = ode(f).set_integrator(sol)
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r.set_initial_value(0, 0)
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r2 = ode(f).set_integrator(sol)
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r2.set_initial_value(0, 0)
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r.integrate(r.t + 0.1)
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r2.integrate(r2.t + 0.1)
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assert_raises(RuntimeError, r.integrate, r.t + 0.1)
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def test_concurrent_ok(self):
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def f(t, y):
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return 1.0
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for k in range(3):
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for sol in ('vode', 'zvode', 'lsoda', 'dopri5', 'dop853'):
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r = ode(f).set_integrator(sol)
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r.set_initial_value(0, 0)
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r2 = ode(f).set_integrator(sol)
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r2.set_initial_value(0, 0)
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r.integrate(r.t + 0.1)
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r2.integrate(r2.t + 0.1)
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r2.integrate(r2.t + 0.1)
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assert_allclose(r.y, 0.1)
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assert_allclose(r2.y, 0.2)
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for sol in ('dopri5', 'dop853'):
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r = ode(f).set_integrator(sol)
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r.set_initial_value(0, 0)
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r2 = ode(f).set_integrator(sol)
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r2.set_initial_value(0, 0)
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r.integrate(r.t + 0.1)
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r.integrate(r.t + 0.1)
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r2.integrate(r2.t + 0.1)
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r.integrate(r.t + 0.1)
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r2.integrate(r2.t + 0.1)
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assert_allclose(r.y, 0.3)
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assert_allclose(r2.y, 0.2)
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class TestComplexOde(TestODEClass):
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ode_class = complex_ode
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def test_vode(self):
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# Check the vode solver
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for problem_cls in PROBLEMS:
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problem = problem_cls()
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if not problem.stiff:
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self._do_problem(problem, 'vode', 'adams')
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else:
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self._do_problem(problem, 'vode', 'bdf')
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def test_lsoda(self):
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# Check the lsoda solver
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for problem_cls in PROBLEMS:
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problem = problem_cls()
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self._do_problem(problem, 'lsoda')
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def test_dopri5(self):
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# Check the dopri5 solver
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for problem_cls in PROBLEMS:
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problem = problem_cls()
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if problem.stiff:
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continue
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if hasattr(problem, 'jac'):
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continue
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self._do_problem(problem, 'dopri5')
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def test_dop853(self):
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# Check the dop853 solver
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for problem_cls in PROBLEMS:
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problem = problem_cls()
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if problem.stiff:
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continue
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if hasattr(problem, 'jac'):
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continue
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self._do_problem(problem, 'dop853')
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class TestSolout:
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# Check integrate.ode correctly handles solout for dopri5 and dop853
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def _run_solout_test(self, integrator):
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# Check correct usage of solout
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ts = []
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ys = []
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t0 = 0.0
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tend = 10.0
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y0 = [1.0, 2.0]
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def solout(t, y):
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ts.append(t)
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ys.append(y.copy())
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def rhs(t, y):
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return [y[0] + y[1], -y[1]**2]
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ig = ode(rhs).set_integrator(integrator)
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ig.set_solout(solout)
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ig.set_initial_value(y0, t0)
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ret = ig.integrate(tend)
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assert_array_equal(ys[0], y0)
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assert_array_equal(ys[-1], ret)
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assert_equal(ts[0], t0)
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assert_equal(ts[-1], tend)
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def test_solout(self):
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for integrator in ('dopri5', 'dop853'):
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self._run_solout_test(integrator)
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def _run_solout_after_initial_test(self, integrator):
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# Check if solout works even if it is set after the initial value.
