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Inzynierka/Lib/site-packages/scipy/optimize/tests/test_nonlin.py

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2023-06-02 12:51:02 +02:00
""" Unit tests for nonlinear solvers
Author: Ondrej Certik
May 2007
"""
from numpy.testing import assert_
import pytest
from scipy.optimize import _nonlin as nonlin, root
from numpy import diag, dot
from numpy.linalg import inv
import numpy as np
from .test_minpack import pressure_network
SOLVERS = {'anderson': nonlin.anderson, 'diagbroyden': nonlin.diagbroyden,
'linearmixing': nonlin.linearmixing, 'excitingmixing': nonlin.excitingmixing,
'broyden1': nonlin.broyden1, 'broyden2': nonlin.broyden2,
'krylov': nonlin.newton_krylov}
MUST_WORK = {'anderson': nonlin.anderson, 'broyden1': nonlin.broyden1,
'broyden2': nonlin.broyden2, 'krylov': nonlin.newton_krylov}
#-------------------------------------------------------------------------------
# Test problems
#-------------------------------------------------------------------------------
def F(x):
x = np.asarray(x).T
d = diag([3,2,1.5,1,0.5])
c = 0.01
f = -d @ x - c * float(x.T @ x) * x
return f
F.xin = [1,1,1,1,1]
F.KNOWN_BAD = {}
F.JAC_KSP_BAD = {}
F.ROOT_JAC_KSP_BAD = {}
def F2(x):
return x
F2.xin = [1,2,3,4,5,6]
F2.KNOWN_BAD = {'linearmixing': nonlin.linearmixing,
'excitingmixing': nonlin.excitingmixing}
F2.JAC_KSP_BAD = {}
F2.ROOT_JAC_KSP_BAD = {}
def F2_lucky(x):
return x
F2_lucky.xin = [0,0,0,0,0,0]
F2_lucky.KNOWN_BAD = {}
F2_lucky.JAC_KSP_BAD = {}
F2_lucky.ROOT_JAC_KSP_BAD = {}
def F3(x):
A = np.array([[-2, 1, 0.], [1, -2, 1], [0, 1, -2]])
b = np.array([1, 2, 3.])
return A @ x - b
F3.xin = [1,2,3]
F3.KNOWN_BAD = {}
F3.JAC_KSP_BAD = {}
F3.ROOT_JAC_KSP_BAD = {}
def F4_powell(x):
A = 1e4
return [A*x[0]*x[1] - 1, np.exp(-x[0]) + np.exp(-x[1]) - (1 + 1/A)]
F4_powell.xin = [-1, -2]
F4_powell.KNOWN_BAD = {'linearmixing': nonlin.linearmixing,
'excitingmixing': nonlin.excitingmixing,
'diagbroyden': nonlin.diagbroyden}
# In the extreme case, it does not converge for nolinear problem solved by
# MINRES and root problem solved by GMRES/BiCGStab/CGS/MINRES/TFQMR when using
# Krylov method to approximate Jacobian
F4_powell.JAC_KSP_BAD = {'minres'}
F4_powell.ROOT_JAC_KSP_BAD = {'gmres', 'bicgstab', 'cgs', 'minres', 'tfqmr'}
def F5(x):
return pressure_network(x, 4, np.array([.5, .5, .5, .5]))
F5.xin = [2., 0, 2, 0]
F5.KNOWN_BAD = {'excitingmixing': nonlin.excitingmixing,
'linearmixing': nonlin.linearmixing,
'diagbroyden': nonlin.diagbroyden}
# In the extreme case, the Jacobian inversion yielded zero vector for nonlinear
# problem solved by CGS/MINRES and it does not converge for root problem solved
# by MINRES and when using Krylov method to approximate Jacobian
F5.JAC_KSP_BAD = {'cgs', 'minres'}
F5.ROOT_JAC_KSP_BAD = {'minres'}
def F6(x):
x1, x2 = x
J0 = np.array([[-4.256, 14.7],
[0.8394989, 0.59964207]])
v = np.array([(x1 + 3) * (x2**5 - 7) + 3*6,
np.sin(x2 * np.exp(x1) - 1)])
return -np.linalg.solve(J0, v)
F6.xin = [-0.5, 1.4]
F6.KNOWN_BAD = {'excitingmixing': nonlin.excitingmixing,
'linearmixing': nonlin.linearmixing,
'diagbroyden': nonlin.diagbroyden}
F6.JAC_KSP_BAD = {}
F6.ROOT_JAC_KSP_BAD = {}
#-------------------------------------------------------------------------------
# Tests
#-------------------------------------------------------------------------------
class TestNonlin:
"""
Check the Broyden methods for a few test problems.
