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CCR/.venv/lib/python3.12/site-packages/scipy/differentiate/tests/test_differentiate.py

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import math
import pytest
import numpy as np
from scipy.conftest import array_api_compatible
import scipy._lib._elementwise_iterative_method as eim
from scipy._lib._array_api_no_0d import xp_assert_close, xp_assert_equal, xp_assert_less
from scipy._lib._array_api import is_numpy, is_torch, array_namespace
from scipy import stats, optimize, special
from scipy.differentiate import derivative, jacobian, hessian
from scipy.differentiate._differentiate import _EERRORINCREASE
pytestmark = [array_api_compatible, pytest.mark.usefixtures("skip_xp_backends")]
array_api_strict_skip_reason = 'Array API does not support fancy indexing assignment.'
jax_skip_reason = 'JAX arrays do not support item assignment.'
@pytest.mark.skip_xp_backends('array_api_strict', reason=array_api_strict_skip_reason)
@pytest.mark.skip_xp_backends('jax.numpy',reason=jax_skip_reason)
class TestDerivative:
def f(self, x):
return special.ndtr(x)
@pytest.mark.parametrize('x', [0.6, np.linspace(-0.05, 1.05, 10)])
def test_basic(self, x, xp):
# Invert distribution CDF and compare against distribution `ppf`
default_dtype = xp.asarray(1.).dtype
res = derivative(self.f, xp.asarray(x, dtype=default_dtype))
ref = xp.asarray(stats.norm().pdf(x), dtype=default_dtype)
xp_assert_close(res.df, ref)
# This would be nice, but doesn't always work out. `error` is an
# estimate, not a bound.
if not is_torch(xp):
xp_assert_less(xp.abs(res.df - ref), res.error)
@pytest.mark.skip_xp_backends(np_only=True)
@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 = derivative(dist.cdf, x)
ref = dist.pdf(x)
xp_assert_close(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, xp):
# 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)
state = {}
@np.vectorize
def _derivative_single(x):
return derivative(self.f, x, order=order)
def f(x, *args, **kwargs):
state['nit'] += 1
state['feval'] += 1 if (x.size == n or x.ndim <=1) else x.shape[-1]
return self.f(x, *args, **kwargs)
state['nit'] = -1
state['feval'] = 0
res = derivative(f, xp.asarray(x, dtype=xp.float64), order=order)
refs = _derivative_single(x).ravel()
ref_x = [ref.x for ref in refs]
xp_assert_close(xp.reshape(res.x, (-1,)), xp.asarray(ref_x))
ref_df = [ref.df for ref in refs]
xp_assert_close(xp.reshape(res.df, (-1,)), xp.asarray(ref_df))
ref_error = [ref.error for ref in refs]
xp_assert_close(xp.reshape(res.error, (-1,)), xp.asarray(ref_error),
atol=1e-12)
ref_success = [bool(ref.success) for ref in refs]
xp_assert_equal(xp.reshape(res.success, (-1,)), xp.asarray(ref_success))
ref_flag = [np.int32(ref.status) for ref in refs]
xp_assert_equal(xp.reshape(res.status, (-1,)), xp.asarray(ref_flag))
ref_nfev = [np.int32(ref.nfev) for ref in refs]
xp_assert_equal(xp.reshape(res.nfev, (-1,)), xp.asarray(ref_nfev))
if is_numpy(xp): # can't expect other backends to be exactly the same
assert xp.max(res.nfev) == state['feval']
ref_nit = [np.int32(ref.nit) for ref in refs]
xp_assert_equal(xp.reshape(res.nit, (-1,)), xp.asarray(ref_nit))
if is_numpy(xp): # can't expect other backends to be exactly the same
assert xp.max(res.nit) == state['nit']
def test_flags(self, xp):
# 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: xp.exp(x)*rng.random(), # error increases
lambda x: xp.exp(x), # reaches maxiter due to order=2
lambda x: xp.full_like(x, xp.nan)] # stops due to NaN
res = [funcs[int(j)](x) for x, j in zip(xs, xp.reshape(js, (-1,)))]
return xp.stack(res)
f.nit = 0
args = (xp.arange(4, dtype=xp.int64),)
res = derivative(f, xp.ones(4, dtype=xp.float64),
tolerances=dict(rtol=1e-14),
order=2, args=args)
ref_flags = xp.asarray([eim._ECONVERGED,
_EERRORINCREASE,
eim._ECONVERR,
eim._EVALUEERR], dtype=xp.int32)
xp_assert_equal(res.status, ref_flags)
def test_flags_preserve_shape(self, xp):
