import pytest
import numpy as np
from numpy.testing import assert_array_less, assert_allclose, assert_equal
import scipy._lib._elementwise_iterative_method as eim
from scipy import stats
from scipy.optimize._differentiate import (_differentiate as differentiate,
_EERRORINCREASE)
class TestDifferentiate:
def f(self, x):
return stats.norm().cdf(x)
@pytest.mark.parametrize('x', [0.6, np.linspace(-0.05, 1.05, 10)])
def test_basic(self, x):
# Invert distribution CDF and compare against distribution `ppf`
res = differentiate(self.f, x)
ref = stats.norm().pdf(x)
np.testing.assert_allclose(res.df, ref)
# This would be nice, but doesn't always work out. `error` is an
# estimate, not a bound.
assert_array_less(abs(res.df - ref), res.error)
assert res.x.shape == ref.shape
@pytest.mark.parametrize('case', stats._distr_params.distcont)
def test_accuracy(self, case):
distname, params = case
dist = getattr(stats, distname)(*params)
x = dist.median() + 0.1
res = differentiate(dist.cdf, x)
ref = dist.pdf(x)
assert_allclose(res.df, ref, atol=1e-10)
@pytest.mark.parametrize('order', [1, 6])
@pytest.mark.parametrize('shape', [tuple(), (12,), (3, 4), (3, 2, 2)])
def test_vectorization(self, order, shape):
# Test for correct functionality, output shapes, and dtypes for various
# input shapes.
x = np.linspace(-0.05, 1.05, 12).reshape(shape) if shape else 0.6
n = np.size(x)
@np.vectorize
def _differentiate_single(x):
return differentiate(self.f, x, order=order)
def f(x, *args, **kwargs):
f.nit += 1
f.feval += 1 if (x.size == n or x.ndim <=1) else x.shape[-1]
return self.f(x, *args, **kwargs)
f.nit = -1
f.feval = 0
res = differentiate(f, x, order=order)
refs = _differentiate_single(x).ravel()
ref_x = [ref.x for ref in refs]
assert_allclose(res.x.ravel(), ref_x)
assert_equal(res.x.shape, shape)
ref_df = [ref.df for ref in refs]
assert_allclose(res.df.ravel(), ref_df)
assert_equal(res.df.shape, shape)
ref_error = [ref.error for ref in refs]
assert_allclose(res.error.ravel(), ref_error, atol=5e-15)
assert_equal(res.error.shape, shape)
ref_success = [ref.success for ref in refs]
assert_equal(res.success.ravel(), ref_success)
assert_equal(res.success.shape, shape)
assert np.issubdtype(res.success.dtype, np.bool_)
ref_flag = [ref.status for ref in refs]
assert_equal(res.status.ravel(), ref_flag)
assert_equal(res.status.shape, shape)
assert np.issubdtype(res.status.dtype, np.integer)
ref_nfev = [ref.nfev for ref in refs]
assert_equal(res.nfev.ravel(), ref_nfev)
assert_equal(np.max(res.nfev), f.feval)
assert_equal(res.nfev.shape, res.x.shape)
assert np.issubdtype(res.nfev.dtype, np.integer)
ref_nit = [ref.nit for ref in refs]
assert_equal(res.nit.ravel(), ref_nit)
assert_equal(np.max(res.nit), f.nit)
assert_equal(res.nit.shape, res.x.shape)
assert np.issubdtype(res.nit.dtype, np.integer)
def test_flags(self):
# Test cases that should produce different status flags; show that all
# can be produced simultaneously.
rng = np.random.default_rng(5651219684984213)
def f(xs, js):
f.nit += 1
funcs = [lambda x: x - 2.5, # converges
lambda x: np.exp(x)*rng.random(), # error increases
lambda x: np.exp(x), # reaches maxiter due to order=2
lambda x: np.full_like(x, np.nan)[()]] # stops due to NaN
res = [funcs[j](x) for x, j in zip(xs, js.ravel())]
return res
f.nit = 0
args = (np.arange(4, dtype=np.int64),)
res = differentiate(f, [1]*4, rtol=1e-14, order=2, args=args)
ref_flags = np.array([eim._ECONVERGED,
_EERRORINCREASE,
eim._ECONVERR,
eim._EVALUEERR])
assert_equal(res.status, ref_flags)
def test_flags_preserve_shape(self):