broyden1, broyden2, and newton_krylov must succeed for
all functions. Some of the others don't -- tests in KNOWN_BAD are skipped.
"""
def _check_nonlin_func(self, f, func, f_tol=1e-2):
# Test all methods mentioned in the class `KrylovJacobian`
if func == SOLVERS['krylov']:
for method in ['gmres', 'bicgstab', 'cgs', 'minres', 'tfqmr']:
if method in f.JAC_KSP_BAD:
continue
x = func(f, f.xin, method=method, line_search=None,
f_tol=f_tol, maxiter=200, verbose=0)
assert_(np.absolute(f(x)).max() < f_tol)
x = func(f, f.xin, f_tol=f_tol, maxiter=200, verbose=0)
assert_(np.absolute(f(x)).max() < f_tol)
def _check_root(self, f, method, f_tol=1e-2):
# Test Krylov methods
if method == 'krylov':
for jac_method in ['gmres', 'bicgstab', 'cgs', 'minres', 'tfqmr']:
if jac_method in f.ROOT_JAC_KSP_BAD:
continue
res = root(f, f.xin, method=method,
options={'ftol': f_tol, 'maxiter': 200,
'disp': 0,
'jac_options': {'method': jac_method}})
assert_(np.absolute(res.fun).max() < f_tol)
res = root(f, f.xin, method=method,
options={'ftol': f_tol, 'maxiter': 200, 'disp': 0})
assert_(np.absolute(res.fun).max() < f_tol)
@pytest.mark.xfail
def _check_func_fail(self, *a, **kw):
pass
def test_problem_nonlin(self):
for f in [F, F2, F2_lucky, F3, F4_powell, F5, F6]:
for func in SOLVERS.values():
if func in f.KNOWN_BAD.values():
if func in MUST_WORK.values():
self._check_func_fail(f, func)
continue
self._check_nonlin_func(f, func)
@pytest.mark.parametrize("method", ['lgmres', 'gmres', 'bicgstab', 'cgs',
'minres', 'tfqmr'])
def test_tol_norm_called(self, method):
# Check that supplying tol_norm keyword to nonlin_solve works
self._tol_norm_used = False
def local_norm_func(x):
self._tol_norm_used = True
return np.absolute(x).max()
nonlin.newton_krylov(F, F.xin, method=method, f_tol=1e-2,
maxiter=200, verbose=0,
tol_norm=local_norm_func)
assert_(self._tol_norm_used)
def test_problem_root(self):
for f in [F, F2, F2_lucky, F3, F4_powell, F5, F6]:
for meth in SOLVERS:
if meth in f.KNOWN_BAD:
if meth in MUST_WORK:
self._check_func_fail(f, meth)
continue
self._check_root(f, meth)
class TestSecant:
"""Check that some Jacobian approximations satisfy the secant condition"""
xs = [np.array([1,2,3,4,5], float),
np.array([2,3,4,5,1], float),
np.array([3,4,5,1,2], float),
np.array([4,5,1,2,3], float),
np.array([9,1,9,1,3], float),
np.array([0,1,9,1,3], float),
np.array([5,5,7,1,1], float),
np.array([1,2,7,5,1], float),]
fs = [x**2 - 1 for x in xs]
def _check_secant(self, jac_cls, npoints=1, **kw):
"""
Check that the given Jacobian approximation satisfies secant
conditions for last `npoints` points.