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ts = []
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ys = []
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t0 = 0.0
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tend = 10.0
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y0 = [1.0, 2.0]
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def solout(t, y):
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ts.append(t)
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ys.append(y.copy())
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def rhs(t, y):
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return [y[0] + y[1], -y[1]**2]
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ig = ode(rhs).set_integrator(integrator)
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ig.set_initial_value(y0, t0)
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ig.set_solout(solout)
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ret = ig.integrate(tend)
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assert_array_equal(ys[0], y0)
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assert_array_equal(ys[-1], ret)
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assert_equal(ts[0], t0)
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assert_equal(ts[-1], tend)
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def test_solout_after_initial(self):
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for integrator in ('dopri5', 'dop853'):
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self._run_solout_after_initial_test(integrator)
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def _run_solout_break_test(self, integrator):
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# Check correct usage of stopping via solout
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ts = []
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ys = []
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t0 = 0.0
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tend = 10.0
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y0 = [1.0, 2.0]
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def solout(t, y):
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ts.append(t)
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ys.append(y.copy())
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if t > tend/2.0:
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return -1
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def rhs(t, y):
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return [y[0] + y[1], -y[1]**2]
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ig = ode(rhs).set_integrator(integrator)
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ig.set_solout(solout)
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ig.set_initial_value(y0, t0)
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ret = ig.integrate(tend)
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assert_array_equal(ys[0], y0)
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assert_array_equal(ys[-1], ret)
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assert_equal(ts[0], t0)
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assert_(ts[-1] > tend/2.0)
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assert_(ts[-1] < tend)
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def test_solout_break(self):
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for integrator in ('dopri5', 'dop853'):
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self._run_solout_break_test(integrator)
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class TestComplexSolout:
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# Check integrate.ode correctly handles solout for dopri5 and dop853
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def _run_solout_test(self, integrator):
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# Check correct usage of solout
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ts = []
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ys = []
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t0 = 0.0
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tend = 20.0
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y0 = [0.0]
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def solout(t, y):
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ts.append(t)
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ys.append(y.copy())
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def rhs(t, y):
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return [1.0/(t - 10.0 - 1j)]
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ig = complex_ode(rhs).set_integrator(integrator)
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ig.set_solout(solout)
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ig.set_initial_value(y0, t0)
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ret = ig.integrate(tend)
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assert_array_equal(ys[0], y0)
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assert_array_equal(ys[-1], ret)
