756 lines
27 KiB
Python
756 lines
27 KiB
Python
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import pytest
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from math import sqrt, exp, sin, cos
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from functools import lru_cache
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from numpy.testing import (assert_warns, assert_,
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assert_allclose,
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assert_equal,
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assert_array_equal,
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suppress_warnings)
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import numpy as np
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from numpy import finfo, power, nan, isclose
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from scipy.optimize import zeros, newton, root_scalar
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from scipy._lib._util import getfullargspec_no_self as _getfullargspec
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# Import testing parameters
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from scipy.optimize._tstutils import get_tests, functions as tstutils_functions, fstrings as tstutils_fstrings
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TOL = 4*np.finfo(float).eps # tolerance
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_FLOAT_EPS = finfo(float).eps
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# A few test functions used frequently:
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# # A simple quadratic, (x-1)^2 - 1
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def f1(x):
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return x ** 2 - 2 * x - 1
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def f1_1(x):
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return 2 * x - 2
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def f1_2(x):
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return 2.0 + 0 * x
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def f1_and_p_and_pp(x):
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return f1(x), f1_1(x), f1_2(x)
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# Simple transcendental function
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def f2(x):
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return exp(x) - cos(x)
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def f2_1(x):
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return exp(x) + sin(x)
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def f2_2(x):
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return exp(x) + cos(x)
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# lru cached function
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@lru_cache()
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def f_lrucached(x):
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return x
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class TestBasic(object):
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def run_check_by_name(self, name, smoothness=0, **kwargs):
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a = .5
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b = sqrt(3)
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xtol = 4*np.finfo(float).eps
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rtol = 4*np.finfo(float).eps
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for function, fname in zip(tstutils_functions, tstutils_fstrings):
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if smoothness > 0 and fname in ['f4', 'f5', 'f6']:
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continue
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r = root_scalar(function, method=name, bracket=[a, b], x0=a,
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xtol=xtol, rtol=rtol, **kwargs)
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zero = r.root
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assert_(r.converged)
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assert_allclose(zero, 1.0, atol=xtol, rtol=rtol,
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err_msg='method %s, function %s' % (name, fname))
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def run_check(self, method, name):
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a = .5
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b = sqrt(3)
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xtol = 4 * _FLOAT_EPS
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rtol = 4 * _FLOAT_EPS
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for function, fname in zip(tstutils_functions, tstutils_fstrings):
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zero, r = method(function, a, b, xtol=xtol, rtol=rtol,
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full_output=True)
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assert_(r.converged)
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assert_allclose(zero, 1.0, atol=xtol, rtol=rtol,
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err_msg='method %s, function %s' % (name, fname))
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def run_check_lru_cached(self, method, name):
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# check that https://github.com/scipy/scipy/issues/10846 is fixed
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a = -1
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b = 1
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zero, r = method(f_lrucached, a, b, full_output=True)
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assert_(r.converged)
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assert_allclose(zero, 0,
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err_msg='method %s, function %s' % (name, 'f_lrucached'))
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def _run_one_test(self, tc, method, sig_args_keys=None,
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sig_kwargs_keys=None, **kwargs):
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method_args = []
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for k in sig_args_keys or []:
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if k not in tc:
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# If a,b not present use x0, x1. Similarly for f and func
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k = {'a': 'x0', 'b': 'x1', 'func': 'f'}.get(k, k)
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method_args.append(tc[k])
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method_kwargs = dict(**kwargs)
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method_kwargs.update({'full_output': True, 'disp': False})
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for k in sig_kwargs_keys or []:
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method_kwargs[k] = tc[k]
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root = tc.get('root')
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func_args = tc.get('args', ())
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try:
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r, rr = method(*method_args, args=func_args, **method_kwargs)
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return root, rr, tc
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except Exception:
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return root, zeros.RootResults(nan, -1, -1, zeros._EVALUEERR), tc
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def run_tests(self, tests, method, name,
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xtol=4 * _FLOAT_EPS, rtol=4 * _FLOAT_EPS,
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known_fail=None, **kwargs):
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r"""Run test-cases using the specified method and the supplied signature.
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Extract the arguments for the method call from the test case
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dictionary using the supplied keys for the method's signature."""
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# The methods have one of two base signatures:
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# (f, a, b, **kwargs) # newton
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# (func, x0, **kwargs) # bisect/brentq/...
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sig = _getfullargspec(method) # FullArgSpec with args, varargs, varkw, defaults, ...
