Source code for numdifftools.core

# !/usr/bin/env python
"""numerical differentiation functions:

Derivative, Gradient, Jacobian, and Hessian

Author:      Per A. Brodtkorb
Created:     01.08.2008
Copyright:   (c) pab 2008
Licence:     New BSD
"""

from __future__ import division, print_function
import numpy as np
from numdifftools.multicomplex import Bicomplex
from numdifftools.extrapolation import Richardson, dea3, convolve
from numdifftools.step_generators import MaxStepGenerator, MinStepGenerator
from numdifftools.limits import _Limit
from numpy import linalg
from scipy import misc

__all__ = ('dea3', 'Derivative', 'Jacobian', 'Gradient', 'Hessian', 'Hessdiag',
           'MinStepGenerator', 'MaxStepGenerator', 'Richardson',
           'directionaldiff')
_TINY = np.finfo(float).tiny
_EPS = np.finfo(float).eps
EPS = np.MachAr().eps
_SQRT_J = (1j + 1.0) / np.sqrt(2.0)  # = 1j**0.5

_cmn_doc = """
    Calculate %(derivative)s with finite difference approximation

    Parameters
    ----------
    f : function
       function of one array f(x, `*args`, `**kwds`)
    step : float, array-like or StepGenerator object, optional
       Defines the spacing used in the approximation.
       Default is MinStepGenerator(base_step=step, step_ratio=None,
                                   num_extrap=0, **step_options)
       if step or method in in ['complex', 'multicomplex'],
       otherwise
           MaxStepGenerator(step_ratio=None, num_extrap=14, **step_options)
       The results are extrapolated if the StepGenerator generate more than 3
       steps.
    method : {'central', 'complex', 'multicomplex', 'forward', 'backward'}
        defines the method used in the approximation%(extra_parameter)s
    full_output : bool, optional
        If `full_output` is False, only the derivative is returned.
        If `full_output` is True, then (der, r) is returned `der` is the
        derivative, and `r` is a Results object.
    **step_options:
        options to pass on to the XXXStepGenerator used.

    Call Parameters
    ---------------
    x : array_like
       value at which function derivative is evaluated
    args : tuple
        Arguments for function `f`.
    kwds : dict
        Keyword arguments for function `f`.
    %(returns)s
    Notes
    -----
    Complex methods are usually the most accurate provided the function to
    differentiate is analytic. The complex-step methods also requires fewer
    steps than the other methods and can work very close to the support of
    a function.
    The complex-step derivative has truncation error O(steps**2) for `n=1` and
    O(steps**4) for `n` larger, so truncation error can be eliminated by
    choosing steps to be very small.
    Especially the first order complex-step derivative avoids the problem of
    round-off error with small steps because there is no subtraction. However,
    this method fails if f(x) does not support complex numbers or involves
    non-analytic functions such as e.g.: abs, max, min.
    Central difference methods are almost as accurate and has no restriction on
    type of function. For this reason the 'central' method is the default
    method, but sometimes one can only allow evaluation in forward or backward
    direction.

    For all methods one should be careful in decreasing the step size too much
    due to round-off errors.
    %(extra_note)s
    Reference
    ---------
    Ridout, M.S. (2009) Statistical applications of the complex-step method
        of numerical differentiation. The American Statistician, 63, 66-74

    K.-L. Lai, J.L. Crassidis, Y. Cheng, J. Kim (2005), New complex step
        derivative approximations with application to second-order
        kalman filtering, AIAA Guidance, Navigation and Control Conference,
        San Francisco, California, August 2005, AIAA-2005-5944.

    Lyness, J. M., Moler, C. B. (1966). Vandermonde Systems and Numerical
                     Differentiation. *Numerische Mathematik*.