# Same test as above but using `preserve_shape` option to simplify.
rng = np.random.default_rng(5651219684984213)
def f(x):
return [x - 2.5, # converges
np.exp(x)*rng.random(), # error increases
np.exp(x), # reaches maxiter due to order=2
np.full_like(x, np.nan)[()]] # stops due to NaN
res = differentiate(f, 1, rtol=1e-14, order=2, preserve_shape=True)
ref_flags = np.array([eim._ECONVERGED,
_EERRORINCREASE,
eim._ECONVERR,
eim._EVALUEERR])
assert_equal(res.status, ref_flags)
def test_preserve_shape(self):
# Test `preserve_shape` option
def f(x):
return [x, np.sin(3*x), x+np.sin(10*x), np.sin(20*x)*(x-1)**2]
x = 0
ref = [1, 3*np.cos(3*x), 1+10*np.cos(10*x),
20*np.cos(20*x)*(x-1)**2 + 2*np.sin(20*x)*(x-1)]
res = differentiate(f, x, preserve_shape=True)
assert_allclose(res.df, ref)
def test_convergence(self):
# Test that the convergence tolerances behave as expected
dist = stats.norm()
x = 1
f = dist.cdf
ref = dist.pdf(x)
kwargs0 = dict(atol=0, rtol=0, order=4)
kwargs = kwargs0.copy()
kwargs['atol'] = 1e-3
res1 = differentiate(f, x, **kwargs)
assert_array_less(abs(res1.df - ref), 1e-3)
kwargs['atol'] = 1e-6
res2 = differentiate(f, x, **kwargs)
assert_array_less(abs(res2.df - ref), 1e-6)
assert_array_less(abs(res2.df - ref), abs(res1.df - ref))
kwargs = kwargs0.copy()
kwargs['rtol'] = 1e-3
res1 = differentiate(f, x, **kwargs)
assert_array_less(abs(res1.df - ref), 1e-3 * np.abs(ref))
kwargs['rtol'] = 1e-6
res2 = differentiate(f, x, **kwargs)
assert_array_less(abs(res2.df - ref), 1e-6 * np.abs(ref))
assert_array_less(abs(res2.df - ref), abs(res1.df - ref))
def test_step_parameters(self):
# Test that step factors have the expected effect on accuracy
dist = stats.norm()
x = 1
f = dist.cdf
ref = dist.pdf(x)
res1 = differentiate(f, x, initial_step=0.5, maxiter=1)
res2 = differentiate(f, x, initial_step=0.05, maxiter=1)
assert abs(res2.df - ref) < abs(res1.df - ref)
res1 = differentiate(f, x, step_factor=2, maxiter=1)
res2 = differentiate(f, x, step_factor=20, maxiter=1)
assert abs(res2.df - ref) < abs(res1.df - ref)
# `step_factor` can be less than 1: `initial_step` is the minimum step
kwargs = dict(order=4, maxiter=1, step_direction=0)
res = differentiate(f, x, initial_step=0.5, step_factor=0.5, **kwargs)
ref = differentiate(f, x, initial_step=1, step_factor=2, **kwargs)
assert_allclose(res.df, ref.df, rtol=5e-15)
# This is a similar test for one-sided difference
kwargs = dict(order=2, maxiter=1, step_direction=1)
res = differentiate(f, x, initial_step=1, step_factor=2, **kwargs)
ref = differentiate(f, x, initial_step=1/np.sqrt(2), step_factor=0.5,
**kwargs)
assert_allclose(res.df, ref.df, rtol=5e-15)
kwargs['step_direction'] = -1
res = differentiate(f, x, initial_step=1, step_factor=2, **kwargs)
ref = differentiate(f, x, initial_step=1/np.sqrt(2), step_factor=0.5,
**kwargs)
assert_allclose(res.df, ref.df, rtol=5e-15)
def test_step_direction(self):
# test that `step_direction` works as expected
def f(x):
y = np.exp(x)
y[(x < 0) + (x > 2)] = np.nan
return y
x = np.linspace(0, 2, 10)
step_direction = np.zeros_like(x)
step_direction[x < 0.6], step_direction[x > 1.4] = 1, -1
res = differentiate(f, x, step_direction=step_direction)
assert_allclose(res.df, np.exp(x))
assert np.all(res.success)
def test_vectorized_step_direction_args(self):
# test that `step_direction` and `args` are vectorized properly
def f(x, p):
return x ** p
def df(x, p):
return p * x ** (p - 1)
x = np.array([1, 2, 3, 4]).reshape(-1, 1, 1)
hdir = np.array([-1, 0, 1]).reshape(1, -1, 1)
p = np.array([2, 3]).reshape(1, 1, -1)
res = differentiate(f, x, step_direction=hdir, args=(p,))
ref = np.broadcast_to(df(x, p), res.df.shape)
assert_allclose(res.df, ref)
def test_maxiter_callback(self):
# Test behavior of `maxiter` parameter and `callback` interface
x = 0.612814
dist = stats.norm()
maxiter = 3
def f(x):
res = dist.cdf(x)
return res
default_order = 8
res = differentiate(f, x, maxiter=maxiter, rtol=1e-15)
assert not np.any(res.success)
assert np.all(res.nfev == default_order + 1 + (maxiter - 1)*2)
assert np.all(res.nit == maxiter)
def callback(res):
callback.iter += 1
callback.res = res
assert hasattr(res, 'x')
assert res.df not in callback.dfs
callback.dfs.add(res.df)
assert res.status == eim._EINPROGRESS
if callback.iter == maxiter:
raise StopIteration
callback.iter = -1 # callback called once before first iteration
callback.res = None
callback.dfs = set()
res2 = differentiate(f, x, callback=callback, rtol=1e-15)
# terminating with callback is identical to terminating due to maxiter
# (except for `status`)
for key in res.keys():
if key == 'status':
assert res[key] == eim._ECONVERR
assert callback.res[key] == eim._EINPROGRESS
assert res2[key] == eim._ECALLBACK
else:
assert res2[key] == callback.res[key] == res[key]
@pytest.mark.parametrize("hdir", (-1, 0, 1))
@pytest.mark.parametrize("x", (0.65, [0.65, 0.7]))
@pytest.mark.parametrize("dtype", (np.float16, np.float32, np.float64))
def test_dtype(self, hdir, x, dtype):
# Test that dtypes are preserved
x = np.asarray(x, dtype=dtype)[()]
def f(x):
assert x.dtype == dtype
return np.exp(x)
def callback(res):
assert res.x.dtype == dtype
assert res.df.dtype == dtype
assert res.error.dtype == dtype
res = differentiate(f, x, order=4, step_direction=hdir,
callback=callback)
assert res.x.dtype == dtype
assert res.df.dtype == dtype
assert res.error.dtype == dtype
eps = np.finfo(dtype).eps
assert_allclose(res.df, np.exp(res.x), rtol=np.sqrt(eps))
def test_input_validation(self):
# Test input validation for appropriate error messages
message = '`func` must be callable.'
