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977 lines
38 KiB
977 lines
38 KiB
import math
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import pytest
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import numpy as np
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from copy import deepcopy
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from scipy import stats, special
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import scipy._lib._elementwise_iterative_method as eim
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import scipy._lib.array_api_extra as xpx
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from scipy._lib._array_api import (array_namespace, is_cupy, is_numpy, xp_ravel,
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xp_size, make_xp_test_case)
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from scipy._lib._array_api_no_0d import (xp_assert_close, xp_assert_equal,
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xp_assert_less)
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from scipy.optimize.elementwise import find_minimum, find_root
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from scipy.optimize._tstutils import _CHANDRUPATLA_TESTS
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from itertools import permutations
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def _vectorize(xp):
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# xp-compatible version of np.vectorize
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# assumes arguments are all arrays of the same shape
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def decorator(f):
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def wrapped(*arg_arrays):
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shape = arg_arrays[0].shape
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arg_arrays = [xp_ravel(arg_array, xp=xp) for arg_array in arg_arrays]
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res = []
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for i in range(math.prod(shape)):
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arg_scalars = [arg_array[i] for arg_array in arg_arrays]
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res.append(f(*arg_scalars))
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return res
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return wrapped
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return decorator
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# These tests were originally written for the private `optimize._chandrupatla`
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# interfaces, but now we want the tests to check the behavior of the public
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# `optimize.elementwise` interfaces. Therefore, rather than importing
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# `_chandrupatla`/`_chandrupatla_minimize` from `_chandrupatla.py`, we import
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# `find_root`/`find_minimum` from `optimize.elementwise` and wrap those
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# functions to conform to the private interface. This may look a little strange,
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# since it effectively just inverts the interface transformation done within the
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# `find_root`/`find_minimum` functions, but it allows us to run the original,
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# unmodified tests on the public interfaces, simplifying the PR that adds
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# the public interfaces. We'll refactor this when we want to @parametrize the
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# tests over multiple `method`s.
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def _wrap_chandrupatla(func):
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def _chandrupatla_wrapper(f, *bracket, **kwargs):
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# avoid passing arguments to `find_minimum` to this function
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tol_keys = {'xatol', 'xrtol', 'fatol', 'frtol'}
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tolerances = {key: kwargs.pop(key) for key in tol_keys if key in kwargs}
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_callback = kwargs.pop('callback', None)
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if callable(_callback):
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def callback(res):
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if func == find_root:
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res.xl, res.xr = res.bracket
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res.fl, res.fr = res.f_bracket
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else:
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res.xl, res.xm, res.xr = res.bracket
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res.fl, res.fm, res.fr = res.f_bracket
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res.fun = res.f_x
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del res.bracket
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del res.f_bracket
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del res.f_x
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return _callback(res)
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else:
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callback = _callback
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res = func(f, bracket, tolerances=tolerances, callback=callback, **kwargs)
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if func == find_root:
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res.xl, res.xr = res.bracket
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res.fl, res.fr = res.f_bracket
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else:
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res.xl, res.xm, res.xr = res.bracket
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res.fl, res.fm, res.fr = res.f_bracket
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res.fun = res.f_x
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del res.bracket
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del res.f_bracket
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del res.f_x
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return res
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return _chandrupatla_wrapper
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_chandrupatla_minimize = _wrap_chandrupatla(find_minimum)
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def f1(x):
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return 100*(1 - x**3.)**2 + (1-x**2.) + 2*(1-x)**2.
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def f2(x):
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return 5 + (x - 2.)**6
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def f3(x):
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xp = array_namespace(x)
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return xp.exp(x) - 5*x
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def f4(x):
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return x**5. - 5*x**3. - 20.*x + 5.
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def f5(x):
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return 8*x**3 - 2*x**2 - 7*x + 3
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def _bracket_minimum(func, x1, x2):
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phi = 1.61803398875
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maxiter = 100
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f1 = func(x1)
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f2 = func(x2)
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step = x2 - x1
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x1, x2, f1, f2, step = ((x2, x1, f2, f1, -step) if f2 > f1
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else (x1, x2, f1, f2, step))
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for i in range(maxiter):
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step *= phi
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x3 = x2 + step
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f3 = func(x3)
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if f3 < f2:
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x1, x2, f1, f2 = x2, x3, f2, f3
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else:
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break
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return x1, x2, x3, f1, f2, f3
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cases = [
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(f1, -1, 11),
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(f1, -2, 13),
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(f1, -4, 13),
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(f1, -8, 15),
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(f1, -16, 16),
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(f1, -32, 19),
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(f1, -64, 20),
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(f1, -128, 21),
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(f1, -256, 21),
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(f1, -512, 19),
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(f1, -1024, 24),
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(f2, -1, 8),
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(f2, -2, 6),
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(f2, -4, 6),
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(f2, -8, 7),
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(f2, -16, 8),
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(f2, -32, 8),
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(f2, -64, 9),
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(f2, -128, 11),
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(f2, -256, 13),
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(f2, -512, 12),
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(f2, -1024, 13),
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(f3, -1, 11),
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(f3, -2, 11),
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(f3, -4, 11),
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(f3, -8, 10),
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(f3, -16, 14),
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(f3, -32, 12),
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(f3, -64, 15),
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(f3, -128, 18),
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(f3, -256, 18),
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(f3, -512, 19),
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(f3, -1024, 19),
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(f4, -0.05, 9),
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(f4, -0.10, 11),
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(f4, -0.15, 11),
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(f4, -0.20, 11),
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(f4, -0.25, 11),
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(f4, -0.30, 9),
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(f4, -0.35, 9),
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(f4, -0.40, 9),
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(f4, -0.45, 10),
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(f4, -0.50, 10),
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(f4, -0.55, 10),
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(f5, -0.05, 6),
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(f5, -0.10, 7),
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(f5, -0.15, 8),
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(f5, -0.20, 10),
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(f5, -0.25, 9),
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(f5, -0.30, 8),
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(f5, -0.35, 7),
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(f5, -0.40, 7),
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(f5, -0.45, 9),
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(f5, -0.50, 9),
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(f5, -0.55, 8)
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]
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@make_xp_test_case(find_minimum)
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class TestChandrupatlaMinimize:
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def f(self, x, loc):
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xp = array_namespace(x, loc)
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res = -xp.exp(-1/2 * (x-loc)**2) / (2*xp.pi)**0.5
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return xp.asarray(res, dtype=x.dtype)[()]
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@pytest.mark.parametrize('dtype', ('float32', 'float64'))
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@pytest.mark.parametrize('loc', [0.6, np.linspace(-1.05, 1.05, 10)])
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def test_basic(self, loc, xp, dtype):
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# Find mode of normal distribution. Compare mode against location
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# parameter and value of pdf at mode against expected pdf.