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assert_(not sig.kwonlyargs)
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nDefaults = len(sig.defaults)
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nRequired = len(sig.args) - nDefaults
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sig_args_keys = sig.args[:nRequired]
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sig_kwargs_keys = []
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if name in ['secant', 'newton', 'halley']:
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if name in ['newton', 'halley']:
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sig_kwargs_keys.append('fprime')
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if name in ['halley']:
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sig_kwargs_keys.append('fprime2')
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kwargs['tol'] = xtol
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else:
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kwargs['xtol'] = xtol
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kwargs['rtol'] = rtol
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results = [list(self._run_one_test(
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tc, method, sig_args_keys=sig_args_keys,
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sig_kwargs_keys=sig_kwargs_keys, **kwargs)) for tc in tests]
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# results= [[true root, full output, tc], ...]
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known_fail = known_fail or []
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notcvgd = [elt for elt in results if not elt[1].converged]
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notcvgd = [elt for elt in notcvgd if elt[-1]['ID'] not in known_fail]
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notcvged_IDS = [elt[-1]['ID'] for elt in notcvgd]
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assert_equal([len(notcvged_IDS), notcvged_IDS], [0, []])
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# The usable xtol and rtol depend on the test
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tols = {'xtol': 4 * _FLOAT_EPS, 'rtol': 4 * _FLOAT_EPS}
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tols.update(**kwargs)
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rtol = tols['rtol']
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atol = tols.get('tol', tols['xtol'])
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cvgd = [elt for elt in results if elt[1].converged]
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approx = [elt[1].root for elt in cvgd]
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correct = [elt[0] for elt in cvgd]
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notclose = [[a] + elt for a, c, elt in zip(approx, correct, cvgd) if
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not isclose(a, c, rtol=rtol, atol=atol)
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and elt[-1]['ID'] not in known_fail]
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# Evaluate the function and see if is 0 at the purported root
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fvs = [tc['f'](aroot, *(tc['args'])) for aroot, c, fullout, tc in notclose]
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notclose = [[fv] + elt for fv, elt in zip(fvs, notclose) if fv != 0]
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assert_equal([notclose, len(notclose)], [[], 0])
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def run_collection(self, collection, method, name, smoothness=None,
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known_fail=None,
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xtol=4 * _FLOAT_EPS, rtol=4 * _FLOAT_EPS,
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**kwargs):
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r"""Run a collection of tests using the specified method.
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The name is used to determine some optional arguments."""
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tests = get_tests(collection, smoothness=smoothness)
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self.run_tests(tests, method, name, xtol=xtol, rtol=rtol,
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known_fail=known_fail, **kwargs)
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def test_bisect(self):
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self.run_check(zeros.bisect, 'bisect')
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self.run_check_lru_cached(zeros.bisect, 'bisect')
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self.run_check_by_name('bisect')
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self.run_collection('aps', zeros.bisect, 'bisect', smoothness=1)
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def test_ridder(self):
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self.run_check(zeros.ridder, 'ridder')
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self.run_check_lru_cached(zeros.ridder, 'ridder')
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self.run_check_by_name('ridder')
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self.run_collection('aps', zeros.ridder, 'ridder', smoothness=1)
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def test_brentq(self):
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self.run_check(zeros.brentq, 'brentq')
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self.run_check_lru_cached(zeros.brentq, 'brentq')
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self.run_check_by_name('brentq')
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# Brentq/h needs a lower tolerance to be specified
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self.run_collection('aps', zeros.brentq, 'brentq', smoothness=1,
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xtol=1e-14, rtol=1e-14)
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def test_brenth(self):
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self.run_check(zeros.brenth, 'brenth')
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self.run_check_lru_cached(zeros.brenth, 'brenth')
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self.run_check_by_name('brenth')
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self.run_collection('aps', zeros.brenth, 'brenth', smoothness=1,
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xtol=1e-14, rtol=1e-14)
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def test_toms748(self):
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self.run_check(zeros.toms748, 'toms748')
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self.run_check_lru_cached(zeros.toms748, 'toms748')
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self.run_check_by_name('toms748')
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self.run_collection('aps', zeros.toms748, 'toms748', smoothness=1)
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def test_newton_collections(self):
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known_fail = ['aps.13.00']
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known_fail += ['aps.12.05', 'aps.12.17'] # fails under Windows Py27
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for collection in ['aps', 'complex']:
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self.run_collection(collection, zeros.newton, 'newton',
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smoothness=2, known_fail=known_fail)
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def test_halley_collections(self):
