535 lines
18 KiB
Python
535 lines
18 KiB
Python
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""" Unit tests for nonlinear solvers
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Author: Ondrej Certik
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May 2007
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"""
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from numpy.testing import assert_
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import pytest
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from scipy.optimize import _nonlin as nonlin, root
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from scipy.sparse import csr_array
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from numpy import diag, dot
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from numpy.linalg import inv
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import numpy as np
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import scipy
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from .test_minpack import pressure_network
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SOLVERS = {'anderson': nonlin.anderson,
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'diagbroyden': nonlin.diagbroyden,
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'linearmixing': nonlin.linearmixing,
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'excitingmixing': nonlin.excitingmixing,
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'broyden1': nonlin.broyden1,
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'broyden2': nonlin.broyden2,
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'krylov': nonlin.newton_krylov}
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MUST_WORK = {'anderson': nonlin.anderson, 'broyden1': nonlin.broyden1,
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'broyden2': nonlin.broyden2, 'krylov': nonlin.newton_krylov}
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# ----------------------------------------------------------------------------
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# Test problems
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# ----------------------------------------------------------------------------
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def F(x):
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x = np.asarray(x).T
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d = diag([3, 2, 1.5, 1, 0.5])
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c = 0.01
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f = -d @ x - c * float(x.T @ x) * x
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return f
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F.xin = [1, 1, 1, 1, 1]
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F.KNOWN_BAD = {}
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F.JAC_KSP_BAD = {}
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F.ROOT_JAC_KSP_BAD = {}
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def F2(x):
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return x
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F2.xin = [1, 2, 3, 4, 5, 6]
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F2.KNOWN_BAD = {'linearmixing': nonlin.linearmixing,
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'excitingmixing': nonlin.excitingmixing}
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F2.JAC_KSP_BAD = {}
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F2.ROOT_JAC_KSP_BAD = {}
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def F2_lucky(x):
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return x
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F2_lucky.xin = [0, 0, 0, 0, 0, 0]
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F2_lucky.KNOWN_BAD = {}
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F2_lucky.JAC_KSP_BAD = {}
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F2_lucky.ROOT_JAC_KSP_BAD = {}
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def F3(x):
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A = np.array([[-2, 1, 0.], [1, -2, 1], [0, 1, -2]])
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b = np.array([1, 2, 3.])
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return A @ x - b
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F3.xin = [1, 2, 3]
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F3.KNOWN_BAD = {}
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F3.JAC_KSP_BAD = {}
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F3.ROOT_JAC_KSP_BAD = {}
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def F4_powell(x):
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A = 1e4
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return [A*x[0]*x[1] - 1, np.exp(-x[0]) + np.exp(-x[1]) - (1 + 1/A)]
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F4_powell.xin = [-1, -2]
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F4_powell.KNOWN_BAD = {'linearmixing': nonlin.linearmixing,
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'excitingmixing': nonlin.excitingmixing,
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'diagbroyden': nonlin.diagbroyden}
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# In the extreme case, it does not converge for nolinear problem solved by
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# MINRES and root problem solved by GMRES/BiCGStab/CGS/MINRES/TFQMR when using
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# Krylov method to approximate Jacobian
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F4_powell.JAC_KSP_BAD = {'minres'}
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F4_powell.ROOT_JAC_KSP_BAD = {'gmres', 'bicgstab', 'cgs', 'minres', 'tfqmr'}
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def F5(x):
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return pressure_network(x, 4, np.array([.5, .5, .5, .5]))
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F5.xin = [2., 0, 2, 0]
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F5.KNOWN_BAD = {'excitingmixing': nonlin.excitingmixing,
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'linearmixing': nonlin.linearmixing,
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'diagbroyden': nonlin.diagbroyden}
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# In the extreme case, the Jacobian inversion yielded zero vector for nonlinear
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# problem solved by CGS/MINRES and it does not converge for root problem solved
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# by MINRES and when using Krylov method to approximate Jacobian
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F5.JAC_KSP_BAD = {'cgs', 'minres'}
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F5.ROOT_JAC_KSP_BAD = {'minres'}
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def F6(x):
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x1, x2 = x
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J0 = np.array([[-4.256, 14.7],
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[0.8394989, 0.59964207]])
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v = np.array([(x1 + 3) * (x2**5 - 7) + 3*6,
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np.sin(x2 * np.exp(x1) - 1)])
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return -np.linalg.solve(J0, v)
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F6.xin = [-0.5, 1.4]
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F6.KNOWN_BAD = {'excitingmixing': nonlin.excitingmixing,
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'linearmixing': nonlin.linearmixing,
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'diagbroyden': nonlin.diagbroyden}
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F6.JAC_KSP_BAD = {}
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F6.ROOT_JAC_KSP_BAD = {}
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# ----------------------------------------------------------------------------
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# Tests
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# ----------------------------------------------------------------------------
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class TestNonlin:
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"""
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Check the Broyden methods for a few test problems.