    Lyness, J. M., Moler, C. B. (1969). Generalized Romberg Methods for
                     Integrals of Derivatives. *Numerische Mathematik*.
    %(example)s
    %(see_also)s
    """


[docs]class Derivative(_Limit): __doc__ = _cmn_doc % dict( derivative='n-th derivative', extra_parameter=""" order : int, optional defines the order of the error term in the Taylor approximation used. For 'central' and 'complex' methods, it must be an even number. n : int, optional Order of the derivative.""", extra_note=""" Higher order approximation methods will generally be more accurate, but may also suffer more from numerical problems. First order methods is usually not recommended. """, returns=""" Returns ------- der : ndarray array of derivatives """, example=""" Example ------- >>> import numpy as np >>> import numdifftools as nd # 1'st derivative of exp(x), at x == 1 >>> fd = nd.Derivative(np.exp) >>> np.allclose(fd(1), 2.71828183) True >>> d2 = fd([1, 2]) >>> np.allclose(d2, [ 2.71828183, 7.3890561 ]) True >>> def f(x): ... return x**3 + x**2 >>> df = nd.Derivative(f) >>> np.allclose(df(1), 5) True >>> ddf = nd.Derivative(f, n=2) >>> np.allclose(ddf(1), 8) True """, see_also=""" See also -------- Gradient, Hessian """)
[docs] def __init__(self, f, step=None, method='central', order=2, n=1, full_output=False, **step_options): self.n = n self.richardson_terms = 2 super(Derivative, self).__init__(f, step=step, method=method, order=order, full_output=full_output, **step_options)
n = property(fget=lambda cls: cls._n, fset=lambda cls, n: (setattr(cls, '_n', n), cls._set_derivative())) def _set_derivative(self): if self.n == 0: self._derivative = self._derivative_zero_order else: self._derivative = self._derivative_nonzero_order def _derivative_zero_order(self, xi, args, kwds): steps = [np.zeros_like(xi)] results = [self.f(xi, *args, **kwds)] self.set_richardson_rule(2, 0) return self._vstack(results, steps) def _derivative_nonzero_order(self, xi, args, kwds): diff, f = self._get_functions() steps, step_ratio = self._get_steps(xi) fxi = self._eval_first(f, xi, *args, **kwds) results = [diff(f, fxi, xi, h, *args, **kwds) for h in steps] self.set_richardson_rule(step_ratio, self.richardson_terms) return self._apply_fd_rule(step_ratio, results, steps) def _make_generator(self, step, step_options): if hasattr(step, '__call__'): return step options = dict(step_ratio=None, num_extrap=14) if step is None and self.method not in ['complex', 'multicomplex']: options.update(**step_options) return MaxStepGenerator(**options) options['num_extrap'] = 0 options.update(**step_options) return MinStepGenerator(base_step=step, **options) @property def _method_order(self): step = self._richardson_step() # Make sure it is even and at least 2 or 4 order = max((self.order // step) * step, step) return order @property def _complex_high_order(self): return self.method == 'complex' and (self.n > 1 or self.order >= 4) def _richardson_step(self): complex_step = 4 if self._complex_high_order else 2 return dict(central=2, central2=2, complex=complex_step, multicomplex=2).get(self.method, 1) def set_richardson_rule(self, step_ratio, num_terms=2): order = self._method_order step = self._richardson_step() self.richardson = Richardson(step_ratio=step_ratio, step=step, order=order, num_terms=num_terms) def _multicomplex_middle_name_or_empty(self): if self.method == 'multicomplex' and self.n > 1: if self.n > 2: raise ValueError('Multicomplex method only support first ' 'and second order derivatives.') return '2' return '' def _get_middle_name(self): if self._even_derivative and self.method in ('central', 'complex'): return '_even' if self._complex_high_order and self._odd_derivative: return '_odd' return self._multicomplex_middle_name_or_empty() def _get_last_name(self): last = '' if (self.method == 'complex' and self._derivative_mod_four_is_zero or self._complex_high_order and self._derivative_mod_four_is_three): last = '_higher' return last def _get_function_name(self): first = '_{0!s}'.format(self.method) middle = self._get_middle_name() last = self._get_last_name() name = first + middle + last return name def _get_functions(self): name = self._get_function_name() return getattr(self, name), self.f def _get_steps(self, xi): method, n, order = self.method, self.n, self._method_order step_gen = self.step.step_generator_function(xi, method, n, order) return [step for step in step_gen()], step_gen.step_ratio @property def _odd_derivative(self): return self.n % 2 == 1 @property def _even_derivative(self): return self.n % 2 == 0 @property def _derivative_mod_four_is_three(self): return self.n % 4 == 3 @property def _derivative_mod_four_is_zero(self): return self.n % 4 == 0 def _eval_first_condition(self): even_derivative = self._even_derivative return ((even_derivative and self.method in ('central', 'central2')) or self.method in ['forward', 'backward'] or self.method == 'complex' and self._derivative_mod_four_is_zero) def _eval_first(self, f, x, *args, **kwds): if self._eval_first_condition(): return f(x, *args, **kwds) return 0.0 def __call__(self, x, *args, **kwds): xi = np.asarray(x) results = self._derivative(xi, args, kwds) derivative, info = self._extrapolate(*results) if self.full_output: return derivative, info return derivative @staticmethod def _fd_matrix(step_ratio, parity, nterms): """ Return matrix for finite difference and complex step derivation. Parameters ---------- step_ratio : real scalar ratio between steps in unequally spaced difference rule. parity : scalar, integer 0 (one sided, all terms included but zeroth order) 1 (only odd terms included) 2 (only even terms included) 3 (only every 4'th order terms included starting from order 2) 4 (only every 4'th order terms included starting from order 4) 5 (only every 4'th order terms included starting from order 1) 6 (only every 4'th order terms included starting from order 3) nterms : scalar, integer number of terms """ try: step = [1, 2, 2, 4, 4, 4, 4][parity] except Exception as e: msg = '{0!s}. Parity must be 0, 1, 2, 3, 4, 5 or 6! ({1:d})' raise ValueError(msg.format(str(e), parity)) inv_sr = 1.0 / step_ratio offset = [1, 1, 2, 2, 4, 1, 3][parity] c0 = [1.0, 1.0, 1.0, 2.0, 24.0, 1.0, 6.0][parity] c = c0 / \ misc.factorial(np.arange(offset, step * nterms + offset, step)) [i, j] = np.ogrid[0:nterms, 0:nterms] return np.atleast_2d(c[j] * inv_sr ** (i * (step * j + offset))) @property def _flip_fd_rule(self): n = self.n return ((self._even_derivative and (self.method == 'backward')) or (self.method == 'complex' and (n % 8 in [3, 4, 5, 6]))) def _parity_complex(self, order, method_order): if self.n == 1 and method_order < 4: return (order % 2) + 1 return (3 + 2 * int(self._odd_derivative) + int(self._derivative_mod_four_is_three) + int(self._derivative_mod_four_is_zero)) def _parity(self, method, order, method_order): if method.startswith('central'): return (order % 2) + 1 if method == 'complex': return self._parity_complex(order, method_order) return 0 def _get_finite_difference_rule(self, step_ratio): """ Generate finite differencing rule in advance. The rule is for a nominal unit step size, and will be scaled later to reflect the local step size. Member methods used ------------------- _fd_matrix Member variables used --------------------- n order method """ method = self.method if method in ('multicomplex', ) or self.n == 0: return np.ones((1,)) order, method_order = self.n - 1, self._method_order parity = self._parity(method, order, method_order) step = self._richardson_step() num_terms, ix = (order + method_order) // step, order // step fd_mat = self._fd_matrix(step_ratio, parity, num_terms) fd_rule = linalg.pinv(fd_mat)[ix] if self._flip_fd_rule: fd_rule *= -1 return fd_rule def _apply_fd_rule(self, step_ratio, sequence, steps): """ Return derivative estimates of f at x0 for a sequence of stepsizes h Member variables used --------------------- n """ f_del, h, original_shape = self._vstack(sequence, steps) fd_rule = self._get_finite_difference_rule(step_ratio) ne = h.shape[0] if ne < fd_rule.size: raise ValueError('num_steps ({0:d}) must be larger than ' '({1:d}) n + order - 1 = {2:d} + {3:d} -1' ' ({4:s})'.format(ne, fd_rule.size, self.n, self.order, self.method) ) nr = fd_rule.size - 1 f_diff = convolve(f_del, fd_rule[::-1], axis=0, origin=nr // 2) der_init = f_diff / (h ** self.n) ne = max(ne - nr, 1) return der_init[:ne], h[:ne], original_shape @staticmethod def _central_even(f, f_x0i, x0i, h, *args, **kwds): return (f(x0i + h, *args, **kwds) + f(x0i - h, *args, **kwds)) / 2.0 - f_x0i @staticmethod def _central(f, f_x0i, x0i, h, *args, **kwds): return (f(x0i + h, *args, **kwds) - f(x0i - h, *args, **kwds)) / 2.0 @staticmethod def _forward(f, f_x0i, x0i, h, *args, **kwds): return f(x0i + h, *args, **kwds) - f_x0i @staticmethod def _backward(f, f_x0i, x0i, h, *args, **kwds): return f_x0i - f(x0i - h, *args, **kwds) @staticmethod def _complex(f, fx, x, h, *args, **kwds): return f(x + 1j * h, *args, **kwds).imag @staticmethod def _complex_odd(f, fx, x, h, *args, **kwds): ih = h * _SQRT_J return ((_SQRT_J / 2.) * (f(x + ih, *args, **kwds) - f(x - ih, *args, **kwds))).imag @staticmethod def _complex_odd_higher(f, fx, x, h, *args, **kwds): ih = h * _SQRT_J return ((3 * _SQRT_J) * (f(x + ih, *args, **kwds) - f(x - ih, *args, **kwds))).real @staticmethod def _complex_even(f, fx, x, h, *args, **kwds): ih = h * _SQRT_J return (f(x + ih, *args, **kwds) + f(x - ih, *args, **kwds)).imag @staticmethod def _complex_even_higher(f, fx, x, h, *args, **kwds): ih = h * _SQRT_J return 12.0 * (f(x + ih, *args, **kwds) + f(x - ih, *args, **kwds) - 2 * fx).real @staticmethod def _multicomplex(f, fx, x, h, *args, **kwds): z = Bicomplex(x + 1j * h, 0) return f(z, *args, **kwds).imag @staticmethod def _multicomplex2(f, fx, x, h, *args, **kwds): z = Bicomplex(x + 1j * h, h) return f(z, *args, **kwds).imag12
[docs]def directionaldiff(f, x0, vec, **options): """ Return directional derivative of a function of n variables Parameters ---------- f: function analytical function to differentiate. x0: array vector location at which to differentiate f. If x0 is an nxm array, then f is assumed to be a function of n*m variables. vec: array vector defining the line along which to take the derivative. It should be the same size as x0, but need not be a vector of unit length. **options: optional arguments to pass on to Derivative. Returns ------- dder: scalar estimate of the first derivative of f in the specified direction. Example ------- At the global minimizer (1,1) of the Rosenbrock function, compute the directional derivative in the direction [1 2] >>> import numpy as np >>> import numdifftools as nd >>> vec = np.r_[1, 2] >>> rosen = lambda x: (1-x[0])**2 + 105*(x[1]-x[0]**2)**2 >>> dd, info = nd.directionaldiff(rosen, [1, 1], vec, full_output=True) >>> np.allclose(dd, 0) True >>> np.abs(info.error_estimate)<1e-14 True See also -------- Derivative, Gradient """ x0 = np.asarray(x0) vec = np.asarray(vec) if x0.size != vec.size: raise ValueError('vec and x0 must be the same shapes') vec = np.reshape(vec / np.linalg.norm(vec.ravel()), x0.shape) return Derivative(lambda t: f(x0 + t * vec), **options)(0)