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known_fail = ['aps.12.06', 'aps.12.07', 'aps.12.08', 'aps.12.09',
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'aps.12.10', 'aps.12.11', 'aps.12.12', 'aps.12.13',
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'aps.12.14', 'aps.12.15', 'aps.12.16', 'aps.12.17',
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'aps.12.18', 'aps.13.00']
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for collection in ['aps', 'complex']:
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self.run_collection(collection, zeros.newton, 'halley',
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smoothness=2, known_fail=known_fail)
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@staticmethod
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def f1(x):
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return x**2 - 2*x - 1 # == (x-1)**2 - 2
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@staticmethod
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def f1_1(x):
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return 2*x - 2
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@staticmethod
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def f1_2(x):
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return 2.0 + 0*x
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@staticmethod
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def f2(x):
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return exp(x) - cos(x)
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@staticmethod
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def f2_1(x):
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return exp(x) + sin(x)
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@staticmethod
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def f2_2(x):
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return exp(x) + cos(x)
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def test_newton(self):
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for f, f_1, f_2 in [(self.f1, self.f1_1, self.f1_2),
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(self.f2, self.f2_1, self.f2_2)]:
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x = zeros.newton(f, 3, tol=1e-6)
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assert_allclose(f(x), 0, atol=1e-6)
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x = zeros.newton(f, 3, x1=5, tol=1e-6) # secant, x0 and x1
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assert_allclose(f(x), 0, atol=1e-6)
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x = zeros.newton(f, 3, fprime=f_1, tol=1e-6) # newton
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assert_allclose(f(x), 0, atol=1e-6)
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x = zeros.newton(f, 3, fprime=f_1, fprime2=f_2, tol=1e-6) # halley
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assert_allclose(f(x), 0, atol=1e-6)
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def test_newton_by_name(self):
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r"""Invoke newton through root_scalar()"""
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for f, f_1, f_2 in [(f1, f1_1, f1_2), (f2, f2_1, f2_2)]:
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r = root_scalar(f, method='newton', x0=3, fprime=f_1, xtol=1e-6)
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assert_allclose(f(r.root), 0, atol=1e-6)
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def test_secant_by_name(self):
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r"""Invoke secant through root_scalar()"""
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for f, f_1, f_2 in [(f1, f1_1, f1_2), (f2, f2_1, f2_2)]:
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r = root_scalar(f, method='secant', x0=3, x1=2, xtol=1e-6)
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assert_allclose(f(r.root), 0, atol=1e-6)
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r = root_scalar(f, method='secant', x0=3, x1=5, xtol=1e-6)
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assert_allclose(f(r.root), 0, atol=1e-6)
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def test_halley_by_name(self):
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r"""Invoke halley through root_scalar()"""
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for f, f_1, f_2 in [(f1, f1_1, f1_2), (f2, f2_1, f2_2)]:
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r = root_scalar(f, method='halley', x0=3,
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fprime=f_1, fprime2=f_2, xtol=1e-6)
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assert_allclose(f(r.root), 0, atol=1e-6)
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def test_root_scalar_fail(self):
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with pytest.raises(ValueError):
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root_scalar(f1, method='secant', x0=3, xtol=1e-6) # no x1
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with pytest.raises(ValueError):
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root_scalar(f1, method='newton', x0=3, xtol=1e-6) # no fprime
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with pytest.raises(ValueError):
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root_scalar(f1, method='halley', fprime=f1_1, x0=3, xtol=1e-6) # no fprime2
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with pytest.raises(ValueError):
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root_scalar(f1, method='halley', fprime2=f1_2, x0=3, xtol=1e-6) # no fprime
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def test_array_newton(self):
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"""test newton with array"""
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def f1(x, *a):
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b = a[0] + x * a[3]
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return a[1] - a[2] * (np.exp(b / a[5]) - 1.0) - b / a[4] - x
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def f1_1(x, *a):
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b = a[3] / a[5]
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return -a[2] * np.exp(a[0] / a[5] + x * b) * b - a[3] / a[4] - 1
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def f1_2(x, *a):
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b = a[3] / a[5]
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return -a[2] * np.exp(a[0] / a[5] + x * b) * b**2
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a0 = np.array([
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5.32725221, 5.48673747, 5.49539973,
|
||
|
|
5.36387202, 4.80237316, 1.43764452,
|
||
|
|
5.23063958, 5.46094772, 5.50512718,
|
||
|
|
5.42046290
|
||
|
|
])
|
||
|
|
a1 = (np.sin(range(10)) + 1.0) * 7.0
|
||
|
|
args = (a0, a1, 1e-09, 0.004, 10, 0.27456)
|
||
|
|
x0 = [7.0] * 10
|
||
|
|
x = zeros.newton(f1, x0, f1_1, args)
|
||
|
|
x_expected = (
|
||
|
|
6.17264965, 11.7702805, 12.2219954,
|
||
|
|
7.11017681, 1.18151293, 0.143707955,
|
||
|
|
4.31928228, 10.5419107, 12.7552490,
|
||
|
|
8.91225749
|
||
|
|
)
|
||
|
|
assert_allclose(x, x_expected)
|
||
|
|
# test halley's
|
||
|
|
x = zeros.newton(f1, x0, f1_1, args, fprime2=f1_2)
|
||
|
|
assert_allclose(x, x_expected)
|
||
|
|
# test secant
|
||
|
|
x = zeros.newton(f1, x0, args=args)
|
||
|
|
assert_allclose(x, x_expected)
|
||
|
|
|
||
|
|
def test_array_newton_complex(self):
|
||
|
|
def f(x):
|
||
|
|
return x + 1+1j
|
||
|
|
|
||
|
|
def fprime(x):
|
||
|
|
return 1.0
|
||
|
|
|
||
|
|
t = np.full(4, 1j)
|
||
|
|
x = zeros.newton(f, t, fprime=fprime)
|
||
|
|
assert_allclose(f(x), 0.)