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broyden1, broyden2, and newton_krylov must succeed for
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all functions. Some of the others don't -- tests in KNOWN_BAD are skipped.
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"""
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def _check_nonlin_func(self, f, func, f_tol=1e-2):
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# Test all methods mentioned in the class `KrylovJacobian`
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if func == SOLVERS['krylov']:
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for method in ['gmres', 'bicgstab', 'cgs', 'minres', 'tfqmr']:
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if method in f.JAC_KSP_BAD:
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continue
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x = func(f, f.xin, method=method, line_search=None,
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f_tol=f_tol, maxiter=200, verbose=0)
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assert_(np.absolute(f(x)).max() < f_tol)
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x = func(f, f.xin, f_tol=f_tol, maxiter=200, verbose=0)
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assert_(np.absolute(f(x)).max() < f_tol)
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def _check_root(self, f, method, f_tol=1e-2):
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# Test Krylov methods
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if method == 'krylov':
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for jac_method in ['gmres', 'bicgstab', 'cgs', 'minres', 'tfqmr']:
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if jac_method in f.ROOT_JAC_KSP_BAD:
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continue
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res = root(f, f.xin, method=method,
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options={'ftol': f_tol, 'maxiter': 200,
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'disp': 0,
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'jac_options': {'method': jac_method}})
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assert_(np.absolute(res.fun).max() < f_tol)
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res = root(f, f.xin, method=method,
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options={'ftol': f_tol, 'maxiter': 200, 'disp': 0})
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assert_(np.absolute(res.fun).max() < f_tol)
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@pytest.mark.xfail
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def _check_func_fail(self, *a, **kw):
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pass
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@pytest.mark.filterwarnings('ignore::DeprecationWarning')
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def test_problem_nonlin(self):
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for f in [F, F2, F2_lucky, F3, F4_powell, F5, F6]:
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for func in SOLVERS.values():
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if func in f.KNOWN_BAD.values():
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if func in MUST_WORK.values():
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self._check_func_fail(f, func)
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continue
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self._check_nonlin_func(f, func)
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@pytest.mark.filterwarnings('ignore::DeprecationWarning')
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@pytest.mark.parametrize("method", ['lgmres', 'gmres', 'bicgstab', 'cgs',
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'minres', 'tfqmr'])
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def test_tol_norm_called(self, method):
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# Check that supplying tol_norm keyword to nonlin_solve works
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self._tol_norm_used = False
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def local_norm_func(x):
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self._tol_norm_used = True
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return np.absolute(x).max()
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nonlin.newton_krylov(F, F.xin, method=method, f_tol=1e-2,
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maxiter=200, verbose=0,
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tol_norm=local_norm_func)
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assert_(self._tol_norm_used)
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@pytest.mark.filterwarnings('ignore::DeprecationWarning')
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def test_problem_root(self):
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for f in [F, F2, F2_lucky, F3, F4_powell, F5, F6]:
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for meth in SOLVERS:
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if meth in f.KNOWN_BAD:
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if meth in MUST_WORK:
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self._check_func_fail(f, meth)
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continue
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self._check_root(f, meth)
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def test_no_convergence(self):
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def wont_converge(x):
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return 1e3 + x
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with pytest.raises(scipy.optimize.NoConvergence):
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nonlin.newton_krylov(wont_converge, xin=[0], maxiter=1)
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class TestSecant:
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"""Check that some Jacobian approximations satisfy the secant condition"""
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xs = [np.array([1., 2., 3., 4., 5.]),
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np.array([2., 3., 4., 5., 1.]),
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np.array([3., 4., 5., 1., 2.]),
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np.array([4., 5., 1., 2., 3.]),
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np.array([9., 1., 9., 1., 3.]),
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np.array([0., 1., 9., 1., 3.]),
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np.array([5., 5., 7., 1., 1.]),
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np.array([1., 2., 7., 5., 1.]),]
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fs = [x**2 - 1 for x in xs]
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def _check_secant(self, jac_cls, npoints=1, **kw):
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"""
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Check that the given Jacobian approximation satisfies secant
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conditions for last `npoints` points.