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rtol = {'float32': 5e-3, 'float64': 5e-7}[dtype]
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dtype = getattr(xp, dtype)
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bracket = (xp.asarray(xi, dtype=dtype) for xi in (-5, 0, 5))
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loc = xp.asarray(loc, dtype=dtype)
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fun = xp.broadcast_to(xp.asarray(-stats.norm.pdf(0), dtype=dtype), loc.shape)
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res = _chandrupatla_minimize(self.f, *bracket, args=(loc,))
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xp_assert_close(res.x, loc, rtol=rtol)
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xp_assert_equal(res.fun, fun)
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@pytest.mark.parametrize('shape', [tuple(), (12,), (3, 4), (3, 2, 2)])
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def test_vectorization(self, shape, xp):
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# Test for correct functionality, output shapes, and dtypes for various
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# input shapes.
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loc = xp.linspace(-0.05, 1.05, 12).reshape(shape) if shape else xp.asarray(0.6)
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args = (loc,)
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bracket = xp.asarray(-5.), xp.asarray(0.), xp.asarray(5.)
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@_vectorize(xp)
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def chandrupatla_single(loc_single):
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return _chandrupatla_minimize(self.f, *bracket, args=(loc_single,))
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def f(*args, **kwargs):
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f.f_evals += 1
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return self.f(*args, **kwargs)
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f.f_evals = 0
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res = _chandrupatla_minimize(f, *bracket, args=args)
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refs = chandrupatla_single(loc)
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attrs = ['x', 'fun', 'success', 'status', 'nfev', 'nit',
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'xl', 'xm', 'xr', 'fl', 'fm', 'fr']
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for attr in attrs:
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ref_attr = xp.stack([getattr(ref, attr) for ref in refs])
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res_attr = xp_ravel(getattr(res, attr))
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xp_assert_equal(res_attr, ref_attr)
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assert getattr(res, attr).shape == shape
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xp_assert_equal(res.fun, self.f(res.x, *args))
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xp_assert_equal(res.fl, self.f(res.xl, *args))
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xp_assert_equal(res.fm, self.f(res.xm, *args))
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xp_assert_equal(res.fr, self.f(res.xr, *args))
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assert xp.max(res.nfev) == f.f_evals
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assert xp.max(res.nit) == f.f_evals - 3
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assert xp.isdtype(res.success.dtype, 'bool')
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assert xp.isdtype(res.status.dtype, 'integral')
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assert xp.isdtype(res.nfev.dtype, 'integral')
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assert xp.isdtype(res.nit.dtype, 'integral')
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def test_flags(self, xp):
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# Test cases that should produce different status flags; show that all
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# can be produced simultaneously.
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def f(xs, js):
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funcs = [lambda x: (x - 2.5) ** 2,
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lambda x: x - 10,
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lambda x: (x - 2.5) ** 4,
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lambda x: xp.full_like(x, xp.asarray(xp.nan))]
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res = []
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for i in range(xp_size(js)):
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x = xs[i, ...]
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j = int(xp_ravel(js)[i])
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res.append(funcs[j](x))
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return xp.stack(res)
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args = (xp.arange(4, dtype=xp.int64),)
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bracket = (xp.asarray([0]*4, dtype=xp.float64),
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xp.asarray([2]*4, dtype=xp.float64),
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xp.asarray([np.pi]*4, dtype=xp.float64))
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res = _chandrupatla_minimize(f, *bracket, args=args, maxiter=10)
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ref_flags = xp.asarray([eim._ECONVERGED, eim._ESIGNERR, eim._ECONVERR,
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eim._EVALUEERR], dtype=xp.int32)
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xp_assert_equal(res.status, ref_flags)
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def test_convergence(self, xp):
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# Test that the convergence tolerances behave as expected
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rng = np.random.default_rng(2585255913088665241)
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p = xp.asarray(rng.random(size=3))
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bracket = (xp.asarray(-5, dtype=xp.float64), xp.asarray(0), xp.asarray(5))
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args = (p,)
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kwargs0 = dict(args=args, xatol=0, xrtol=0, fatol=0, frtol=0)
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kwargs = kwargs0.copy()
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kwargs['xatol'] = 1e-3
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res1 = _chandrupatla_minimize(self.f, *bracket, **kwargs)
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j1 = xp.abs(res1.xr - res1.xl)
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tol = xp.asarray(4*kwargs['xatol'], dtype=p.dtype)
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xp_assert_less(j1, xp.full((3,), tol, dtype=p.dtype))
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kwargs['xatol'] = 1e-6
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res2 = _chandrupatla_minimize(self.f, *bracket, **kwargs)
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j2 = xp.abs(res2.xr - res2.xl)
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tol = xp.asarray(4*kwargs['xatol'], dtype=p.dtype)
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xp_assert_less(j2, xp.full((3,), tol, dtype=p.dtype))
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xp_assert_less(j2, j1)
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kwargs = kwargs0.copy()
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kwargs['xrtol'] = 1e-3
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res1 = _chandrupatla_minimize(self.f, *bracket, **kwargs)
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j1 = xp.abs(res1.xr - res1.xl)
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tol = xp.asarray(4*kwargs['xrtol']*xp.abs(res1.x), dtype=p.dtype)
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xp_assert_less(j1, tol)
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kwargs['xrtol'] = 1e-6
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res2 = _chandrupatla_minimize(self.f, *bracket, **kwargs)
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j2 = xp.abs(res2.xr - res2.xl)
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tol = xp.asarray(4*kwargs['xrtol']*xp.abs(res2.x), dtype=p.dtype)
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xp_assert_less(j2, tol)
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xp_assert_less(j2, j1)
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kwargs = kwargs0.copy()
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kwargs['fatol'] = 1e-3