|
||
|
|
|
||
|
|
# should work even if x0 is not complex
|
||
|
|
t = np.ones(4)
|
||
|
|
x = zeros.newton(f, t, fprime=fprime)
|
||
|
|
assert_allclose(f(x), 0.)
|
||
|
|
|
||
|
|
x = zeros.newton(f, t)
|
||
|
|
assert_allclose(f(x), 0.)
|
||
|
|
|
||
|
|
def test_array_secant_active_zero_der(self):
|
||
|
|
"""test secant doesn't continue to iterate zero derivatives"""
|
||
|
|
x = zeros.newton(lambda x, *a: x*x - a[0], x0=[4.123, 5],
|
||
|
|
args=[np.array([17, 25])])
|
||
|
|
assert_allclose(x, (4.123105625617661, 5.0))
|
||
|
|
|
||
|
|
def test_array_newton_integers(self):
|
||
|
|
# test secant with float
|
||
|
|
x = zeros.newton(lambda y, z: z - y ** 2, [4.0] * 2,
|
||
|
|
args=([15.0, 17.0],))
|
||
|
|
assert_allclose(x, (3.872983346207417, 4.123105625617661))
|
||
|
|
# test integer becomes float
|
||
|
|
x = zeros.newton(lambda y, z: z - y ** 2, [4] * 2, args=([15, 17],))
|
||
|
|
assert_allclose(x, (3.872983346207417, 4.123105625617661))
|
||
|
|
|
||
|
|
def test_array_newton_zero_der_failures(self):
|
||
|
|
# test derivative zero warning
|
||
|
|
assert_warns(RuntimeWarning, zeros.newton,
|
||
|
|
lambda y: y**2 - 2, [0., 0.], lambda y: 2 * y)
|
||
|
|
# test failures and zero_der
|
||
|
|
with pytest.warns(RuntimeWarning):
|
||
|
|
results = zeros.newton(lambda y: y**2 - 2, [0., 0.],
|
||
|
|
lambda y: 2*y, full_output=True)
|
||
|
|
assert_allclose(results.root, 0)
|
||
|
|
assert results.zero_der.all()
|
||
|
|
assert not results.converged.any()
|
||
|
|
|
||
|
|
def test_newton_combined(self):
|
||
|
|
f1 = lambda x: x**2 - 2*x - 1
|
||
|
|
f1_1 = lambda x: 2*x - 2
|
||
|
|
f1_2 = lambda x: 2.0 + 0*x
|
||
|
|
|
||
|
|
def f1_and_p_and_pp(x):
|
||
|
|
return x**2 - 2*x-1, 2*x-2, 2.0
|
||
|
|
|
||
|
|
sol0 = root_scalar(f1, method='newton', x0=3, fprime=f1_1)
|
||
|
|
sol = root_scalar(f1_and_p_and_pp, method='newton', x0=3, fprime=True)
|
||
|
|
assert_allclose(sol0.root, sol.root, atol=1e-8)
|
||
|
|
assert_equal(2*sol.function_calls, sol0.function_calls)
|
||
|
|
|
||
|
|
sol0 = root_scalar(f1, method='halley', x0=3, fprime=f1_1, fprime2=f1_2)
|
||
|
|
sol = root_scalar(f1_and_p_and_pp, method='halley', x0=3, fprime2=True)
|
||
|
|
assert_allclose(sol0.root, sol.root, atol=1e-8)
|
||
|
|
assert_equal(3*sol.function_calls, sol0.function_calls)
|
||
|
|
|
||
|
|
def test_newton_full_output(self):