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assert_equal(ts[0], t0)
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assert_equal(ts[-1], tend)
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def test_solout(self):
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for integrator in ('dopri5', 'dop853'):
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self._run_solout_test(integrator)
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def _run_solout_break_test(self, integrator):
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# Check correct usage of stopping via solout
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ts = []
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ys = []
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t0 = 0.0
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tend = 20.0
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y0 = [0.0]
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def solout(t, y):
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ts.append(t)
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ys.append(y.copy())
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if t > tend/2.0:
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return -1
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def rhs(t, y):
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return [1.0/(t - 10.0 - 1j)]
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ig = complex_ode(rhs).set_integrator(integrator)
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ig.set_solout(solout)
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ig.set_initial_value(y0, t0)
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ret = ig.integrate(tend)
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assert_array_equal(ys[0], y0)
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assert_array_equal(ys[-1], ret)
|
||
|
|
assert_equal(ts[0], t0)
|
||
|
|
assert_(ts[-1] > tend/2.0)
|
||
|
|
assert_(ts[-1] < tend)
|
||
|
|
|
||
|
|
def test_solout_break(self):
|
||
|
|
for integrator in ('dopri5', 'dop853'):
|
||
|
|
self._run_solout_break_test(integrator)
|
||
|
|
|
||
|
|
|
||
|
|
#------------------------------------------------------------------------------
|
||
|
|
# Test problems
|
||
|
|
#------------------------------------------------------------------------------
|
||
|
|
|
||
|
|
|
||
|
|
class ODE:
|
||
|
|
"""
|
||
|
|
ODE problem
|
||
|
|
"""
|
||
|
|
stiff = False
|
||
|
|
cmplx = False
|
||
|
|
stop_t = 1
|
||
|
|
z0 = []
|
||
|
|
|
||
|
|
lband = None
|
||
|
|
uband = None
|
||
|
|
|
||
|
|
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 * np.arctan(10)
|
||
|
|
return allclose(u, zs[-1, :], atol=self.atol, rtol=self.rtol)
|
||
|
|
|
||
|
|
|
||
|
|
class CoupledDecay(ODE):
|
||
|
|
r"""
|
||
|
|
3 coupled decays suited for banded treatment
|
||
|
|
(banded mode makes it necessary when N>>3)
|
||
|
|
"""
|
||
|
|
|
||
|
|
stiff = True
|
||
|
|
stop_t = 0.5
|
||
|
|
z0 = [5.0, 7.0, 13.0]
|
||
|
|
lband = 1
|
||
|
|
uband = 0
|
||
|
|
|
||
|
|
lmbd = [0.17, 0.23, 0.29] # fictitious decay constants
|
||
|
|
|
||
|
|
def f(self, z, t):
|
||
|
|
lmbd = self.lmbd
|
||
|
|
return np.array([-lmbd[0]*z[0],
|
||
|
|
-lmbd[1]*z[1] + lmbd[0]*z[0],
|
||
|
|
-lmbd[2]*z[2] + lmbd[1]*z[1]])
|
||
|
|
|
||
|
|
def jac(self, z, t):
|
||
|
|
# The full Jacobian is
|
||
|
|
#
|
||
|
|
# [-lmbd[0] 0 0 ]
|
||
|
|
# [ lmbd[0] -lmbd[1] 0 ]
|
||
|
|
# [ 0 lmbd[1] -lmbd[2]]
|
||
|
|
#
|
||
|
|
# The lower and upper bandwidths are lband=1 and uband=0, resp.
|
||
|
|
# The representation of this array in packed format is
|
||
|
|
#
|
||
|
|
# [-lmbd[0] -lmbd[1] -lmbd[2]]
|
||
|
|
# [ lmbd[0] lmbd[1] 0 ]
|
||
|
|
|
||
|
|
lmbd = self.lmbd
|
||
|
|
j = np.zeros((self.lband + self.uband + 1, 3), order='F')
|
||
|
|
|
||
|
|
def set_j(ri, ci, val):
|
||
|
|
j[self.uband + ri - ci, ci] = val
|
||
|
|
set_j(0, 0, -lmbd[0])
|
||
|
|
set_j(1, 0, lmbd[0])
|
||
|
|
set_j(1, 1, -lmbd[1])
|
||
|
|
set_j(2, 1, lmbd[1])
|
||
|
|
set_j(2, 2, -lmbd[2])
|
||
|
|
return j
|
||
|
|
|
||
|
|
def verify(self, zs, t):
|
||
|
|
# Formulae derived by hand
|
||
|
|
lmbd = np.array(self.lmbd)
|
||
|
|
d10 = lmbd[1] - lmbd[0]
|
||
|
|
d21 = lmbd[2] - lmbd[1]
|
||
|
|
d20 = lmbd[2] - lmbd[0]
|
||
|
|
e0 = np.exp(-lmbd[0] * t)
|
||
|
|
e1 = np.exp(-lmbd[1] * t)
|
||
|
|
e2 = np.exp(-lmbd[2] * t)
|
||
|
|
u = np.vstack((
|
||
|
|
self.z0[0] * e0,
|
||
|
|
self.z0[1] * e1 + self.z0[0] * lmbd[0] / d10 * (e0 - e1),
|
||
|
|
self.z0[2] * e2 + self.z0[1] * lmbd[1] / d21 * (e1 - e2) +
|
||
|
|
lmbd[1] * lmbd[0] * self.z0[0] / d10 *
|
||
|
|
(1 / d20 * (e0 - e2) - 1 / d21 * (e1 - e2)))).transpose()
|
||
|
|
return allclose(u, zs, atol=self.atol, rtol=self.rtol)
|
||
|
|
|
||
|
|
|
||
|
|
PROBLEMS = [SimpleOscillator, ComplexExp, Pi, CoupledDecay]
|
||
|
|
|
||
|
|
#------------------------------------------------------------------------------
|
||
|
|
|
||
|
|
|
||
|
|
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:
|
||
|
|
"""Call an ode-class solver with several cases of parameter use."""