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res1 = _chandrupatla_minimize(self.f, *bracket, **kwargs)
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h1 = xp.abs(res1.fl - 2 * res1.fm + res1.fr)
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tol = xp.asarray(2*kwargs['fatol'], dtype=p.dtype)
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xp_assert_less(h1, xp.full((3,), tol, dtype=p.dtype))
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kwargs['fatol'] = 1e-6
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res2 = _chandrupatla_minimize(self.f, *bracket, **kwargs)
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h2 = xp.abs(res2.fl - 2 * res2.fm + res2.fr)
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tol = xp.asarray(2*kwargs['fatol'], dtype=p.dtype)
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xp_assert_less(h2, xp.full((3,), tol, dtype=p.dtype))
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xp_assert_less(h2, h1)
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kwargs = kwargs0.copy()
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kwargs['frtol'] = 1e-3
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res1 = _chandrupatla_minimize(self.f, *bracket, **kwargs)
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h1 = xp.abs(res1.fl - 2 * res1.fm + res1.fr)
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tol = xp.asarray(2*kwargs['frtol']*xp.abs(res1.fun), dtype=p.dtype)
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xp_assert_less(h1, tol)
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kwargs['frtol'] = 1e-6
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res2 = _chandrupatla_minimize(self.f, *bracket, **kwargs)
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h2 = xp.abs(res2.fl - 2 * res2.fm + res2.fr)
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tol = xp.asarray(2*kwargs['frtol']*abs(res2.fun), dtype=p.dtype)
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xp_assert_less(h2, tol)
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xp_assert_less(h2, h1)
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def test_maxiter_callback(self, xp):
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# Test behavior of `maxiter` parameter and `callback` interface
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loc = xp.asarray(0.612814)
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bracket = (xp.asarray(-5), xp.asarray(0), xp.asarray(5))
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maxiter = 5
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res = _chandrupatla_minimize(self.f, *bracket, args=(loc,),
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maxiter=maxiter)
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assert not xp.any(res.success)
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assert xp.all(res.nfev == maxiter+3)
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assert xp.all(res.nit == maxiter)
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def callback(res):
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callback.iter += 1
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callback.res = res
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assert hasattr(res, 'x')
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if callback.iter == 0:
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# callback is called once with initial bracket
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assert (res.xl, res.xm, res.xr) == bracket
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else:
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changed_xr = (res.xl == callback.xl) & (res.xr != callback.xr)
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changed_xl = (res.xl != callback.xl) & (res.xr == callback.xr)
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assert xp.all(changed_xr | changed_xl)
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callback.xl = res.xl
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callback.xr = res.xr
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assert res.status == eim._EINPROGRESS
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xp_assert_equal(self.f(res.xl, loc), res.fl)
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xp_assert_equal(self.f(res.xm, loc), res.fm)
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xp_assert_equal(self.f(res.xr, loc), res.fr)
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xp_assert_equal(self.f(res.x, loc), res.fun)
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if callback.iter == maxiter:
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raise StopIteration
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callback.xl = xp.nan
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callback.xr = xp.nan
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callback.iter = -1 # callback called once before first iteration
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callback.res = None
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res2 = _chandrupatla_minimize(self.f, *bracket, args=(loc,),
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callback=callback)
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# terminating with callback is identical to terminating due to maxiter
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# (except for `status`)
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for key in res.keys():
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if key == 'status':
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assert res[key] == eim._ECONVERR
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# assert callback.res[key] == eim._EINPROGRESS
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assert res2[key] == eim._ECALLBACK
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else:
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assert res2[key] == callback.res[key] == res[key]
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@pytest.mark.parametrize('case', cases)
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def test_nit_expected(self, case, xp):
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# Test that `_chandrupatla` implements Chandrupatla's algorithm:
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# in all 55 test cases, the number of iterations performed
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# matches the number reported in the original paper.
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func, x1, nit = case
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# Find bracket using the algorithm in the paper
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step = 0.2
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x2 = x1 + step
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x1, x2, x3, f1, f2, f3 = _bracket_minimum(func, x1, x2)
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# Use tolerances from original paper
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xatol = 0.0001
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fatol = 0.000001
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xrtol = 1e-16
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frtol = 1e-16
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bracket = xp.asarray(x1), xp.asarray(x2), xp.asarray(x3, dtype=xp.float64)
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res = _chandrupatla_minimize(func, *bracket, xatol=xatol,
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fatol=fatol, xrtol=xrtol, frtol=frtol)
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xp_assert_equal(res.nit, xp.asarray(nit, dtype=xp.int32))
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@pytest.mark.parametrize("loc", (0.65, [0.65, 0.7]))
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@pytest.mark.parametrize("dtype", ('float16', 'float32', 'float64'))
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def test_dtype(self, loc, dtype, xp):
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# Test that dtypes are preserved
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dtype = getattr(xp, dtype)
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loc = xp.asarray(loc, dtype=dtype)
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|
bracket = (xp.asarray(-3, dtype=dtype),
|
|
xp.asarray(1, dtype=dtype),
|
|
xp.asarray(5, dtype=dtype))
|
|
|
|
def f(x, loc):
|
|
assert x.dtype == dtype
|
|
return xp.astype((x - loc)**2, dtype)
|
|
|
|
res = _chandrupatla_minimize(f, *bracket, args=(loc,))
|
|
assert res.x.dtype == dtype
|
|
xp_assert_close(res.x, loc, rtol=math.sqrt(xp.finfo(dtype).eps))
|
|
|
|
def test_input_validation(self, xp):
|
|
# Test input validation for appropriate error messages
|
|
|
|
message = '`func` must be callable.'