|
||
|
|
# Test the full_output capability, both when converging and not.
|
||
|
|
# Use simple polynomials, to avoid hitting platform dependencies
|
||
|
|
# (e.g., exp & trig) in number of iterations
|
||
|
|
|
||
|
|
x0 = 3
|
||
|
|
expected_counts = [(6, 7), (5, 10), (3, 9)]
|
||
|
|
|
||
|
|
for derivs in range(3):
|
||
|
|
kwargs = {'tol': 1e-6, 'full_output': True, }
|
||
|
|
for k, v in [['fprime', self.f1_1], ['fprime2', self.f1_2]][:derivs]:
|
||
|
|
kwargs[k] = v
|
||
|
|
|
||
|
|
x, r = zeros.newton(self.f1, x0, disp=False, **kwargs)
|
||
|
|
assert_(r.converged)
|
||
|
|
assert_equal(x, r.root)
|
||
|
|
assert_equal((r.iterations, r.function_calls), expected_counts[derivs])
|
||
|
|
if derivs == 0:
|
||
|
|
assert(r.function_calls <= r.iterations + 1)
|
||
|
|
else:
|
||
|
|
assert_equal(r.function_calls, (derivs + 1) * r.iterations)
|
||
|
|
|
||
|
|
# Now repeat, allowing one fewer iteration to force convergence failure
|
||
|
|
iters = r.iterations - 1
|
||
|
|
x, r = zeros.newton(self.f1, x0, maxiter=iters, disp=False, **kwargs)
|
||
|
|
assert_(not r.converged)
|
||
|
|
assert_equal(x, r.root)
|
||
|
|
assert_equal(r.iterations, iters)
|
||
|
|
|
||
|
|
if derivs == 1:
|
||
|
|
# Check that the correct Exception is raised and
|
||
|
|
# validate the start of the message.
|
||
|
|
with pytest.raises(
|
||
|
|
RuntimeError,
|
||
|
|
match='Failed to converge after %d iterations, value is .*' % (iters)):
|
||
|
|
x, r = zeros.newton(self.f1, x0, maxiter=iters, disp=True, **kwargs)
|
||
|
|
|
||
|
|
def test_deriv_zero_warning(self):
|
||
|
|
func = lambda x: x**2 - 2.0
|
||
|
|
dfunc = lambda x: 2*x
|
||
|
|
assert_warns(RuntimeWarning, zeros.newton, func, 0.0, dfunc, disp=False)
|
||
|
|
with pytest.raises(RuntimeError, match='Derivative was zero'):
|
||
|
|
zeros.newton(func, 0.0, dfunc)
|
||
|
|
|
||
|
|
def test_newton_does_not_modify_x0(self):
|
||
|
|
# https://github.com/scipy/scipy/issues/9964
|
||
|
|
x0 = np.array([0.1, 3])
|
||
|
|
x0_copy = x0.copy() # Copy to test for equality.
|
||
|
|
newton(np.sin, x0, np.cos)
|
||
|
|
assert_array_equal(x0, x0_copy)
|
||
|
|
|
||
|
|
def test_maxiter_int_check(self):
|
||
|
|
for method in [zeros.bisect, zeros.newton, zeros.ridder, zeros.brentq,
|
||
|
|
zeros.brenth, zeros.toms748]:
|
||
|
|
with pytest.raises(TypeError,
|
||
|
|
match="'float' object cannot be interpreted as an integer"):
|
||
|
|
method(f1, 0.0, 1.0, maxiter=72.45)
|
||
|
|
|
||
|
|
|
||
|
|
def test_gh_5555():
|
||
|
|
root = 0.1
|
||
|
|
|
||
|
|
def f(x):
|
||
|
|
return x - root
|
||
|
|
|
||
|
|
methods = [zeros.bisect, zeros.ridder]
|
||
|
|
xtol = rtol = TOL
|
||
|
|
for method in methods:
|
||
|
|
res = method(f, -1e8, 1e7, xtol=xtol, rtol=rtol)
|
||
|
|
assert_allclose(root, res, atol=xtol, rtol=rtol,
|
||
|
|
err_msg='method %s' % method.__name__)
|
||
|
|
|
||
|
|
|
||
|
|
def test_gh_5557():