|
||
|
|
|
||
|
|
# 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)
|
||
|
|
|
||
|
|
def test_warns_on_failure(self):
|
||
|
|
# Set nsteps small to ensure failure
|
||
|
|
solver = self._get_solver(f, jac)
|
||
|
|
solver.set_integrator(self.solver_name, nsteps=1)
|
||
|
|
ic = [1.0, 0.0]
|
||
|
|
solver.set_initial_value(ic, 0.0)
|
||
|
|
assert_warns(UserWarning, solver.integrate, pi)
|
||
|
|
|
||
|
|
|
||
|
|
class TestDOPRI5CheckParameterUse(ODECheckParameterUse):
|
||
|
|
solver_name = 'dopri5'
|
||
|
|
solver_uses_jac = False
|
||
|
|
|
||
|
|
|
||
|
|
class TestDOP853CheckParameterUse(ODECheckParameterUse):
|
||
|
|
solver_name = 'dop853'
|
||
|
|
solver_uses_jac = False
|
||
|
|
|
||
|
|
|
||
|
|
class TestVODECheckParameterUse(ODECheckParameterUse):
|
||
|
|
solver_name = 'vode'
|
||
|
|
solver_uses_jac = True
|
||
|
|
|
||
|
|
|
||
|
|
class TestZVODECheckParameterUse(ODECheckParameterUse):
|
||
|
|
solver_name = 'zvode'
|
||
|
|
solver_uses_jac = True
|
||
|
|
|
||
|
|
|
||
|
|
class TestLSODACheckParameterUse(ODECheckParameterUse):
|
||
|
|
solver_name = 'lsoda'
|
||
|
|
solver_uses_jac = True
|
||
|
|
|
||
|
|
|
||
|
|
def test_odeint_trivial_time():
|
||
|
|
# Test that odeint succeeds when given a single time point
|
||
|
|
# and full_output=True. This is a regression test for gh-4282.
|
||
|
|
y0 = 1
|
||
|
|
t = [0]
|
||
|
|
y, info = odeint(lambda y, t: -y, y0, t, full_output=True)
|
||
|
|
assert_array_equal(y, np.array([[y0]]))
|
||
|
|
|
||
|
|
|
||
|
|
def test_odeint_banded_jacobian():
|
||
|
|
# Test the use of the `Dfun`, `ml` and `mu` options of odeint.
|
||
|
|
|
||
|
|
def func(y, t, c):
|
||
|
|
return c.dot(y)
|
||
|
|
|
||
|
|
def jac(y, t, c):
|
||
|
|
return c
|
||
|
|
|
||
|
|
def jac_transpose(y, t, c):
|
||
|
|
return c.T.copy(order='C')
|
||
|
|
|
||
|
|
def bjac_rows(y, t, c):
|
||
|
|
jac = np.vstack((np.r_[0, np.diag(c, 1)],
|
||
|
|
np.diag(c),
|
||
|
|
np.r_[np.diag(c, -1), 0],
|
||
|
|
np.r_[np.diag(c, -2), 0, 0]))
|
||
|
|
return jac
|
||
|
|
|
||
|
|
def bjac_cols(y, t, c):
|
||
|
|
return bjac_rows(y, t, c).T.copy(order='C')
|
||
|
|
|
||
|
|
c = array([[-205, 0.01, 0.00, 0.0],
|
||
|
|
[0.1, -2.50, 0.02, 0.0],
|
||
|
|
[1e-3, 0.01, -2.0, 0.01],
|
||
|
|
[0.00, 0.00, 0.1, -1.0]])
|
||
|
|
|
||
|
|
y0 = np.ones(4)
|
||
|
|
t = np.array([0, 5, 10, 100])