# Same test as above but using `preserve_shape` option to simplify.
rng = np.random.default_rng(5651219684984213)
def f(x):
out = [x - 2.5, # converges
xp.exp(x)*rng.random(), # error increases
xp.exp(x), # reaches maxiter due to order=2
xp.full_like(x, xp.nan)] # stops due to NaN
return xp.stack(out)
res = derivative(f, xp.asarray(1, dtype=xp.float64),
tolerances=dict(rtol=1e-14),
order=2, preserve_shape=True)
ref_flags = xp.asarray([eim._ECONVERGED,
_EERRORINCREASE,
eim._ECONVERR,
eim._EVALUEERR], dtype=xp.int32)
xp_assert_equal(res.status, ref_flags)
def test_preserve_shape(self, xp):
# Test `preserve_shape` option
def f(x):
out = [x, xp.sin(3*x), x+xp.sin(10*x), xp.sin(20*x)*(x-1)**2]
return xp.stack(out)
x = xp.asarray(0.)
ref = xp.asarray([xp.asarray(1), 3*xp.cos(3*x), 1+10*xp.cos(10*x),
20*xp.cos(20*x)*(x-1)**2 + 2*xp.sin(20*x)*(x-1)])
res = derivative(f, x, preserve_shape=True)
xp_assert_close(res.df, ref)
def test_convergence(self, xp):
# Test that the convergence tolerances behave as expected
x = xp.asarray(1., dtype=xp.float64)
f = special.ndtr
ref = float(stats.norm.pdf(1.))
tolerances0 = dict(atol=0, rtol=0)
tolerances = tolerances0.copy()
tolerances['atol'] = 1e-3
res1 = derivative(f, x, tolerances=tolerances, order=4)
assert abs(res1.df - ref) < 1e-3
tolerances['atol'] = 1e-6
res2 = derivative(f, x, tolerances=tolerances, order=4)
assert abs(res2.df - ref) < 1e-6
assert abs(res2.df - ref) < abs(res1.df - ref)
tolerances = tolerances0.copy()
tolerances['rtol'] = 1e-3
res1 = derivative(f, x, tolerances=tolerances, order=4)
assert abs(res1.df - ref) < 1e-3 * ref
tolerances['rtol'] = 1e-6
res2 = derivative(f, x, tolerances=tolerances, order=4)
assert abs(res2.df - ref) < 1e-6 * ref
assert abs(res2.df - ref) < abs(res1.df - ref)
def test_step_parameters(self, xp):
# Test that step factors have the expected effect on accuracy
x = xp.asarray(1., dtype=xp.float64)
f = special.ndtr
ref = float(stats.norm.pdf(1.))
res1 = derivative(f, x, initial_step=0.5, maxiter=1)
res2 = derivative(f, x, initial_step=0.05, maxiter=1)
assert abs(res2.df - ref) < abs(res1.df - ref)
res1 = derivative(f, x, step_factor=2, maxiter=1)
res2 = derivative(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 = derivative(f, x, initial_step=0.5, step_factor=0.5, **kwargs)
ref = derivative(f, x, initial_step=1, step_factor=2, **kwargs)
xp_assert_close(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 = derivative(f, x, initial_step=1, step_factor=2, **kwargs)
ref = derivative(f, x, initial_step=1/np.sqrt(2), step_factor=0.5, **kwargs)
xp_assert_close(res.df, ref.df, rtol=5e-15)
kwargs['step_direction'] = -1
res = derivative(f, x, initial_step=1, step_factor=2, **kwargs)
ref = derivative(f, x, initial_step=1/np.sqrt(2), step_factor=0.5, **kwargs)
xp_assert_close(res.df, ref.df, rtol=5e-15)
def test_step_direction(self, xp):
# test that `step_direction` works as expected
def f(x):
y = xp.exp(x)
y[(x < 0) + (x > 2)] = xp.nan
return y
x = xp.linspace(0, 2, 10)
step_direction = xp.zeros_like(x)
step_direction[x < 0.6], step_direction[x > 1.4] = 1, -1
res = derivative(f, x, step_direction=step_direction)
xp_assert_close(res.df, xp.exp(x))
assert xp.all(res.success)
def test_vectorized_step_direction_args(self, xp):
# 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 = xp.reshape(xp.asarray([1, 2, 3, 4]), (-1, 1, 1))
hdir = xp.reshape(xp.asarray([-1, 0, 1]), (1, -1, 1))
p = xp.reshape(xp.asarray([2, 3]), (1, 1, -1))
res = derivative(f, x, step_direction=hdir, args=(p,))
ref = xp.broadcast_to(df(x, p), res.df.shape)
ref = xp.asarray(ref, dtype=xp.asarray(1.).dtype)
xp_assert_close(res.df, ref)
def test_initial_step(self, xp):
# Test that `initial_step` works as expected and is vectorized
def f(x):
return xp.exp(x)
x = xp.asarray(0., dtype=xp.float64)
step_direction = xp.asarray([-1, 0, 1])
h0 = xp.reshape(xp.logspace(-3, 0, 10), (-1, 1))
res = derivative(f, x, initial_step=h0, order=2, maxiter=1,
step_direction=step_direction)
err = xp.abs(res.df - f(x))
# error should be smaller for smaller step sizes
assert xp.all(err[:-1, ...] < err[1:, ...])