|
|
bracket = xp.asarray(-4), xp.asarray(0), xp.asarray(4)
|
|
with pytest.raises(ValueError, match=message):
|
|
_chandrupatla_minimize(None, *bracket)
|
|
|
|
message = 'Abscissae and function output must be real numbers.'
|
|
bracket = xp.asarray(-4 + 1j), xp.asarray(0), xp.asarray(4)
|
|
with pytest.raises(ValueError, match=message):
|
|
_chandrupatla_minimize(lambda x: x, *bracket)
|
|
|
|
message = "...be broadcast..."
|
|
bracket = xp.asarray([-2, -3]), xp.asarray([0, 0]), xp.asarray([3, 4, 5])
|
|
# raised by `np.broadcast, but the traceback is readable IMO
|
|
with pytest.raises((ValueError, RuntimeError), match=message):
|
|
_chandrupatla_minimize(lambda x: x, *bracket)
|
|
|
|
message = "The shape of the array returned by `func` must be the same"
|
|
bracket = xp.asarray([-3, -3]), xp.asarray([0, 0]), xp.asarray([5, 5])
|
|
with pytest.raises(ValueError, match=message):
|
|
_chandrupatla_minimize(lambda x: [x[0, ...], x[1, ...], x[1, ...]],
|
|
*bracket)
|
|
|
|
message = 'Tolerances must be non-negative scalars.'
|
|
bracket = xp.asarray(-4), xp.asarray(0), xp.asarray(4)
|
|
with pytest.raises(ValueError, match=message):
|
|
_chandrupatla_minimize(lambda x: x, *bracket, xatol=-1)
|
|
with pytest.raises(ValueError, match=message):
|
|
_chandrupatla_minimize(lambda x: x, *bracket, xrtol=xp.nan)
|
|
with pytest.raises(ValueError, match=message):
|
|
_chandrupatla_minimize(lambda x: x, *bracket, fatol='ekki')
|
|
with pytest.raises(ValueError, match=message):
|
|
_chandrupatla_minimize(lambda x: x, *bracket, frtol=xp.nan)
|
|
|
|
message = '`maxiter` must be a non-negative integer.'
|
|
with pytest.raises(ValueError, match=message):
|
|
_chandrupatla_minimize(lambda x: x, *bracket, maxiter=1.5)
|
|
with pytest.raises(ValueError, match=message):
|
|
_chandrupatla_minimize(lambda x: x, *bracket, maxiter=-1)
|
|
|
|
message = '`callback` must be callable.'
|
|
with pytest.raises(ValueError, match=message):
|
|
_chandrupatla_minimize(lambda x: x, *bracket, callback='shrubbery')
|
|
|
|
def test_bracket_order(self, xp):
|
|
# Confirm that order of points in bracket doesn't
|
|
loc = xp.linspace(-1, 1, 6)[:, xp.newaxis]
|
|
brackets = xp.asarray(list(permutations([-5, 0, 5]))).T
|
|
res = _chandrupatla_minimize(self.f, *brackets, args=(loc,))
|
|
assert xp.all(xpx.isclose(res.x, loc) | (res.fun == self.f(loc, loc)))
|
|
ref = res.x[:, 0] # all columns should be the same
|
|
xp_assert_close(*xp.broadcast_arrays(res.x.T, ref), rtol=1e-15)
|
|
|
|
def test_special_cases(self, xp):
|
|
# Test edge cases and other special cases
|
|
|
|
# Test that integers are not passed to `f`
|
|
def f(x):
|
|
assert xp.isdtype(x.dtype, "real floating")
|
|
return (x - 1)**2
|
|
|
|
bracket = xp.asarray(-7), xp.asarray(0), xp.asarray(8)
|
|
with np.errstate(invalid='ignore'):
|
|
res = _chandrupatla_minimize(f, *bracket, fatol=0, frtol=0)
|
|
assert res.success
|
|
xp_assert_close(res.x, xp.asarray(1.), rtol=1e-3)
|
|
xp_assert_close(res.fun, xp.asarray(0.), atol=1e-200)
|
|
|
|
# Test that if all elements of bracket equal minimizer, algorithm
|
|
# reports convergence
|
|
def f(x):
|
|
return (x-1)**2
|
|
|
|
bracket = xp.asarray(1), xp.asarray(1), xp.asarray(1)
|
|
res = _chandrupatla_minimize(f, *bracket)
|
|
assert res.success
|
|
xp_assert_equal(res.x, xp.asarray(1.))
|
|
|
|
# Test maxiter = 0. Should do nothing to bracket.
|
|
def f(x):
|
|
return (x-1)**2
|
|
|
|
bracket = xp.asarray(-3), xp.asarray(1.1), xp.asarray(5)
|
|
res = _chandrupatla_minimize(f, *bracket, maxiter=0)
|
|
assert res.xl, res.xr == bracket
|
|
assert res.nit == 0
|
|
assert res.nfev == 3
|
|
assert res.status == -2
|
|
assert res.x == 1.1 # best so far
|
|
|
|
# Test scalar `args` (not in tuple)
|
|
def f(x, c):
|
|
return (x-c)**2 - 1
|
|
|
|
bracket = xp.asarray(-1), xp.asarray(0), xp.asarray(1)
|
|
c = xp.asarray(1/3)
|
|
res = _chandrupatla_minimize(f, *bracket, args=(c,))
|
|
xp_assert_close(res.x, c)
|
|
|
|
# Test zero tolerances
|
|
def f(x):
|
|
return -xp.sin(x)
|
|
|
|
bracket = xp.asarray(0), xp.asarray(1), xp.asarray(xp.pi)
|
|
res = _chandrupatla_minimize(f, *bracket, xatol=0, xrtol=0, fatol=0, frtol=0)
|
|
assert res.success
|
|
# found a minimum exactly (according to floating point arithmetic)
|
|
assert res.xl < res.xm < res.xr
|
|
assert f(res.xl) == f(res.xm) == f(res.xr)
|
|
|
|
|
|
@make_xp_test_case(find_root)
|
|
class TestFindRoot:
|
|
|
|
def f(self, q, p):
|
|
return special.ndtr(q) - p
|
|
|
|
@pytest.mark.parametrize('p', [0.6, np.linspace(-0.05, 1.05, 10)])
|
|
def test_basic(self, p, xp):
|
|
# Invert distribution CDF and compare against distribution `ppf`
|
|
a, b = xp.asarray(-5.), xp.asarray(5.)