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"""
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jac = jac_cls(**kw)
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jac.setup(self.xs[0], self.fs[0], None)
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for j, (x, f) in enumerate(zip(self.xs[1:], self.fs[1:])):
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jac.update(x, f)
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for k in range(min(npoints, j+1)):
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dx = self.xs[j-k+1] - self.xs[j-k]
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df = self.fs[j-k+1] - self.fs[j-k]
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assert_(np.allclose(dx, jac.solve(df)))
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# Check that the `npoints` secant bound is strict
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if j >= npoints:
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dx = self.xs[j-npoints+1] - self.xs[j-npoints]
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df = self.fs[j-npoints+1] - self.fs[j-npoints]
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assert_(not np.allclose(dx, jac.solve(df)))
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def test_broyden1(self):
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self._check_secant(nonlin.BroydenFirst)
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def test_broyden2(self):
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self._check_secant(nonlin.BroydenSecond)
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def test_broyden1_update(self):
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# Check that BroydenFirst update works as for a dense matrix
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jac = nonlin.BroydenFirst(alpha=0.1)
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jac.setup(self.xs[0], self.fs[0], None)
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B = np.identity(5) * (-1/0.1)
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for last_j, (x, f) in enumerate(zip(self.xs[1:], self.fs[1:])):
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df = f - self.fs[last_j]
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dx = x - self.xs[last_j]
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B += (df - dot(B, dx))[:, None] * dx[None, :] / dot(dx, dx)
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jac.update(x, f)
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assert_(np.allclose(jac.todense(), B, rtol=1e-10, atol=1e-13))
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def test_broyden2_update(self):
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# Check that BroydenSecond update works as for a dense matrix
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jac = nonlin.BroydenSecond(alpha=0.1)
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jac.setup(self.xs[0], self.fs[0], None)
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H = np.identity(5) * (-0.1)
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for last_j, (x, f) in enumerate(zip(self.xs[1:], self.fs[1:])):
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df = f - self.fs[last_j]
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dx = x - self.xs[last_j]
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H += (dx - dot(H, df))[:, None] * df[None, :] / dot(df, df)
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jac.update(x, f)
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assert_(np.allclose(jac.todense(), inv(H), rtol=1e-10, atol=1e-13))
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def test_anderson(self):
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# Anderson mixing (with w0=0) satisfies secant conditions
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# for the last M iterates, see [Ey]_
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#
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# .. [Ey] V. Eyert, J. Comp. Phys., 124, 271 (1996).