# results of vectorized call should match results with
# initial_step taken one at a time
for i in range(h0.shape[0]):
ref = derivative(f, x, initial_step=h0[i, 0], order=2, maxiter=1,
step_direction=step_direction)
xp_assert_close(res.df[i, :], ref.df, rtol=1e-14)
def test_maxiter_callback(self, xp):
# Test behavior of `maxiter` parameter and `callback` interface
x = xp.asarray(0.612814, dtype=xp.float64)
maxiter = 3
def f(x):
res = special.ndtr(x)
return res
default_order = 8
res = derivative(f, x, maxiter=maxiter, tolerances=dict(rtol=1e-15))
assert not xp.any(res.success)
assert xp.all(res.nfev == default_order + 1 + (maxiter - 1)*2)
assert xp.all(res.nit == maxiter)
def callback(res):
callback.iter += 1
callback.res = res
assert hasattr(res, 'x')
assert float(res.df) not in callback.dfs
callback.dfs.add(float(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 = derivative(f, x, callback=callback, tolerances=dict(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 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", ('float16', 'float32', 'float64'))
def test_dtype(self, hdir, x, dtype, xp):
if dtype == 'float16' and not is_numpy(xp):
pytest.skip('float16 not tested for alternative backends')
# Test that dtypes are preserved
dtype = getattr(xp, dtype)
x = xp.asarray(x, dtype=dtype)
def f(x):
assert x.dtype == dtype
return xp.exp(x)
def callback(res):
assert res.x.dtype == dtype
assert res.df.dtype == dtype
assert res.error.dtype == dtype
res = derivative(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 = xp.finfo(dtype).eps
# not sure why torch is less accurate here; might be worth investigating
rtol = eps**0.5 * 50 if is_torch(xp) else eps**0.5
xp_assert_close(res.df, xp.exp(res.x), rtol=rtol)
def test_input_validation(self, xp):
# Test input validation for appropriate error messages
one = xp.asarray(1)
message = '`f` must be callable.'
with pytest.raises(ValueError, match=message):
derivative(None, one)
message = 'Abscissae and function output must be real numbers.'
with pytest.raises(ValueError, match=message):
derivative(lambda x: x, xp.asarray(-4+1j))
message = "When `preserve_shape=False`, the shape of the array..."
with pytest.raises(ValueError, match=message):
derivative(lambda x: [1, 2, 3], xp.asarray([-2, -3]))
message = 'Tolerances and step parameters must be non-negative...'
with pytest.raises(ValueError, match=message):
derivative(lambda x: x, one, tolerances=dict(atol=-1))
with pytest.raises(ValueError, match=message):
derivative(lambda x: x, one, tolerances=dict(rtol='ekki'))
with pytest.raises(ValueError, match=message):
derivative(lambda x: x, one, step_factor=object())
message = '`maxiter` must be a positive integer.'
with pytest.raises(ValueError, match=message):
derivative(lambda x: x, one, maxiter=1.5)
with pytest.raises(ValueError, match=message):
derivative(lambda x: x, one, maxiter=0)
message = '`order` must be a positive integer'
with pytest.raises(ValueError, match=message):
derivative(lambda x: x, one, order=1.5)
with pytest.raises(ValueError, match=message):
derivative(lambda x: x, one, order=0)
message = '`preserve_shape` must be True or False.'