with pytest.raises(ValueError, match=message):
differentiate(None, 1)
message = 'Abscissae and function output must be real numbers.'
with pytest.raises(ValueError, match=message):
differentiate(lambda x: x, -4+1j)
message = "When `preserve_shape=False`, the shape of the array..."
with pytest.raises(ValueError, match=message):
differentiate(lambda x: [1, 2, 3], [-2, -3])
message = 'Tolerances and step parameters must be non-negative...'
with pytest.raises(ValueError, match=message):
differentiate(lambda x: x, 1, atol=-1)
with pytest.raises(ValueError, match=message):
differentiate(lambda x: x, 1, rtol='ekki')
with pytest.raises(ValueError, match=message):
differentiate(lambda x: x, 1, initial_step=None)
with pytest.raises(ValueError, match=message):
differentiate(lambda x: x, 1, step_factor=object())
message = '`maxiter` must be a positive integer.'
with pytest.raises(ValueError, match=message):
differentiate(lambda x: x, 1, maxiter=1.5)
with pytest.raises(ValueError, match=message):
differentiate(lambda x: x, 1, maxiter=0)
message = '`order` must be a positive integer'
with pytest.raises(ValueError, match=message):
differentiate(lambda x: x, 1, order=1.5)
with pytest.raises(ValueError, match=message):
differentiate(lambda x: x, 1, order=0)
message = '`preserve_shape` must be True or False.'
with pytest.raises(ValueError, match=message):
differentiate(lambda x: x, 1, preserve_shape='herring')
message = '`callback` must be callable.'
with pytest.raises(ValueError, match=message):
differentiate(lambda x: x, 1, callback='shrubbery')
def test_special_cases(self):
# Test edge cases and other special cases
# Test that integers are not passed to `f`
# (otherwise this would overflow)
def f(x):
assert np.issubdtype(x.dtype, np.floating)
return x ** 99 - 1
res = differentiate(f, 7, rtol=1e-10)
assert res.success
assert_allclose(res.df, 99*7.**98)
# Test that if success is achieved in the correct number
# of iterations if function is a polynomial. Ideally, all polynomials
# of order 0-2 would get exact result with 0 refinement iterations,
# all polynomials of order 3-4 would be differentiated exactly after
# 1 iteration, etc. However, it seems that _differentiate needs an
# extra iteration to detect convergence based on the error estimate.
for n in range(6):
x = 1.5
def f(x):
return 2*x**n
ref = 2*n*x**(n-1)
res = differentiate(f, x, maxiter=1, order=max(1, n))
assert_allclose(res.df, ref, rtol=1e-15)
assert_equal(res.error, np.nan)
res = differentiate(f, x, order=max(1, n))
assert res.success
assert res.nit == 2
assert_allclose(res.df, ref, rtol=1e-15)
# Test scalar `args` (not in tuple)
def f(x, c):
return c*x - 1
res = differentiate(f, 2, args=3)
assert_allclose(res.df, 3)
@pytest.mark.xfail
@pytest.mark.parametrize("case", ( # function, evaluation point
(lambda x: (x - 1) ** 3, 1),
(lambda x: np.where(x > 1, (x - 1) ** 5, (x - 1) ** 3), 1)
))
def test_saddle_gh18811(self, case):
# With default settings, _differentiate will not always converge when
# the true derivative is exactly zero. This tests that specifying a
# (tight) `atol` alleviates the problem. See discussion in gh-18811.
atol = 1e-16
res = differentiate(*case, step_direction=[-1, 0, 1], atol=atol)
assert np.all(res.success)
assert_allclose(res.df, 0, atol=atol)