>>> c, info = ndf.derivative(fun, z0=0, n=6, full_output=True)
>>> np.allclose(c, [1, 1, 2, 6, 24, 120, 720, 5040])
True
>>> np.all(info.error_estimate < 1e-9*c.real)
True
>>> (info.function_count, info.iterations, info.failed) == (144, 18, False)
True
References
----------
[1] Fornberg, B. (1981).
Numerical Differentiation of Analytic Functions.
ACM Transactions on Mathematical Software (TOMS),
7(4), 512-526. http://doi.org/10.1145/355972.355979
"""
result = taylor(fun, z0, n=n, **kwds)
# convert taylor series --> actual derivatives.
m = _num_taylor_coefficients(n)
fact = factorial(np.arange(m))
if kwds.get('full_output'):
coefs, info_ = result
info = _INFO(info_.error_estimate * fact, *info_[1:])
return coefs * fact, info
return result * fact
if __name__ == '__main__':
from numdifftools.testing import test_docstrings
test_docstrings()
True
'''
kwargs = {} if kwargs is None else kwargs
n = len(x)
# TODO: add scaled stepsize
f0 = f(*(x,) + args, **kwargs)
dim = np.atleast_1d(f0).shape # it could be a scalar
grad = np.zeros((n,) + dim, float)
ei = np.zeros(np.shape(x), float)
if not centered:
epsilon = _get_epsilon(x, 2, epsilon, n)
for k in range(n):
ei[k] = epsilon[k]
grad[k, :] = (f(*(x + ei,) + args, **kwargs) - f0) / epsilon[k]
ei[k] = 0.0
else:
epsilon = _get_epsilon(x, 3, epsilon, n) / 2.
for k in range(n):
ei[k] = epsilon[k]
grad[k, :] = (f(*(x + ei,) + args, **kwargs) -
f(*(x - ei,) + args, **kwargs)) / (2 * epsilon[k])
ei[k] = 0.0
grad = grad.squeeze()
axes = [0, 1, 2][:grad.ndim]
axes[:2] = axes[1::-1]
return np.transpose(grad, axes=axes).squeeze()
if __name__ == '__main__':
from numdifftools.testing import test_docstrings
test_docstrings()
>>> z = lambda xy: sin(xy[0]-xy[1]) + xy[1]*exp(xy[0])
>>> dz = nd.Gradient(z)
>>> grad2 = dz([1, 1])
>>> np.allclose(grad2, [ 3.71828183, 1.71828183])
True
# At the global minimizer (1,1) of the Rosenbrock function,
# compute the gradient. It should be essentially zero.
>>> rosen = lambda x : (1-x[0])**2 + 105.*(x[1]-x[0]**2)**2
>>> rd = nd.Gradient(rosen)
>>> grad3 = rd([1,1])
>>> np.allclose(grad3,[0, 0])
True
See also
--------
Hessian, Jacobian
"""
def __call__(self, x, *args, **kwds):
return super(Gradient,
self).__call__(np.atleast_1d(x).ravel(), *args,
**kwds).squeeze()
if __name__ == '__main__':
from numdifftools.testing import test_docstrings
test_docstrings(__file__)
# print(np.log(_EPS)/np.log(1e-6))
# print(_EPS**(1./2.5))
>>> def h(z): return 1.0/np.sin(z)**2
>>> res_h, info = Residue(h, full_output=True, pole_order=2)([0, np.pi])
>>> np.allclose(res_h, 1)
True
>>> (info.error_estimate < 1e-10).all()
True
"""
def __init__(self, f, step=None, method="above", order=None, pole_order=1, full_output=False, **options):
if order is None:
# MethodOrder will always = pole_order + 2
order = pole_order + 2
_assert(pole_order < order, "order must be at least pole_order+1.")
self.pole_order = pole_order
super(Residue, self).__init__(f, step=step, method=method, order=order, full_output=full_output, **options)
def _fun(self, z, dz, *args, **kwds):
return self.fun(z + dz, *args, **kwds) * (dz ** self.pole_order)
def __call__(self, x, *args, **kwds):
return self.limit(x, *args, **kwds)
if __name__ == "__main__":
from numdifftools.testing import test_docstrings
test_docstrings(__file__)