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self._check_secant(nonlin.Anderson, M=3, w0=0, npoints=3)
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class TestLinear:
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"""Solve a linear equation;
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some methods find the exact solution in a finite number of steps"""
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def _check(self, jac, N, maxiter, complex=False, **kw):
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np.random.seed(123)
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A = np.random.randn(N, N)
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if complex:
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A = A + 1j*np.random.randn(N, N)
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b = np.random.randn(N)
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if complex:
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b = b + 1j*np.random.randn(N)
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def func(x):
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return dot(A, x) - b
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sol = nonlin.nonlin_solve(func, np.zeros(N), jac, maxiter=maxiter,
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f_tol=1e-6, line_search=None, verbose=0)
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assert_(np.allclose(dot(A, sol), b, atol=1e-6))
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def test_broyden1(self):
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# Broyden methods solve linear systems exactly in 2*N steps
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self._check(nonlin.BroydenFirst(alpha=1.0), 20, 41, False)
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self._check(nonlin.BroydenFirst(alpha=1.0), 20, 41, True)
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def test_broyden2(self):
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# Broyden methods solve linear systems exactly in 2*N steps
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self._check(nonlin.BroydenSecond(alpha=1.0), 20, 41, False)
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self._check(nonlin.BroydenSecond(alpha=1.0), 20, 41, True)
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def test_anderson(self):
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# Anderson is rather similar to Broyden, if given enough storage space
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self._check(nonlin.Anderson(M=50, alpha=1.0), 20, 29, False)
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self._check(nonlin.Anderson(M=50, alpha=1.0), 20, 29, True)
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def test_krylov(self):
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# Krylov methods solve linear systems exactly in N inner steps
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self._check(nonlin.KrylovJacobian, 20, 2, False, inner_m=10)
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self._check(nonlin.KrylovJacobian, 20, 2, True, inner_m=10)
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|
|
||
|
|
def _check_autojac(self, A, b):
|
||
|
|
def func(x):
|
||
|
|
return A.dot(x) - b
|
||
|
|
|
||
|
|
def jac(v):
|
||
|
|
return A
|
||
|
|
|
||
|
|
sol = nonlin.nonlin_solve(func, np.zeros(b.shape[0]), jac, maxiter=2,
|
||
|
|
f_tol=1e-6, line_search=None, verbose=0)
|
||
|
|
np.testing.assert_allclose(A @ sol, b, atol=1e-6)
|
||
|
|
# test jac input as array -- not a function
|
||
|
|
sol = nonlin.nonlin_solve(func, np.zeros(b.shape[0]), A, maxiter=2,
|
||
|
|
f_tol=1e-6, line_search=None, verbose=0)
|
||
|
|
np.testing.assert_allclose(A @ sol, b, atol=1e-6)
|
||
|
|
|
||
|
|
def test_jac_sparse(self):
|
||
|
|
A = csr_array([[1, 2], [2, 1]])
|