|
||
|
|
# Show that without the changes in 5557 brentq and brenth might
|
||
|
|
# only achieve a tolerance of 2*(xtol + rtol*|res|).
|
||
|
|
|
||
|
|
# f linearly interpolates (0, -0.1), (0.5, -0.1), and (1,
|
||
|
|
# 0.4). The important parts are that |f(0)| < |f(1)| (so that
|
||
|
|
# brent takes 0 as the initial guess), |f(0)| < atol (so that
|
||
|
|
# brent accepts 0 as the root), and that the exact root of f lies
|
||
|
|
# more than atol away from 0 (so that brent doesn't achieve the
|
||
|
|
# desired tolerance).
|
||
|
|
def f(x):
|
||
|
|
if x < 0.5:
|
||
|
|
return -0.1
|
||
|
|
else:
|
||
|
|
return x - 0.6
|
||
|
|
|
||
|
|
atol = 0.51
|
||
|
|
rtol = 4 * _FLOAT_EPS
|
||
|
|
methods = [zeros.brentq, zeros.brenth]
|
||
|
|
for method in methods:
|
||
|
|
res = method(f, 0, 1, xtol=atol, rtol=rtol)
|
||
|
|
assert_allclose(0.6, res, atol=atol, rtol=rtol)
|
||
|
|
|
||
|
|
|
||
|
|
class TestRootResults:
|
||
|
|
def test_repr(self):
|
||
|
|
r = zeros.RootResults(root=1.0,
|
||
|
|
iterations=44,
|
||
|
|
function_calls=46,
|
||
|
|
flag=0)
|
||
|
|
expected_repr = (" converged: True\n flag: 'converged'"
|
||
|
|
"\n function_calls: 46\n iterations: 44\n"
|
||
|
|
" root: 1.0")
|
||
|
|
assert_equal(repr(r), expected_repr)
|
||
|
|
|
||
|
|
|
||
|
|
def test_complex_halley():
|
||
|
|
"""Test Halley's works with complex roots"""
|
||
|
|
def f(x, *a):
|
||
|
|
return a[0] * x**2 + a[1] * x + a[2]
|
||
|
|
|
||
|
|
def f_1(x, *a):
|
||
|
|
return 2 * a[0] * x + a[1]
|
||
|
|
|
||
|
|
def f_2(x, *a):
|
||
|
|
retval = 2 * a[0]
|
||
|
|
try:
|
||
|
|
size = len(x)
|
||
|
|
except TypeError:
|
||
|
|
return retval
|
||
|
|
else:
|
||
|
|
return [retval] * size
|
||
|
|
|
||
|
|
z = complex(1.0, 2.0)
|
||
|
|
coeffs = (2.0, 3.0, 4.0)
|
||
|
|
y = zeros.newton(f, z, args=coeffs, fprime=f_1, fprime2=f_2, tol=1e-6)
|
||
|
|
# (-0.75000000000000078+1.1989578808281789j)
|
||
|
|
assert_allclose(f(y, *coeffs), 0, atol=1e-6)
|
||
|
|
z = [z] * 10
|
||
|
|
coeffs = (2.0, 3.0, 4.0)
|
||
|
|
y = zeros.newton(f, z, args=coeffs, fprime=f_1, fprime2=f_2, tol=1e-6)
|
||
|
|
assert_allclose(f(y, *coeffs), 0, atol=1e-6)
|
||
|
|
|
||
|
|
|
||
|
|
def test_zero_der_nz_dp():
|
||
|
|
"""Test secant method with a non-zero dp, but an infinite newton step"""
|
||
|
|
# pick a symmetrical functions and choose a point on the side that with dx
|
||
|
|
# makes a secant that is a flat line with zero slope, EG: f = (x - 100)**2,
|
||
|
|
# which has a root at x = 100 and is symmetrical around the line x = 100
|
||
|
|
# we have to pick a really big number so that it is consistently true
|
||
|
|
# now find a point on each side so that the secant has a zero slope
|
||
|
|
dx = np.finfo(float).eps ** 0.33
|
||
|
|
# 100 - p0 = p1 - 100 = p0 * (1 + dx) + dx - 100
|
||
|
|
# -> 200 = p0 * (2 + dx) + dx
|
||
|
|
p0 = (200.0 - dx) / (2.0 + dx)
|
||
|
|
with suppress_warnings() as sup:
|
||
|
|
sup.filter(RuntimeWarning, "RMS of")
|
||
|
|
x = zeros.newton(lambda y: (y - 100.0)**2, x0=[p0] * 10)
|
||
|
|
assert_allclose(x, [100] * 10)
|
||
|
|
# test scalar cases too
|
||
|
|
p0 = (2.0 - 1e-4) / (2.0 + 1e-4)
|
||
|
|
with suppress_warnings() as sup:
|
||
|
|
sup.filter(RuntimeWarning, "Tolerance of")
|
||
|
|
x = zeros.newton(lambda y: (y - 1.0) ** 2, x0=p0, disp=False)
|
||
|
|
assert_allclose(x, 1)
|
||
|
|
with pytest.raises(RuntimeError, match='Tolerance of'):
|
||
|
|
x = zeros.newton(lambda y: (y - 1.0) ** 2, x0=p0, disp=True)
|
||
|
|
p0 = (-2.0 + 1e-4) / (2.0 + 1e-4)
|
||
|
|
with suppress_warnings() as sup:
|
||
|
|
sup.filter(RuntimeWarning, "Tolerance of")
|
||
|
|
x = zeros.newton(lambda y: (y + 1.0) ** 2, x0=p0, disp=False)
|
||
|
|
assert_allclose(x, -1)
|
||
|