with pytest.raises(ValueError, match=message):
derivative(lambda x: x, one, preserve_shape='herring')
message = '`callback` must be callable.'
with pytest.raises(ValueError, match=message):
derivative(lambda x: x, one, callback='shrubbery')
def test_special_cases(self, xp):
# Test edge cases and other special cases
# Test that integers are not passed to `f`
# (otherwise this would overflow)
def f(x):
xp_test = array_namespace(x) # needs `isdtype`
assert xp_test.isdtype(x.dtype, 'real floating')
return x ** 99 - 1
if not is_torch(xp): # torch defaults to float32
res = derivative(f, xp.asarray(7), tolerances=dict(rtol=1e-10))
assert res.success
xp_assert_close(res.df, xp.asarray(99*7.**98))
# Test invalid step size and direction
res = derivative(xp.exp, xp.asarray(1), step_direction=xp.nan)
xp_assert_equal(res.df, xp.asarray(xp.nan))
xp_assert_equal(res.status, xp.asarray(-3, dtype=xp.int32))
res = derivative(xp.exp, xp.asarray(1), initial_step=0)
xp_assert_equal(res.df, xp.asarray(xp.nan))
xp_assert_equal(res.status, xp.asarray(-3, dtype=xp.int32))
# 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 `derivative` needs an
# extra iteration to detect convergence based on the error estimate.
for n in range(6):
x = xp.asarray(1.5, dtype=xp.float64)
def f(x):
return 2*x**n
ref = 2*n*x**(n-1)
res = derivative(f, x, maxiter=1, order=max(1, n))
xp_assert_close(res.df, ref, rtol=1e-15)
xp_assert_equal(res.error, xp.asarray(xp.nan, dtype=xp.float64))
res = derivative(f, x, order=max(1, n))
assert res.success
assert res.nit == 2
xp_assert_close(res.df, ref, rtol=1e-15)
# Test scalar `args` (not in tuple)
def f(x, c):
return c*x - 1
res = derivative(f, xp.asarray(2), args=xp.asarray(3))
xp_assert_close(res.df, xp.asarray(3.))
# no need to run a test on multiple backends if it's xfailed
@pytest.mark.skip_xp_backends(np_only=True)
@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, `derivative` 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 = derivative(*case, step_direction=[-1, 0, 1], atol=atol)
assert np.all(res.success)
xp_assert_close(res.df, 0, atol=atol)
class JacobianHessianTest:
def test_iv(self, xp):
jh_func = self.jh_func.__func__
# Test input validation
message = "Argument `x` must be at least 1-D."
with pytest.raises(ValueError, match=message):
jh_func(xp.sin, 1, tolerances=dict(atol=-1))
# Confirm that other parameters are being passed to `derivative`,
# which raises an appropriate error message.
x = xp.ones(3)
func = optimize.rosen
message = 'Tolerances and step parameters must be non-negative scalars.'
with pytest.raises(ValueError, match=message):
jh_func(func, x, tolerances=dict(atol=-1))
with pytest.raises(ValueError, match=message):
jh_func(func, x, tolerances=dict(rtol=-1))
with pytest.raises(ValueError, match=message):
jh_func(func, x, step_factor=-1)
message = '`order` must be a positive integer.'
with pytest.raises(ValueError, match=message):
jh_func(func, x, order=-1)
message = '`maxiter` must be a positive integer.'