||
|
|
b = np.array([1, -1])
|
||
|
|
self._check_autojac(A, b)
|
||
|
|
self._check_autojac((1 + 2j) * A, (2 + 2j) * b)
|
||
|
|
|
||
|
|
def test_jac_ndarray(self):
|
||
|
|
A = np.array([[1, 2], [2, 1]])
|
||
|
|
b = np.array([1, -1])
|
||
|
|
self._check_autojac(A, b)
|
||
|
|
self._check_autojac((1 + 2j) * A, (2 + 2j) * b)
|
||
|
|
|
||
|
|
|
||
|
|
class TestJacobianDotSolve:
|
||
|
|
"""
|
||
|
|
Check that solve/dot methods in Jacobian approximations are consistent
|
||
|
|
"""
|
||
|
|
|
||
|
|
def _func(self, x):
|
||
|
|
return x**2 - 1 + np.dot(self.A, x)
|
||
|
|
|
||
|
|
def _check_dot(self, jac_cls, complex=False, tol=1e-6, **kw):
|
||
|
|
np.random.seed(123)
|
||
|
|
|
||
|
|
N = 7
|
||
|
|
|
||
|
|
def rand(*a):
|
||
|
|
q = np.random.rand(*a)
|
||
|
|
if complex:
|
||
|
|
q = q + 1j*np.random.rand(*a)
|
||
|
|
return q
|
||
|
|
|
||
|
|
def assert_close(a, b, msg):
|
||
|
|
d = abs(a - b).max()
|
||
|
|
f = tol + abs(b).max()*tol
|
||
|
|
if d > f:
|
||
|
|
raise AssertionError(f'{msg}: err {d:g}')
|
||
|
|
|
||
|
|
self.A = rand(N, N)
|
||
|
|
|
||
|
|
# initialize
|
||
|
|
x0 = np.random.rand(N)
|
||
|
|
jac = jac_cls(**kw)
|
||
|
|
jac.setup(x0, self._func(x0), self._func)
|
||
|
|
|
||
|
|
# check consistency
|
||
|
|
for k in range(2*N):
|
||
|
|
v = rand(N)
|
||
|
|
|
||
|
|
if hasattr(jac, '__array__'):
|
||
|
|
Jd = np.array(jac)
|
||
|
|
if hasattr(jac, 'solve'):
|
||
|
|
Gv = jac.solve(v)
|
||
|
|
Gv2 = np.linalg.solve(Jd, v)
|
||
|
|
assert_close(Gv, Gv2, 'solve vs array')
|
||
|
|
if hasattr(jac, 'rsolve'):
|
||
|
|
Gv = jac.rsolve(v)
|
||
|
|
Gv2 = np.linalg.solve(Jd.T.conj(), v)
|
||
|
|
assert_close(Gv, Gv2, 'rsolve vs array')
|
||
|
|
if hasattr(jac, 'matvec'):
|
||
|
|
Jv = jac.matvec(v)
|
||
|
|
Jv2 = np.dot(Jd, v)
|
||
|
|
assert_close(Jv, Jv2, 'dot vs array')
|
||
|
|
if hasattr(jac, 'rmatvec'):
|
||
|
|
Jv = jac.rmatvec(v)
|
||
|
|
Jv2 = np.dot(Jd.T.conj(), v)
|
||
|
|
assert_close(Jv, Jv2, 'rmatvec vs array')
|
||
|
|
|
||
|
|
if hasattr(jac, 'matvec') and hasattr(jac, 'solve'):
|
||
|
|
Jv = jac.matvec(v)
|
||
|
|
Jv2 = jac.solve(jac.matvec(Jv))
|
||
|
|
assert_close(Jv, Jv2, 'dot vs solve')
|
||
|
|
|
||
|
|
if hasattr(jac, 'rmatvec') and hasattr(jac, 'rsolve'):
|
||
|
|
Jv = jac.rmatvec(v)
|
||
|
|
Jv2 = jac.rmatvec(jac.rsolve(Jv))
|
||
|
|
assert_close(Jv, Jv2, 'rmatvec vs rsolve')
|
||
|
|
|
||
|
|
x = rand(N)
|
||
|
|
jac.update(x, self._func(x))
|
||
|
|
|
||
|
|
def test_broyden1(self):
|
||
|
|
self._check_dot(nonlin.BroydenFirst, complex=False)
|
||
|
|
self._check_dot(nonlin.BroydenFirst, complex=True)
|
||
|
|
|
||
|
|
def test_broyden2(self):
|
||
|
|
self._check_dot(nonlin.BroydenSecond, complex=False)
|
||
|
|
self._check_dot(nonlin.BroydenSecond, complex=True)
|
||
|
|
|
||
|
|
def test_anderson(self):
|
||
|
|
self._check_dot(nonlin.Anderson, complex=False)
|
||
|
|
self._check_dot(nonlin.Anderson, complex=True)
|
||
|
|
|
||
|
|
def test_diagbroyden(self):
|
||
|
|
self._check_dot(nonlin.DiagBroyden, complex=False)
|
||
|
|
self._check_dot(nonlin.DiagBroyden, complex=True)
|
||
|
|
|
||
|
|
def test_linearmixing(self):
|
||
|
|
self._check_dot(nonlin.LinearMixing, complex=False)
|
||
|
|
self._check_dot(nonlin.LinearMixing, complex=True)
|
||
|
|
|
||
|
|
def test_excitingmixing(self):
|
||
|
|
self._check_dot(nonlin.ExcitingMixing, complex=False)
|
||
|
|
self._check_dot(nonlin.ExcitingMixing, complex=True)
|
||
|
|
|
||
|
|
def test_krylov(self):
|
||
|
|
self._check_dot(nonlin.KrylovJacobian, complex=False, tol=1e-3)
|
||
|
|
self._check_dot(nonlin.KrylovJacobian, complex=True, tol=1e-3)
|
||
|
|
|
||
|
|
|
||
|
|
class TestNonlinOldTests:
|
||
|
|
""" Test case for a simple constrained entropy maximization problem
|
||
|
|
(the machine translation example of Berger et al in
|
||
|
|
Computational Linguistics, vol 22, num 1, pp 39--72, 1996.)