|
with pytest.raises(RuntimeError, match='Tolerance of'):
|
||
|
|
x = zeros.newton(lambda y: (y + 1.0) ** 2, x0=p0, disp=True)
|
||
|
|
|
||
|
|
|
||
|
|
def test_array_newton_failures():
|
||
|
|
"""Test that array newton fails as expected"""
|
||
|
|
# p = 0.68 # [MPa]
|
||
|
|
# dp = -0.068 * 1e6 # [Pa]
|
||
|
|
# T = 323 # [K]
|
||
|
|
diameter = 0.10 # [m]
|
||
|
|
# L = 100 # [m]
|
||
|
|
roughness = 0.00015 # [m]
|
||
|
|
rho = 988.1 # [kg/m**3]
|
||
|
|
mu = 5.4790e-04 # [Pa*s]
|
||
|
|
u = 2.488 # [m/s]
|
||
|
|
reynolds_number = rho * u * diameter / mu # Reynolds number
|
||
|
|
|
||
|
|
def colebrook_eqn(darcy_friction, re, dia):
|
||
|
|
return (1 / np.sqrt(darcy_friction) +
|
||
|
|
2 * np.log10(roughness / 3.7 / dia +
|
||
|
|
2.51 / re / np.sqrt(darcy_friction)))
|
||
|
|
|
||
|
|
# only some failures
|
||
|
|
with pytest.warns(RuntimeWarning):
|
||
|
|
result = zeros.newton(
|
||
|
|
colebrook_eqn, x0=[0.01, 0.2, 0.02223, 0.3], maxiter=2,
|
||
|
|
args=[reynolds_number, diameter], full_output=True
|
||
|
|
)
|
||
|
|
assert not result.converged.all()
|
||
|
|
# they all fail
|
||
|
|
with pytest.raises(RuntimeError):
|
||
|
|
result = zeros.newton(
|
||
|
|
colebrook_eqn, x0=[0.01] * 2, maxiter=2,
|
||
|
|
args=[reynolds_number, diameter], full_output=True
|
||
|
|
)
|
||
|
|
|
||
|
|
|
||
|
|
# this test should **not** raise a RuntimeWarning
|
||
|
|
def test_gh8904_zeroder_at_root_fails():
|
||
|
|
"""Test that Newton or Halley don't warn if zero derivative at root"""
|
||
|
|
|
||
|
|
# a function that has a zero derivative at it's root
|
||
|
|
def f_zeroder_root(x):
|
||
|
|
return x**3 - x**2
|
||
|
|
|
||
|
|
# should work with secant
|
||
|
|
r = zeros.newton(f_zeroder_root, x0=0)
|
||
|
|
assert_allclose(r, 0, atol=zeros._xtol, rtol=zeros._rtol)
|
||
|
|
# test again with array
|
||
|
|
r = zeros.newton(f_zeroder_root, x0=[0]*10)
|
||
|
|
assert_allclose(r, 0, atol=zeros._xtol, rtol=zeros._rtol)
|
||
|
|
|
||
|
|
# 1st derivative
|
||
|
|
def fder(x):
|
||
|
|
return 3 * x**2 - 2 * x
|
||
|
|
|
||
|
|
# 2nd derivative
|
||
|
|
def fder2(x):
|
||
|
|
return 6*x - 2
|
||
|
|
|
||
|
|
# should work with newton and halley
|
||
|
|
r = zeros.newton(f_zeroder_root, x0=0, fprime=fder)
|
||
|
|
assert_allclose(r, 0, atol=zeros._xtol, rtol=zeros._rtol)
|
||
|
|
r = zeros.newton(f_zeroder_root, x0=0, fprime=fder,
|
||
|
|
fprime2=fder2)
|
||
|
|
assert_allclose(r, 0, atol=zeros._xtol, rtol=zeros._rtol)
|
||
|
|
# test again with array
|
||
|
|
r = zeros.newton(f_zeroder_root, x0=[0]*10, fprime=fder)
|
||
|
|
assert_allclose(r, 0, atol=zeros._xtol, rtol=zeros._rtol)
|
||
|
|
r = zeros.newton(f_zeroder_root, x0=[0]*10, fprime=fder,
|
||
|
|
fprime2=fder2)
|
||
|
|
assert_allclose(r, 0, atol=zeros._xtol, rtol=zeros._rtol)
|
||
|
|
|
||
|
|
# also test that if a root is found we do not raise RuntimeWarning even if
|
||
|
|
# the derivative is zero, EG: at x = 0.5, then fval = -0.125 and
|
||
|
|
# fder = -0.25 so the next guess is 0.5 - (-0.125/-0.5) = 0 which is the
|
||
|
|
# root, but if the solver continued with that guess, then it will calculate
|
||
|
|
# a zero derivative, so it should return the root w/o RuntimeWarning
|
||
|
|
r = zeros.newton(f_zeroder_root, x0=0.5, fprime=fder)
|
||
|
|
assert_allclose(r, 0, atol=zeros._xtol, rtol=zeros._rtol)
|
||
|
|
# test again with array
|
||
|
|
r = zeros.newton(f_zeroder_root, x0=[0.5]*10, fprime=fder)
|
||
|
|
assert_allclose(r, 0, atol=zeros._xtol, rtol=zeros._rtol)
|
||
|
|
# doesn't apply to halley
|
||
|
|
|
||
|
|
|
||
|
|
def test_gh_8881():
|
||
|
|
r"""Test that Halley's method realizes that the 2nd order adjustment
|
||
|
|
is too big and drops off to the 1st order adjustment."""