|
|
res = find_root(self.f, (a, b), args=(xp.asarray(p),))
|
|
ref = xp.asarray(stats.norm().ppf(p), dtype=xp.asarray(p).dtype)
|
|
xp_assert_close(res.x, ref)
|
|
|
|
@pytest.mark.parametrize('shape', [tuple(), (12,), (3, 4), (3, 2, 2)])
|
|
def test_vectorization(self, shape, xp):
|
|
# Test for correct functionality, output shapes, and dtypes for various
|
|
# input shapes.
|
|
p = (np.linspace(-0.05, 1.05, 12).reshape(shape) if shape
|
|
else np.float64(0.6))
|
|
p_xp = xp.asarray(p)
|
|
args_xp = (p_xp,)
|
|
dtype = p_xp.dtype
|
|
|
|
@np.vectorize
|
|
def find_root_single(p):
|
|
return find_root(self.f, (-5, 5), args=(p,))
|
|
|
|
def f(*args, **kwargs):
|
|
f.f_evals += 1
|
|
return self.f(*args, **kwargs)
|
|
f.f_evals = 0
|
|
|
|
bracket = xp.asarray(-5., dtype=xp.float64), xp.asarray(5., dtype=xp.float64)
|
|
res = find_root(f, bracket, args=args_xp)
|
|
refs = find_root_single(p).ravel()
|
|
|
|
ref_x = [ref.x for ref in refs]
|
|
ref_x = xp.reshape(xp.asarray(ref_x, dtype=dtype), shape)
|
|
xp_assert_close(res.x, ref_x)
|
|
|
|
ref_f = [ref.f_x for ref in refs]
|
|
ref_f = xp.reshape(xp.asarray(ref_f, dtype=dtype), shape)
|
|
xp_assert_close(res.f_x, ref_f, atol=1e-15)
|
|
xp_assert_equal(res.f_x, self.f(res.x, *args_xp))
|
|
|
|
ref_success = [bool(ref.success) for ref in refs]
|
|
ref_success = xp.reshape(xp.asarray(ref_success, dtype=xp.bool), shape)
|
|
xp_assert_equal(res.success, ref_success)
|
|
|
|
ref_status = [ref.status for ref in refs]
|
|
ref_status = xp.reshape(xp.asarray(ref_status, dtype=xp.int32), shape)
|
|
xp_assert_equal(res.status, ref_status)
|
|
|
|
ref_nfev = [ref.nfev for ref in refs]
|
|
ref_nfev = xp.reshape(xp.asarray(ref_nfev, dtype=xp.int32), shape)
|
|
if is_numpy(xp):
|
|
xp_assert_equal(res.nfev, ref_nfev)
|
|
assert xp.max(res.nfev) == f.f_evals
|
|
else: # different backend may lead to different nfev
|
|
assert res.nfev.shape == shape
|
|
assert res.nfev.dtype == xp.int32
|
|
|
|
ref_nit = [ref.nit for ref in refs]
|
|
ref_nit = xp.reshape(xp.asarray(ref_nit, dtype=xp.int32), shape)
|
|
if is_numpy(xp):
|
|
xp_assert_equal(res.nit, ref_nit)
|
|
assert xp.max(res.nit) == f.f_evals-2
|
|
else:
|
|
assert res.nit.shape == shape
|
|
assert res.nit.dtype == xp.int32
|
|
|
|
ref_xl = [ref.bracket[0] for ref in refs]
|
|
ref_xl = xp.reshape(xp.asarray(ref_xl, dtype=dtype), shape)
|
|
xp_assert_close(res.bracket[0], ref_xl)
|
|
|
|
ref_xr = [ref.bracket[1] for ref in refs]
|
|
ref_xr = xp.reshape(xp.asarray(ref_xr, dtype=dtype), shape)
|
|
xp_assert_close(res.bracket[1], ref_xr)
|
|
|
|
xp_assert_less(res.bracket[0], res.bracket[1])
|
|
finite = xp.isfinite(res.x)
|
|
assert xp.all((res.x[finite] == res.bracket[0][finite])
|
|
| (res.x[finite] == res.bracket[1][finite]))
|
|
|
|
# PyTorch and CuPy don't solve to the same accuracy as NumPy - that's OK.
|
|
atol = 1e-15 if is_numpy(xp) else 1e-9
|
|
|
|
ref_fl = [ref.f_bracket[0] for ref in refs]
|
|
ref_fl = xp.reshape(xp.asarray(ref_fl, dtype=dtype), shape)
|
|
xp_assert_close(res.f_bracket[0], ref_fl, atol=atol)
|
|
xp_assert_equal(res.f_bracket[0], self.f(res.bracket[0], *args_xp))
|
|
|
|
ref_fr = [ref.f_bracket[1] for ref in refs]
|
|
ref_fr = xp.reshape(xp.asarray(ref_fr, dtype=dtype), shape)
|
|
xp_assert_close(res.f_bracket[1], ref_fr, atol=atol)
|
|
xp_assert_equal(res.f_bracket[1], self.f(res.bracket[1], *args_xp))
|
|
|
|
assert xp.all(xp.abs(res.f_x[finite]) ==
|
|
xp.minimum(xp.abs(res.f_bracket[0][finite]),
|
|
xp.abs(res.f_bracket[1][finite])))
|
|
|
|
def test_flags(self, xp):