|
||
|
|
|
||
|
|
# Use the full Jacobian.
|
||
|
|
sol1, info1 = odeint(func, y0, t, args=(c,), full_output=True,
|
||
|
|
atol=1e-13, rtol=1e-11, mxstep=10000,
|
||
|
|
Dfun=jac)
|
||
|
|
|
||
|
|
# Use the transposed full Jacobian, with col_deriv=True.
|
||
|
|
sol2, info2 = odeint(func, y0, t, args=(c,), full_output=True,
|
||
|
|
atol=1e-13, rtol=1e-11, mxstep=10000,
|
||
|
|
Dfun=jac_transpose, col_deriv=True)
|
||
|
|
|
||
|
|
# Use the banded Jacobian.
|
||
|
|
sol3, info3 = odeint(func, y0, t, args=(c,), full_output=True,
|
||
|
|
atol=1e-13, rtol=1e-11, mxstep=10000,
|
||
|
|
Dfun=bjac_rows, ml=2, mu=1)
|
||
|
|
|
||
|
|
# Use the transposed banded Jacobian, with col_deriv=True.
|
||
|
|
sol4, info4 = odeint(func, y0, t, args=(c,), full_output=True,
|
||
|
|
atol=1e-13, rtol=1e-11, mxstep=10000,
|
||
|
|
Dfun=bjac_cols, ml=2, mu=1, col_deriv=True)
|
||
|
|
|
||
|
|
assert_allclose(sol1, sol2, err_msg="sol1 != sol2")
|
||
|
|
assert_allclose(sol1, sol3, atol=1e-12, err_msg="sol1 != sol3")
|
||
|
|
assert_allclose(sol3, sol4, err_msg="sol3 != sol4")
|
||
|
|
|
||
|
|
# Verify that the number of jacobian evaluations was the same for the
|
||
|
|
# calls of odeint with a full jacobian and with a banded jacobian. This is
|
||
|
|
# a regression test--there was a bug in the handling of banded jacobians
|
||
|
|
# that resulted in an incorrect jacobian matrix being passed to the LSODA
|
||
|
|
# code. That would cause errors or excessive jacobian evaluations.
|
||
|
|
assert_array_equal(info1['nje'], info2['nje'])
|
||
|
|
assert_array_equal(info3['nje'], info4['nje'])
|
||
|
|
|
||
|
|
# Test the use of tfirst
|
||
|
|
sol1ty, info1ty = odeint(lambda t, y, c: func(y, t, c), y0, t, args=(c,),
|
||
|
|
full_output=True, atol=1e-13, rtol=1e-11,
|
||
|
|
mxstep=10000,
|
||
|
|
Dfun=lambda t, y, c: jac(y, t, c), tfirst=True)