|
||
|
|
n = 9
|
||
|
|
|
||
|
|
def f(x):
|
||
|
|
return power(x, 1.0/n) - power(n, 1.0/n)
|
||
|
|
|
||
|
|
def fp(x):
|
||
|
|
return power(x, (1.0-n)/n)/n
|
||
|
|
|
||
|
|
def fpp(x):
|
||
|
|
return power(x, (1.0-2*n)/n) * (1.0/n) * (1.0-n)/n
|
||
|
|
|
||
|
|
x0 = 0.1
|
||
|
|
# The root is at x=9.
|
||
|
|
# The function has positive slope, x0 < root.
|
||
|
|
# Newton succeeds in 8 iterations
|
||
|
|
rt, r = newton(f, x0, fprime=fp, full_output=True)
|
||
|
|
assert(r.converged)
|
||
|
|
# Before the Issue 8881/PR 8882, halley would send x in the wrong direction.
|
||
|
|
# Check that it now succeeds.
|
||
|
|
rt, r = newton(f, x0, fprime=fp, fprime2=fpp, full_output=True)
|
||
|
|
assert(r.converged)
|
||
|
|
|
||
|
|
|
||
|
|
def test_gh_9608_preserve_array_shape():
|
||
|
|
"""
|
||
|
|
Test that shape is preserved for array inputs even if fprime or fprime2 is
|
||
|
|
scalar
|
||
|
|
"""
|
||
|
|
def f(x):
|
||
|
|
return x**2
|
||
|
|
|
||
|
|
def fp(x):
|
||
|
|
return 2 * x
|
||
|
|
|
||
|
|
def fpp(x):
|
||
|
|
return 2
|
||
|
|
|
||
|
|
x0 = np.array([-2], dtype=np.float32)
|
||
|
|
rt, r = newton(f, x0, fprime=fp, fprime2=fpp, full_output=True)
|
||
|
|
assert(r.converged)
|
||
|
|
|
||
|
|
x0_array = np.array([-2, -3], dtype=np.float32)
|
||
|
|
# This next invocation should fail
|
||
|
|
with pytest.raises(IndexError):
|
||
|
|
result = zeros.newton(
|
||
|
|
f, x0_array, fprime=fp, fprime2=fpp, full_output=True
|
||
|
|
)
|
||
|
|
|
||
|
|
def fpp_array(x):
|
||
|
|
return np.full(np.shape(x), 2, dtype=np.float32)
|
||
|
|
|
||
|
|
result = zeros.newton(
|
||
|
|
f, x0_array, fprime=fp, fprime2=fpp_array, full_output=True
|
||
|
|
)
|
||
|
|
assert result.converged.all()
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize(
|
||
|
|
"maximum_iterations,flag_expected",
|
||
|
|
[(10, zeros.CONVERR), (100, zeros.CONVERGED)])
|
||
|
|
def test_gh9254_flag_if_maxiter_exceeded(maximum_iterations, flag_expected):
|
||
|
|
"""
|
||
|
|
Test that if the maximum iterations is exceeded that the flag is not
|
||
|
|
converged.
|
||
|
|
"""
|
||
|
|
result = zeros.brentq(
|
||
|
|
lambda x: ((1.2*x - 2.3)*x + 3.4)*x - 4.5,
|
||
|
|
-30, 30, (), 1e-6, 1e-6, maximum_iterations,
|
||
|
|
full_output=True, disp=False)
|
||
|
|
assert result[1].flag == flag_expected
|
||
|
|
if flag_expected == zeros.CONVERR:
|
||
|
|
# didn't converge because exceeded maximum iterations
|
||
|
|
assert result[1].iterations == maximum_iterations
|
||
|
|
elif flag_expected == zeros.CONVERGED:
|
||
|
|
# converged before maximum iterations
|
||
|
|
assert result[1].iterations < maximum_iterations
|
||
|
|
|
||
|
|
|
||
|
|
def test_gh9551_raise_error_if_disp_true():
|
||
|
|
"""Test that if disp is true then zero derivative raises RuntimeError"""
|
||
|
|
|
||
|
|
def f(x):
|
||
|
|
return x*x + 1
|
||
|
|
|
||
|
|
def f_p(x):
|
||
|
|
return 2*x
|
||
|
|
|
||
|
|
assert_warns(RuntimeWarning, zeros.newton, f, 1.0, f_p, disp=False)
|
||
|
|
with pytest.raises(
|
||
|
|
RuntimeError,
|
||
|
|
match=r'^Derivative was zero\. Failed to converge after \d+ iterations, value is [+-]?\d*\.\d+\.$'):
|
||
|
|
zeros.newton(f, 1.0, f_p)
|
||
|
|
root = zeros.newton(f, complex(10.0, 10.0), f_p)
|
||
|
|
assert_allclose(root, complex(0.0, 1.0))
|