"""
jac = jac_cls(**kw)
jac.setup(self.xs[0], self.fs[0], None)
for j, (x, f) in enumerate(zip(self.xs[1:], self.fs[1:])):
jac.update(x, f)
for k in range(min(npoints, j+1)):
dx = self.xs[j-k+1] - self.xs[j-k]
df = self.fs[j-k+1] - self.fs[j-k]
assert_(np.allclose(dx, jac.solve(df)))
# Check that the `npoints` secant bound is strict
if j >= npoints:
dx = self.xs[j-npoints+1] - self.xs[j-npoints]
df = self.fs[j-npoints+1] - self.fs[j-npoints]
assert_(not np.allclose(dx, jac.solve(df)))
def test_broyden1(self):
self._check_secant(nonlin.BroydenFirst)
def test_broyden2(self):
self._check_secant(nonlin.BroydenSecond)
def test_broyden1_update(self):
# Check that BroydenFirst update works as for a dense matrix
jac = nonlin.BroydenFirst(alpha=0.1)
jac.setup(self.xs[0], self.fs[0], None)
B = np.identity(5) * (-1/0.1)
for last_j, (x, f) in enumerate(zip(self.xs[1:], self.fs[1:])):
df = f - self.fs[last_j]
dx = x - self.xs[last_j]
B += (df - dot(B, dx))[:,None] * dx[None,:] / dot(dx, dx)
jac.update(x, f)
assert_(np.allclose(jac.todense(), B, rtol=1e-10, atol=1e-13))
def test_broyden2_update(self):
# Check that BroydenSecond update works as for a dense matrix
jac = nonlin.BroydenSecond(alpha=0.1)
jac.setup(self.xs[0], self.fs[0], None)
H = np.identity(5) * (-0.1)
for last_j, (x, f) in enumerate(zip(self.xs[1:], self.fs[1:])):
df = f - self.fs[last_j]
dx = x - self.xs[last_j]
H += (dx - dot(H, df))[:,None] * df[None,:] / dot(df, df)
jac.update(x, f)
assert_(np.allclose(jac.todense(), inv(H), rtol=1e-10, atol=1e-13))
def test_anderson(self):
# Anderson mixing (with w0=0) satisfies secant conditions
# for the last M iterates, see [Ey]_
#
# .. [Ey] V. Eyert, J. Comp. Phys., 124, 271 (1996).
self._check_secant(nonlin.Anderson, M=3, w0=0, npoints=3)
class TestLinear:
"""Solve a linear equation;
some methods find the exact solution in a finite number of steps"""
def _check(self, jac, N, maxiter, complex=False, **kw):
np.random.seed(123)
A = np.random.randn(N, N)
if complex:
A = A + 1j*np.random.randn(N, N)
b = np.random.randn(N)
if complex:
b = b + 1j*np.random.randn(N)
def func(x):
return dot(A, x) - b
sol = nonlin.nonlin_solve(func, np.zeros(N), jac, maxiter=maxiter,
f_tol=1e-6, line_search=None, verbose=0)
assert_(np.allclose(dot(A, sol), b, atol=1e-6))
def test_broyden1(self):
# Broyden methods solve linear systems exactly in 2*N steps
self._check(nonlin.BroydenFirst(alpha=1.0), 20, 41, False)
self._check(nonlin.BroydenFirst(alpha=1.0), 20, 41, True)
def test_broyden2(self):
# Broyden methods solve linear systems exactly in 2*N steps
self._check(nonlin.BroydenSecond(alpha=1.0), 20, 41, False)
self._check(nonlin.BroydenSecond(alpha=1.0), 20, 41, True)
def test_anderson(self):
# Anderson is rather similar to Broyden, if given enough storage space
self._check(nonlin.Anderson(M=50, alpha=1.0), 20, 29, False)
self._check(nonlin.Anderson(M=50, alpha=1.0), 20, 29, True)
def test_krylov(self):
# Krylov methods solve linear systems exactly in N inner steps
self._check(nonlin.KrylovJacobian, 20, 2, False, inner_m=10)
self._check(nonlin.KrylovJacobian, 20, 2, True, inner_m=10)
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('%s: err %g' % (msg, d))
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)
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