|
||
|
|
"""
|
||
|
|
|
||
|
|
def test_broyden1(self):
|
||
|
|
x = nonlin.broyden1(F, F.xin, iter=12, alpha=1)
|
||
|
|
assert_(nonlin.norm(x) < 1e-9)
|
||
|
|
assert_(nonlin.norm(F(x)) < 1e-9)
|
||
|
|
|
||
|
|
def test_broyden2(self):
|
||
|
|
x = nonlin.broyden2(F, F.xin, iter=12, alpha=1)
|
||
|
|
assert_(nonlin.norm(x) < 1e-9)
|
||
|
|
assert_(nonlin.norm(F(x)) < 1e-9)
|
||
|
|
|
||
|
|
def test_anderson(self):
|
||
|
|
x = nonlin.anderson(F, F.xin, iter=12, alpha=0.03, M=5)
|
||
|
|
assert_(nonlin.norm(x) < 0.33)
|
||
|
|
|
||
|
|
def test_linearmixing(self):
|
||
|
|
x = nonlin.linearmixing(F, F.xin, iter=60, alpha=0.5)
|
||
|
|
assert_(nonlin.norm(x) < 1e-7)
|
||
|
|
assert_(nonlin.norm(F(x)) < 1e-7)
|
||
|
|
|
||
|
|
def test_exciting(self):
|
||
|
|
x = nonlin.excitingmixing(F, F.xin, iter=20, alpha=0.5)
|
||
|
|
assert_(nonlin.norm(x) < 1e-5)
|
||
|
|
assert_(nonlin.norm(F(x)) < 1e-5)
|
||
|
|
|
||
|
|
def test_diagbroyden(self):
|
||
|
|
x = nonlin.diagbroyden(F, F.xin, iter=11, alpha=1)
|
||
|
|
assert_(nonlin.norm(x) < 1e-8)
|
||
|
|
assert_(nonlin.norm(F(x)) < 1e-8)
|
||
|
|
|
||
|
|
def test_root_broyden1(self):
|
||
|
|
res = root(F, F.xin, method='broyden1',
|
||
|
|
options={'nit': 12, 'jac_options': {'alpha': 1}})
|
||
|
|
assert_(nonlin.norm(res.x) < 1e-9)
|
||
|
|
assert_(nonlin.norm(res.fun) < 1e-9)
|
||
|
|
|
||
|
|
def test_root_broyden2(self):
|
||
|
|
res = root(F, F.xin, method='broyden2',
|
||
|
|
options={'nit': 12, 'jac_options': {'alpha': 1}})
|
||
|
|
assert_(nonlin.norm(res.x) < 1e-9)
|
||
|
|
assert_(nonlin.norm(res.fun) < 1e-9)
|
||
|
|
|
||
|
|
def test_root_anderson(self):
|
||
|
|
res = root(F, F.xin, method='anderson',
|
||
|
|
options={'nit': 12,
|
||
|
|
'jac_options': {'alpha': 0.03, 'M': 5}})
|
||
|
|
assert_(nonlin.norm(res.x) < 0.33)
|
||
|
|
|
||
|
|
def test_root_linearmixing(self):
|
||
|
|
res = root(F, F.xin, method='linearmixing',
|
||
|
|
options={'nit': 60,
|
||
|
|
'jac_options': {'alpha': 0.5}})
|
||
|
|
assert_(nonlin.norm(res.x) < 1e-7)
|
||
|
|
assert_(nonlin.norm(res.fun) < 1e-7)
|
||
|
|
|
||
|
|
def test_root_excitingmixing(self):
|
||
|
|
res = root(F, F.xin, method='excitingmixing',
|
||
|
|
options={'nit': 20,
|
||
|
|
'jac_options': {'alpha': 0.5}})
|
||
|
|
assert_(nonlin.norm(res.x) < 1e-5)
|
||
|
|
assert_(nonlin.norm(res.fun) < 1e-5)
|
||
|
|
|
||
|
|
def test_root_diagbroyden(self):
|
||
|
|
res = root(F, F.xin, method='diagbroyden',
|
||
|
|
options={'nit': 11,
|
||
|
|
'jac_options': {'alpha': 1}})
|
||
|
|
assert_(nonlin.norm(res.x) < 1e-8)
|
||
|
|
assert_(nonlin.norm(res.fun) < 1e-8)
|