446 lines
15 KiB
Python
446 lines
15 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, root
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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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from .test_minpack import pressure_network
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SOLVERS = {'anderson': nonlin.anderson, 'diagbroyden': nonlin.diagbroyden,
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'linearmixing': nonlin.linearmixing, 'excitingmixing': nonlin.excitingmixing,
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'broyden1': nonlin.broyden1, '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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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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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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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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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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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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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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#-------------------------------------------------------------------------------
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# Tests
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#-------------------------------------------------------------------------------
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class TestNonlin(object):
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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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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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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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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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def test_tol_norm_called(self):
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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, f_tol=1e-2, 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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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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class TestSecant(object):
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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], float),
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np.array([2,3,4,5,1], float),
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np.array([3,4,5,1,2], float),
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np.array([4,5,1,2,3], float),
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np.array([9,1,9,1,3], float),
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np.array([0,1,9,1,3], float),
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np.array([5,5,7,1,1], float),
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np.array([1,2,7,5,1], float),]
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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(object):
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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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class TestJacobianDotSolve(object):
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"""Check that solve/dot methods in Jacobian approximations are consistent"""
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def _func(self, x):
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return x**2 - 1 + np.dot(self.A, x)
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def _check_dot(self, jac_cls, complex=False, tol=1e-6, **kw):
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np.random.seed(123)
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N = 7
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def rand(*a):
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q = np.random.rand(*a)
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if complex:
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q = q + 1j*np.random.rand(*a)
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return q
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def assert_close(a, b, msg):
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d = abs(a - b).max()
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f = tol + abs(b).max()*tol
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if d > f:
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raise AssertionError('%s: err %g' % (msg, d))
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self.A = rand(N, N)
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# initialize
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x0 = np.random.rand(N)
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jac = jac_cls(**kw)
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jac.setup(x0, self._func(x0), self._func)
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# check consistency
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for k in range(2*N):
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v = rand(N)
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if hasattr(jac, '__array__'):
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Jd = np.array(jac)
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if hasattr(jac, 'solve'):
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Gv = jac.solve(v)
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Gv2 = np.linalg.solve(Jd, v)
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assert_close(Gv, Gv2, 'solve vs array')
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if hasattr(jac, 'rsolve'):
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Gv = jac.rsolve(v)
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Gv2 = np.linalg.solve(Jd.T.conj(), v)
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assert_close(Gv, Gv2, 'rsolve vs array')
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if hasattr(jac, 'matvec'):
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Jv = jac.matvec(v)
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Jv2 = np.dot(Jd, v)
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assert_close(Jv, Jv2, 'dot vs array')
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if hasattr(jac, 'rmatvec'):
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Jv = jac.rmatvec(v)
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Jv2 = np.dot(Jd.T.conj(), v)
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assert_close(Jv, Jv2, 'rmatvec vs array')
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if hasattr(jac, 'matvec') and hasattr(jac, 'solve'):
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Jv = jac.matvec(v)
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Jv2 = jac.solve(jac.matvec(Jv))
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assert_close(Jv, Jv2, 'dot vs solve')
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if hasattr(jac, 'rmatvec') and hasattr(jac, 'rsolve'):
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Jv = jac.rmatvec(v)
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Jv2 = jac.rmatvec(jac.rsolve(Jv))
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assert_close(Jv, Jv2, 'rmatvec vs rsolve')
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x = rand(N)
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jac.update(x, self._func(x))
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def test_broyden1(self):
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self._check_dot(nonlin.BroydenFirst, complex=False)
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self._check_dot(nonlin.BroydenFirst, complex=True)
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def test_broyden2(self):
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self._check_dot(nonlin.BroydenSecond, complex=False)
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self._check_dot(nonlin.BroydenSecond, complex=True)
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def test_anderson(self):
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self._check_dot(nonlin.Anderson, complex=False)
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self._check_dot(nonlin.Anderson, complex=True)
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def test_diagbroyden(self):
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self._check_dot(nonlin.DiagBroyden, complex=False)
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self._check_dot(nonlin.DiagBroyden, complex=True)
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def test_linearmixing(self):
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self._check_dot(nonlin.LinearMixing, complex=False)
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self._check_dot(nonlin.LinearMixing, complex=True)
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def test_excitingmixing(self):
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self._check_dot(nonlin.ExcitingMixing, complex=False)
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self._check_dot(nonlin.ExcitingMixing, complex=True)
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def test_krylov(self):
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self._check_dot(nonlin.KrylovJacobian, complex=False, tol=1e-3)
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self._check_dot(nonlin.KrylovJacobian, complex=True, tol=1e-3)
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class TestNonlinOldTests(object):
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""" Test case for a simple constrained entropy maximization problem
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(the machine translation example of Berger et al in
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|
Computational Linguistics, vol 22, num 1, pp 39--72, 1996.)
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"""
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def test_broyden1(self):
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x = nonlin.broyden1(F,F.xin,iter=12,alpha=1)
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assert_(nonlin.norm(x) < 1e-9)
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assert_(nonlin.norm(F(x)) < 1e-9)
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|
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def test_broyden2(self):
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x = nonlin.broyden2(F,F.xin,iter=12,alpha=1)
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assert_(nonlin.norm(x) < 1e-9)
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assert_(nonlin.norm(F(x)) < 1e-9)
|
||
|
|
|
||
|
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def test_anderson(self):
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x = nonlin.anderson(F,F.xin,iter=12,alpha=0.03,M=5)
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assert_(nonlin.norm(x) < 0.33)
|
||
|
|
|
||
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def test_linearmixing(self):
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||
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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)
|