with pytest.raises(ValueError, match=message):
jh_func(func, x, maxiter=-1)
@pytest.mark.skip_xp_backends('array_api_strict', reason=array_api_strict_skip_reason)
@pytest.mark.skip_xp_backends('jax.numpy',reason=jax_skip_reason)
class TestJacobian(JacobianHessianTest):
jh_func = jacobian
# Example functions and Jacobians from Wikipedia:
# https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant#Examples
def f1(z, xp):
x, y = z
return xp.stack([x ** 2 * y, 5 * x + xp.sin(y)])
def df1(z):
x, y = z
return [[2 * x * y, x ** 2], [np.full_like(x, 5), np.cos(y)]]
f1.mn = 2, 2 # type: ignore[attr-defined]
f1.ref = df1 # type: ignore[attr-defined]
def f2(z, xp):
r, phi = z
return xp.stack([r * xp.cos(phi), r * xp.sin(phi)])
def df2(z):
r, phi = z
return [[np.cos(phi), -r * np.sin(phi)],
[np.sin(phi), r * np.cos(phi)]]
f2.mn = 2, 2 # type: ignore[attr-defined]
f2.ref = df2 # type: ignore[attr-defined]
def f3(z, xp):
r, phi, th = z
return xp.stack([r * xp.sin(phi) * xp.cos(th), r * xp.sin(phi) * xp.sin(th),
r * xp.cos(phi)])
def df3(z):
r, phi, th = z
return [[np.sin(phi) * np.cos(th), r * np.cos(phi) * np.cos(th),
-r * np.sin(phi) * np.sin(th)],
[np.sin(phi) * np.sin(th), r * np.cos(phi) * np.sin(th),
r * np.sin(phi) * np.cos(th)],
[np.cos(phi), -r * np.sin(phi), np.zeros_like(r)]]
f3.mn = 3, 3 # type: ignore[attr-defined]
f3.ref = df3 # type: ignore[attr-defined]
def f4(x, xp):
x1, x2, x3 = x
return xp.stack([x1, 5 * x3, 4 * x2 ** 2 - 2 * x3, x3 * xp.sin(x1)])
def df4(x):
x1, x2, x3 = x
one = np.ones_like(x1)
return [[one, 0 * one, 0 * one],
[0 * one, 0 * one, 5 * one],
[0 * one, 8 * x2, -2 * one],
[x3 * np.cos(x1), 0 * one, np.sin(x1)]]
f4.mn = 3, 4 # type: ignore[attr-defined]
f4.ref = df4 # type: ignore[attr-defined]
def f5(x, xp):
x1, x2, x3 = x
return xp.stack([5 * x2, 4 * x1 ** 2 - 2 * xp.sin(x2 * x3), x2 * x3])
def df5(x):
x1, x2, x3 = x
one = np.ones_like(x1)
return [[0 * one, 5 * one, 0 * one],
[8 * x1, -2 * x3 * np.cos(x2 * x3), -2 * x2 * np.cos(x2 * x3)],
[0 * one, x3, x2]]
f5.mn = 3, 3 # type: ignore[attr-defined]
f5.ref = df5 # type: ignore[attr-defined]
def rosen(x, _): return optimize.rosen(x)
rosen.mn = 5, 1 # type: ignore[attr-defined]
rosen.ref = optimize.rosen_der # type: ignore[attr-defined]
@pytest.mark.parametrize('dtype', ('float32', 'float64'))
@pytest.mark.parametrize('size', [(), (6,), (2, 3)])
@pytest.mark.parametrize('func', [f1, f2, f3, f4, f5, rosen])
def test_examples(self, dtype, size, func, xp):
atol = 1e-10 if dtype == 'float64' else 1.99e-3
dtype = getattr(xp, dtype)
rng = np.random.default_rng(458912319542)
m, n = func.mn
x = rng.random(size=(m,) + size)
res = jacobian(lambda x: func(x , xp), xp.asarray(x, dtype=dtype))
# convert list of arrays to single array before converting to xp array
ref = xp.asarray(np.asarray(func.ref(x)), dtype=dtype)
xp_assert_close(res.df, ref, atol=atol)
def test_attrs(self, xp):
# Test attributes of result object
z = xp.asarray([0.5, 0.25])
# case in which some elements of the Jacobian are harder
# to calculate than others
def df1(z):
x, y = z
return xp.stack([xp.cos(0.5*x) * xp.cos(y), xp.sin(2*x) * y**2])
def df1_0xy(x, y):
return xp.cos(0.5*x) * xp.cos(y)
def df1_1xy(x, y):
return xp.sin(2*x) * y**2
res = jacobian(df1, z, initial_step=10)
if is_numpy(xp):
assert len(np.unique(res.nit)) == 4
assert len(np.unique(res.nfev)) == 4
res00 = jacobian(lambda x: df1_0xy(x, z[1]), z[0:1], initial_step=10)
res01 = jacobian(lambda y: df1_0xy(z[0], y), z[1:2], initial_step=10)
res10 = jacobian(lambda x: df1_1xy(x, z[1]), z[0:1], initial_step=10)
res11 = jacobian(lambda y: df1_1xy(z[0], y), z[1:2], initial_step=10)
ref = optimize.OptimizeResult()
for attr in ['success', 'status', 'df', 'nit', 'nfev']:
ref_attr = xp.asarray([[getattr(res00, attr), getattr(res01, attr)],
[getattr(res10, attr), getattr(res11, attr)]])
ref[attr] = xp.squeeze(ref_attr)
rtol = 1.5e-5 if res[attr].dtype == xp.float32 else 1.5e-14
xp_assert_close(res[attr], ref[attr], rtol=rtol)
def test_step_direction_size(self, xp):
# Check that `step_direction` and `initial_step` can be used to ensure that
# the usable domain of a function is respected.
rng = np.random.default_rng(23892589425245)
b = rng.random(3)
eps = 1e-7 # torch needs wiggle room?