|
||
|
|
# The code should execute the exact same sequence of floating point
|
||
|
|
# calculations, so these should be exactly equal. We'll be safe and use
|
||
|
|
# a small tolerance.
|
||
|
|
assert_allclose(sol1, sol1ty, rtol=1e-12, err_msg="sol1 != sol1ty")
|
||
|
|
|
||
|
|
|
||
|
|
def test_odeint_errors():
|
||
|
|
def sys1d(x, t):
|
||
|
|
return -100*x
|
||
|
|
|
||
|
|
def bad1(x, t):
|
||
|
|
return 1.0/0
|
||
|
|
|
||
|
|
def bad2(x, t):
|
||
|
|
return "foo"
|
||
|
|
|
||
|
|
def bad_jac1(x, t):
|
||
|
|
return 1.0/0
|
||
|
|
|
||
|
|
def bad_jac2(x, t):
|
||
|
|
return [["foo"]]
|
||
|
|
|
||
|
|
def sys2d(x, t):
|
||
|
|
return [-100*x[0], -0.1*x[1]]
|
||
|
|
|
||
|
|
def sys2d_bad_jac(x, t):
|
||
|
|
return [[1.0/0, 0], [0, -0.1]]
|
||
|
|
|
||
|
|
assert_raises(ZeroDivisionError, odeint, bad1, 1.0, [0, 1])
|
||
|
|
assert_raises(ValueError, odeint, bad2, 1.0, [0, 1])
|
||
|
|
|
||
|
|
assert_raises(ZeroDivisionError, odeint, sys1d, 1.0, [0, 1], Dfun=bad_jac1)
|
||
|
|
assert_raises(ValueError, odeint, sys1d, 1.0, [0, 1], Dfun=bad_jac2)
|
||
|
|
|
||
|
|
assert_raises(ZeroDivisionError, odeint, sys2d, [1.0, 1.0], [0, 1],
|
||
|
|
Dfun=sys2d_bad_jac)
|
||
|
|
|
||
|
|
|
||
|
|
def test_odeint_bad_shapes():
|
||
|
|
# Tests of some errors that can occur with odeint.
|
||
|
|
|
||
|
|
def badrhs(x, t):
|
||
|
|
return [1, -1]
|
||
|
|
|
||
|
|
def sys1(x, t):
|
||
|
|
return -100*x
|
||
|
|
|
||
|
|
def badjac(x, t):
|
||
|
|
return [[0, 0, 0]]
|
||
|
|
|
||
|
|
# y0 must be at most 1-d.
|
||
|
|
bad_y0 = [[0, 0], [0, 0]]
|
||
|
|
assert_raises(ValueError, odeint, sys1, bad_y0, [0, 1])
|
||
|
|
|
||
|
|
# t must be at most 1-d.
|
||
|
|
bad_t = [[0, 1], [2, 3]]
|
||
|
|
assert_raises(ValueError, odeint, sys1, [10.0], bad_t)
|
||
|
|
|
||
|
|
# y0 is 10, but badrhs(x, t) returns [1, -1].
|
||
|
|
assert_raises(RuntimeError, odeint, badrhs, 10, [0, 1])
|
||
|
|
|
||
|
|
# shape of array returned by badjac(x, t) is not correct.
|
||
|
|
assert_raises(RuntimeError, odeint, sys1, [10, 10], [0, 1], Dfun=badjac)
|
||
|
|
|
||
|
|
|
||
|
|
def test_repeated_t_values():
|
||
|
|
"""Regression test for gh-8217."""
|
||
|
|
|
||
|
|
def func(x, t):
|
||
|
|
return -0.25*x
|
||
|
|
|
||
|
|
t = np.zeros(10)
|
||
|
|
sol = odeint(func, [1.], t)
|
||
|
|
assert_array_equal(sol, np.ones((len(t), 1)))
|
||
|
|
|
||
|
|
tau = 4*np.log(2)
|
||
|
|
t = [0]*9 + [tau, 2*tau, 2*tau, 3*tau]
|
||
|
|
sol = odeint(func, [1, 2], t, rtol=1e-12, atol=1e-12)
|
||
|
|
expected_sol = np.array([[1.0, 2.0]]*9 +
|
||
|
|
[[0.5, 1.0],
|
||
|
|
[0.25, 0.5],
|
||
|
|
[0.25, 0.5],
|
||
|
|
[0.125, 0.25]])
|
||
|
|
assert_allclose(sol, expected_sol)
|
||
|
|
|
||
|
|
# Edge case: empty t sequence.
|
||
|
|
sol = odeint(func, [1.], [])
|
||
|
|
assert_array_equal(sol, np.array([], dtype=np.float64).reshape((0, 1)))
|
||
|
|
|
||
|
|
# t values are not monotonic.
|
||
|
|
assert_raises(ValueError, odeint, func, [1.], [0, 1, 0.5, 0])
|
||
|
|
assert_raises(ValueError, odeint, func, [1, 2, 3], [0, -1, -2, 3])
|