|
|
# Test cases that should produce different status flags; show that all
|
|
# can be produced simultaneously.
|
|
def f(xs, js):
|
|
# Note that full_like and int(j) shouldn't really be required. CuPy
|
|
# is just really picky here, so I'm making it a special case to
|
|
# make sure the other backends work when the user is less careful.
|
|
assert js.dtype == xp.int64
|
|
if is_cupy(xp):
|
|
funcs = [lambda x: x - 2.5,
|
|
lambda x: x - 10,
|
|
lambda x: (x - 0.1)**3,
|
|
lambda x: xp.full_like(x, xp.asarray(xp.nan))]
|
|
return [funcs[int(j)](x) for x, j in zip(xs, js)]
|
|
|
|
funcs = [lambda x: x - 2.5,
|
|
lambda x: x - 10,
|
|
lambda x: (x - 0.1) ** 3,
|
|
lambda x: xp.nan]
|
|
return [funcs[j](x) for x, j in zip(xs, js)]
|
|
|
|
args = (xp.arange(4, dtype=xp.int64),)
|
|
a, b = xp.asarray([0.]*4), xp.asarray([xp.pi]*4)
|
|
res = find_root(f, (a, b), args=args, maxiter=2)
|
|
|
|
ref_flags = xp.asarray([eim._ECONVERGED,
|
|
eim._ESIGNERR,
|
|
eim._ECONVERR,
|
|
eim._EVALUEERR], dtype=xp.int32)
|
|
xp_assert_equal(res.status, ref_flags)
|
|
|
|
def test_convergence(self, xp):
|
|
# Test that the convergence tolerances behave as expected
|
|
rng = np.random.default_rng(2585255913088665241)
|
|
p = xp.asarray(rng.random(size=3))
|
|
bracket = (-xp.asarray(5.), xp.asarray(5.))
|
|
args = (p,)
|
|
kwargs0 = dict(args=args, tolerances=dict(xatol=0, xrtol=0, fatol=0, frtol=0))
|
|
|
|
kwargs = deepcopy(kwargs0)
|
|
kwargs['tolerances']['xatol'] = 1e-3
|
|
res1 = find_root(self.f, bracket, **kwargs)
|
|
xp_assert_less(res1.bracket[1] - res1.bracket[0],
|
|
xp.full_like(p, xp.asarray(1e-3)))
|
|
kwargs['tolerances']['xatol'] = 1e-6
|
|
res2 = find_root(self.f, bracket, **kwargs)
|
|
xp_assert_less(res2.bracket[1] - res2.bracket[0],
|
|
xp.full_like(p, xp.asarray(1e-6)))
|
|
xp_assert_less(res2.bracket[1] - res2.bracket[0],
|
|
res1.bracket[1] - res1.bracket[0])
|
|
|
|
kwargs = deepcopy(kwargs0)
|
|
kwargs['tolerances']['xrtol'] = 1e-3
|
|
res1 = find_root(self.f, bracket, **kwargs)
|
|
xp_assert_less(res1.bracket[1] - res1.bracket[0], 1e-3 * xp.abs(res1.x))
|
|
kwargs['tolerances']['xrtol'] = 1e-6
|
|
res2 = find_root(self.f, bracket, **kwargs)
|
|
xp_assert_less(res2.bracket[1] - res2.bracket[0],
|
|
1e-6 * xp.abs(res2.x))
|
|
xp_assert_less(res2.bracket[1] - res2.bracket[0],
|
|
res1.bracket[1] - res1.bracket[0])
|
|
|
|
kwargs = deepcopy(kwargs0)
|
|
kwargs['tolerances']['fatol'] = 1e-3
|
|
res1 = find_root(self.f, bracket, **kwargs)
|
|
xp_assert_less(xp.abs(res1.f_x), xp.full_like(p, xp.asarray(1e-3)))
|
|
kwargs['tolerances']['fatol'] = 1e-6
|
|
res2 = find_root(self.f, bracket, **kwargs)
|
|
xp_assert_less(xp.abs(res2.f_x), xp.full_like(p, xp.asarray(1e-6)))
|
|
xp_assert_less(xp.abs(res2.f_x), xp.abs(res1.f_x))
|
|
|
|
kwargs = deepcopy(kwargs0)
|
|
kwargs['tolerances']['frtol'] = 1e-3
|
|
x1, x2 = bracket
|
|
f0 = xp.minimum(xp.abs(self.f(x1, *args)), xp.abs(self.f(x2, *args)))
|
|
res1 = find_root(self.f, bracket, **kwargs)
|
|
xp_assert_less(xp.abs(res1.f_x), 1e-3*f0)
|
|
kwargs['tolerances']['frtol'] = 1e-6
|
|
res2 = find_root(self.f, bracket, **kwargs)
|
|
xp_assert_less(xp.abs(res2.f_x), 1e-6*f0)
|
|
xp_assert_less(xp.abs(res2.f_x), xp.abs(res1.f_x))
|
|
|
|
def test_maxiter_callback(self, xp):
|
|
# Test behavior of `maxiter` parameter and `callback` interface
|
|
p = xp.asarray(0.612814)
|
|
bracket = (xp.asarray(-5.), xp.asarray(5.))