def f(x):
x[0, x[0] < b[0]] = xp.nan
x[0, x[0] > b[0] + 0.25] = xp.nan
x[1, x[1] > b[1]] = xp.nan
x[1, x[1] < b[1] - 0.1-eps] = xp.nan
return TestJacobian.f5(x, xp)
dir = [1, -1, 0]
h0 = [0.25, 0.1, 0.5]
atol = {'atol': 1e-8}
res = jacobian(f, xp.asarray(b, dtype=xp.float64), initial_step=h0,
step_direction=dir, tolerances=atol)
ref = xp.asarray(TestJacobian.df5(b), dtype=xp.float64)
xp_assert_close(res.df, ref, atol=1e-8)
assert xp.all(xp.isfinite(ref))
@pytest.mark.skip_xp_backends('array_api_strict', reason=array_api_strict_skip_reason)
@pytest.mark.skip_xp_backends('jax.numpy',reason=jax_skip_reason)
class TestHessian(JacobianHessianTest):
jh_func = hessian
@pytest.mark.parametrize('shape', [(), (4,), (2, 4)])
def test_example(self, shape, xp):
rng = np.random.default_rng(458912319542)
m = 3
x = xp.asarray(rng.random((m,) + shape), dtype=xp.float64)
res = hessian(optimize.rosen, x)
if shape:
x = xp.reshape(x, (m, -1))
ref = xp.stack([optimize.rosen_hess(xi) for xi in x.T])
ref = xp.moveaxis(ref, 0, -1)
ref = xp.reshape(ref, (m, m,) + shape)
else:
ref = optimize.rosen_hess(x)
xp_assert_close(res.ddf, ref, atol=1e-8)
# # Removed symmetry enforcement; consider adding back in as a feature
# # check symmetry
# for key in ['ddf', 'error', 'nfev', 'success', 'status']:
# assert_equal(res[key], np.swapaxes(res[key], 0, 1))
def test_float32(self, xp):
rng = np.random.default_rng(458912319542)
x = xp.asarray(rng.random(3), dtype=xp.float32)
res = hessian(optimize.rosen, x)
ref = optimize.rosen_hess(x)
mask = (ref != 0)
xp_assert_close(res.ddf[mask], ref[mask])
atol = 1e-2 * xp.abs(xp.min(ref[mask]))
xp_assert_close(res.ddf[~mask], ref[~mask], atol=atol)
def test_nfev(self, xp):
z = xp.asarray([0.5, 0.25])
xp_test = array_namespace(z)
def f1(z):
x, y = xp_test.broadcast_arrays(*z)
f1.nfev = f1.nfev + (math.prod(x.shape[2:]) if x.ndim > 2 else 1)
return xp.sin(x) * y ** 3
f1.nfev = 0
res = hessian(f1, z, initial_step=10)
f1.nfev = 0
res00 = hessian(lambda x: f1([x[0], z[1]]), z[0:1], initial_step=10)
assert res.nfev[0, 0] == f1.nfev == res00.nfev[0, 0]
f1.nfev = 0
res11 = hessian(lambda y: f1([z[0], y[0]]), z[1:2], initial_step=10)
assert res.nfev[1, 1] == f1.nfev == res11.nfev[0, 0]
# Removed symmetry enforcement; consider adding back in as a feature
# assert_equal(res.nfev, res.nfev.T) # check symmetry
# assert np.unique(res.nfev).size == 3
@pytest.mark.thread_unsafe
@pytest.mark.skip_xp_backends(np_only=True,
reason='Python list input uses NumPy backend')
def test_small_rtol_warning(self, xp):
message = 'The specified `rtol=1e-15`, but...'
with pytest.warns(RuntimeWarning, match=message):
hessian(xp.sin, [1.], tolerances=dict(rtol=1e-15))
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