|
|
maxiter = 5
|
|
|
|
def f(q, p):
|
|
res = special.ndtr(q) - p
|
|
f.x = q
|
|
f.f_x = res
|
|
return res
|
|
f.x = None
|
|
f.f_x = None
|
|
|
|
res = find_root(f, bracket, args=(p,), maxiter=maxiter)
|
|
assert not xp.any(res.success)
|
|
assert xp.all(res.nfev == maxiter+2)
|
|
assert xp.all(res.nit == maxiter)
|
|
|
|
def callback(res):
|
|
callback.iter += 1
|
|
callback.res = res
|
|
assert hasattr(res, 'x')
|
|
if callback.iter == 0:
|
|
# callback is called once with initial bracket
|
|
assert (res.bracket[0], res.bracket[1]) == bracket
|
|
else:
|
|
changed = (((res.bracket[0] == callback.bracket[0])
|
|
& (res.bracket[1] != callback.bracket[1]))
|
|
| ((res.bracket[0] != callback.bracket[0])
|
|
& (res.bracket[1] == callback.bracket[1])))
|
|
assert xp.all(changed)
|
|
|
|
callback.bracket[0] = res.bracket[0]
|
|
callback.bracket[1] = res.bracket[1]
|
|
assert res.status == eim._EINPROGRESS
|
|
xp_assert_equal(self.f(res.bracket[0], p), res.f_bracket[0])
|
|
xp_assert_equal(self.f(res.bracket[1], p), res.f_bracket[1])
|
|
xp_assert_equal(self.f(res.x, p), res.f_x)
|
|
if callback.iter == maxiter:
|
|
raise StopIteration
|
|
callback.iter = -1 # callback called once before first iteration
|
|
callback.res = None
|
|
callback.bracket = [None, None]
|
|
|
|
res2 = find_root(f, bracket, args=(p,), callback=callback)
|
|
|
|
# terminating with callback is identical to terminating due to maxiter
|
|
# (except for `status`)
|
|
for key in res.keys():
|
|
if key == 'status':
|
|
xp_assert_equal(res[key], xp.asarray(eim._ECONVERR, dtype=xp.int32))
|
|
xp_assert_equal(res2[key], xp.asarray(eim._ECALLBACK, dtype=xp.int32))
|
|
elif key in {'bracket', 'f_bracket'}:
|
|
xp_assert_equal(res2[key][0], res[key][0])
|
|
xp_assert_equal(res2[key][1], res[key][1])
|
|
elif key.startswith('_'):
|
|
continue
|
|
else:
|
|
xp_assert_equal(res2[key], res[key])
|
|
|
|
@pytest.mark.parametrize('case', _CHANDRUPATLA_TESTS)
|
|
def test_nit_expected(self, case, xp):
|
|
# Test that `_chandrupatla` implements Chandrupatla's algorithm:
|
|
# in all 40 test cases, the number of iterations performed
|
|
# matches the number reported in the original paper.
|
|
f, bracket, root, nfeval, id = case
|
|
# Chandrupatla's criterion is equivalent to
|
|
# abs(x2-x1) < 4*abs(xmin)*xrtol + xatol, but we use the more standard
|
|
# abs(x2-x1) < abs(xmin)*xrtol + xatol. Therefore, set xrtol to 4x
|
|
# that used by Chandrupatla in tests.
|
|
bracket = (xp.asarray(bracket[0], dtype=xp.float64),
|
|
xp.asarray(bracket[1], dtype=xp.float64))
|
|
root = xp.asarray(root, dtype=xp.float64)
|
|
|
|
res = find_root(f, bracket, tolerances=dict(xrtol=4e-10, xatol=1e-5))
|
|
xp_assert_close(res.f_x, xp.asarray(f(root), dtype=xp.float64),
|
|
rtol=1e-8, atol=2e-3)
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xp_assert_equal(res.nfev, xp.asarray(nfeval, dtype=xp.int32))
|
|
|
|
@pytest.mark.parametrize("root", (0.622, [0.622, 0.623]))
|
|
@pytest.mark.parametrize("dtype", ('float16', 'float32', 'float64'))
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|
def test_dtype(self, root, dtype, xp):
|
|
# Test that dtypes are preserved
|
|
not_numpy = not is_numpy(xp)
|
|
if not_numpy and dtype == 'float16':
|
|
pytest.skip("`float16` dtype only supported for NumPy arrays.")
|
|
|
|
dtype = getattr(xp, dtype, None)
|
|
if dtype is None:
|
|
pytest.skip(f"{xp} does not support {dtype}")
|
|
|
|
def f(x, root):
|
|
res = (x - root) ** 3.
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|
if is_numpy(xp): # NumPy does not preserve dtype
|
|
return xp.asarray(res, dtype=dtype)
|
|
return res
|
|
|
|
a, b = xp.asarray(-3, dtype=dtype), xp.asarray(3, dtype=dtype)
|
|
root = xp.asarray(root, dtype=dtype)
|
|
res = find_root(f, (a, b), args=(root,), tolerances={'xatol': 1e-3})
|
|
try:
|
|
xp_assert_close(res.x, root, atol=1e-3)
|
|
except AssertionError:
|
|
assert res.x.dtype == dtype
|
|
xp.all(res.f_x == 0)
|
|
|
|
def test_input_validation(self, xp):
|
|
# Test input validation for appropriate error messages
|
|
|
|
def func(x):
|
|
return x
|
|
|
|
message = '`func` must be callable.'
|
|
with pytest.raises(ValueError, match=message):
|
|
bracket = xp.asarray(-4), xp.asarray(4)
|
|
find_root(None, bracket)
|
|
|
|
message = 'Abscissae and function output must be real numbers.'
|
|
with pytest.raises(ValueError, match=message):
|
|
bracket = xp.asarray(-4+1j), xp.asarray(4)
|
|
find_root(func, bracket)
|
|
|
|
# raised by `np.broadcast, but the traceback is readable IMO
|
|
# all messages include this part
|
|
message = "(not be broadcast|Attempting to broadcast a dimension of length)"
|
|
with pytest.raises((ValueError, RuntimeError), match=message):
|
|
bracket = xp.asarray([-2, -3]), xp.asarray([3, 4, 5])
|
|
find_root(func, bracket)
|
|
|
|
message = "The shape of the array returned by `func`..."
|
|
with pytest.raises(ValueError, match=message):
|
|
bracket = xp.asarray([-3, -3]), xp.asarray([5, 5])
|
|
find_root(lambda x: [x[0], x[1], x[1]], bracket)
|
|
|
|
message = 'Tolerances must be non-negative scalars.'
|
|
bracket = xp.asarray(-4), xp.asarray(4)
|
|
with pytest.raises(ValueError, match=message):
|
|
find_root(func, bracket, tolerances=dict(xatol=-1))
|
|
with pytest.raises(ValueError, match=message):
|
|
find_root(func, bracket, tolerances=dict(xrtol=xp.nan))
|
|
with pytest.raises(ValueError, match=message):
|
|
find_root(func, bracket, tolerances=dict(fatol='ekki'))
|
|
with pytest.raises(ValueError, match=message):
|
|
find_root(func, bracket, tolerances=dict(frtol=xp.nan))
|
|
|
|
message = '`maxiter` must be a non-negative integer.'
|
|
with pytest.raises(ValueError, match=message):
|
|
find_root(func, bracket, maxiter=1.5)
|
|
with pytest.raises(ValueError, match=message):
|
|
find_root(func, bracket, maxiter=-1)
|
|
|
|
message = '`callback` must be callable.'
|
|
with pytest.raises(ValueError, match=message):
|
|
find_root(func, bracket, callback='shrubbery')
|
|
|
|
def test_special_cases(self, xp):
|
|
# Test edge cases and other special cases
|
|
|
|
# Test infinite function values
|
|
def f(x):
|
|
return 1 / x + 1 - 1 / (-x + 1)
|
|
|
|
a, b = xp.asarray([0.1, 0., 0., 0.1]), xp.asarray([0.9, 1.0, 0.9, 1.0])
|
|
|
|
with np.errstate(divide='ignore', invalid='ignore'):
|
|
res = find_root(f, (a, b))
|
|
|
|
assert xp.all(res.success)
|
|
xp_assert_close(res.x[1:], xp.full((3,), res.x[0]))
|
|
|
|
# Test that integers are not passed to `f`
|
|
# (otherwise this would overflow)
|
|
def f(x):
|
|
assert xp.isdtype(x.dtype, "real floating")
|
|
# this would overflow if x were an xp integer dtype
|
|
return x ** 31 - 1
|
|
|
|
# note that all inputs are integer type; result is automatically default float
|
|
res = find_root(f, (xp.asarray(-7), xp.asarray(5)))
|
|
assert res.success
|
|
xp_assert_close(res.x, xp.asarray(1.))
|
|
|
|
# Test that if both ends of bracket equal root, algorithm reports
|
|
# convergence.
|
|
def f(x, root):
|
|
return x**2 - root
|
|
|
|
root = xp.asarray([0, 1])
|
|
res = find_root(f, (xp.asarray(1), xp.asarray(1)), args=(root,))
|
|
xp_assert_equal(res.success, xp.asarray([False, True]))
|
|
xp_assert_equal(res.x, xp.asarray([xp.nan, 1.]))
|
|
|
|
def f(x):
|
|
return 1/x
|
|
|
|
with np.errstate(invalid='ignore'):
|
|
inf = xp.asarray(xp.inf)
|
|
res = find_root(f, (inf, inf))
|
|
assert res.success
|
|
xp_assert_equal(res.x, xp.asarray(xp.inf))
|
|
|
|
# Test maxiter = 0. Should do nothing to bracket.
|
|
def f(x):
|
|
return x**3 - 1
|
|
|
|
a, b = xp.asarray(-3.), xp.asarray(5.)
|
|
res = find_root(f, (a, b), maxiter=0)
|
|
xp_assert_equal(res.success, xp.asarray(False))
|
|
xp_assert_equal(res.status, xp.asarray(-2, dtype=xp.int32))
|
|
xp_assert_equal(res.nit, xp.asarray(0, dtype=xp.int32))
|
|
xp_assert_equal(res.nfev, xp.asarray(2, dtype=xp.int32))
|
|
xp_assert_equal(res.bracket[0], a)
|
|
xp_assert_equal(res.bracket[1], b)
|
|
# The `x` attribute is the one with the smaller function value
|
|
xp_assert_equal(res.x, a)
|
|
# Reverse bracket; check that this is still true
|
|
res = find_root(f, (-b, -a), maxiter=0)
|
|
xp_assert_equal(res.x, -a)
|
|
|
|
# Test maxiter = 1
|
|
res = find_root(f, (a, b), maxiter=1)
|
|
xp_assert_equal(res.success, xp.asarray(True))
|
|
xp_assert_equal(res.status, xp.asarray(0, dtype=xp.int32))
|
|
xp_assert_equal(res.nit, xp.asarray(1, dtype=xp.int32))
|
|
xp_assert_equal(res.nfev, xp.asarray(3, dtype=xp.int32))
|
|
xp_assert_close(res.x, xp.asarray(1.))
|
|
|
|
# Test scalar `args` (not in tuple)
|
|
def f(x, c):
|
|
return c*x - 1
|
|
|
|
res = find_root(f, (xp.asarray(-1), xp.asarray(1)), args=xp.asarray(3))
|
|
xp_assert_close(res.x, xp.asarray(1/3))
|
|
|
|
# # TODO: Test zero tolerance
|
|
# # ~~What's going on here - why are iterations repeated?~~
|
|
# # tl goes to zero when xatol=xrtol=0. When function is nearly linear,
|
|
# # this causes convergence issues.
|
|
# def f(x):
|
|
# return np.cos(x)
|
|
#
|
|
# res = _chandrupatla_root(f, 0, np.pi, xatol=0, xrtol=0)
|
|
# assert res.nit < 100
|
|
# xp = np.nextafter(res.x, np.inf)
|
|
# xm = np.nextafter(res.x, -np.inf)
|
|
# assert np.abs(res.fun) < np.abs(f(xp))
|
|
# assert np.abs(res.fun) < np.abs(f(xm))
|