[docs]class Jacobian(Derivative): __doc__ = _cmn_doc % dict( derivative='Jacobian', extra_parameter=""" order : int, optional defines the order of the error term in the Taylor approximation used. For 'central' and 'complex' methods, it must be an even number.""", returns=""" Returns ------- jacob : array Jacobian """, extra_note=""" Higher order approximation methods will generally be more accurate, but may also suffer more from numerical problems. First order methods is usually not recommended. If f returns a 1d array, it returns a Jacobian. If a 2d array is returned by f (e.g., with a value for each observation), it returns a 3d array with the Jacobian of each observation with shape xk x nobs x xk. I.e., the Jacobian of the first observation would be [:, 0, :] """, example=""" Example ------- >>> import numdifftools as nd #(nonlinear least squares) >>> xdata = np.reshape(np.arange(0,1,0.1),(-1,1)) >>> ydata = 1+2*np.exp(0.75*xdata) >>> fun = lambda c: (c[0]+c[1]*np.exp(c[2]*xdata) - ydata)**2 >>> Jfun = nd.Jacobian(fun) >>> val = Jfun([1,2,0.75]) >>> np.allclose(val, np.zeros((10,3))) True >>> fun2 = lambda x : x[0]*x[1]*x[2] + np.exp(x[0])*x[1] >>> Jfun3 = nd.Jacobian(fun2) >>> Jfun3([3.,5.,7.]) array([[ 135.42768462, 41.08553692, 15. ]]) """, see_also=""" See also -------- Derivative, Hessian, Gradient """) n = property(fget=lambda cls: 1, fset=lambda cls, val: cls._set_derivative()) @staticmethod def _check_equal_size(f_del, h): if f_del.size != h.size: raise ValueError('fun did not return data of correct size ' + '(it must be vectorized)') @staticmethod def _atleast_2d(original_shape, ndim): if ndim == 1: original_shape = (1, ) + tuple(original_shape) return tuple(original_shape) @staticmethod def _vstack_steps(steps, original_shape, axes, n): h_shape = (n, ) + steps[0].shape h = [np.atleast_2d(step).repeat(n, axis=0).reshape(h_shape) for step in steps] one = np.ones(original_shape) if len(h_shape) < 3: h = np.vstack([(one * hi).ravel()] for hi in h) else: h = np.vstack([(one * hi.transpose(axes)).ravel()] for hi in h) return h def _vstack(self, sequence, steps): original_shape = list(np.shape(np.atleast_1d(sequence[0].squeeze()))) ndim = len(original_shape) axes = [0, 1, 2][:ndim] axes[:2] = axes[1::-1] original_shape[:2] = original_shape[1::-1] n = 1 if ndim < 2 else original_shape[0] f_del = np.vstack([np.atleast_1d(r.squeeze()).transpose(axes).ravel()] for r in sequence) h = self._vstack_steps(steps, original_shape, axes, n) self._check_equal_size(f_del, h) return f_del, h, self._atleast_2d(original_shape, ndim) @staticmethod def _identity(n): m = np.zeros((n, n, n)) np.put(m, np.arange(0, n ** 3, n * (n + 1) + 1), 1) return m def _increments(self, n, h): return np.dot(self._identity(n), h) def _central(self, f, fx, x, h, *args, **kwds): n = len(x) increments = self._increments(n, h) partials = [(f(x + hi, *args, **kwds) - f(x - hi, *args, **kwds)) / 2.0 for hi in increments] return np.array(partials) def _backward(self, f, fx, x, h, *args, **kwds): n = len(x) increments = self._increments(n, h) partials = [fx - f(x - hi, *args, **kwds) for hi in increments] return np.array(partials) def _forward(self, f, fx, x, h, *args, **kwds): n = len(x) increments = self._increments(n, h) partials = [f(x + hi, *args, **kwds) - fx for hi in increments] return np.array(partials) def _complex(self, f, fx, x, h, *args, **kwds): n = len(x) increments = self._increments(n, 1j * h) partials = [f(x + ih, *args, **kwds).imag for ih in increments] return np.array(partials) def _complex_odd(self, f, fx, x, h, *args, **kwds): n = len(x) increments = self._increments(n, _SQRT_J * h) partials = [((_SQRT_J / 2.) * (f(x + ih, *args, **kwds) - f(x - ih, *args, **kwds))).imag for ih in increments] return np.array(partials) def _multicomplex(self, f, fx, x, h, *args, **kwds): n = len(x) increments = self._increments(n, 1j * h) partials = [f(Bicomplex(x + hi, 0), *args, **kwds).imag for hi in increments] return np.array(partials) def __call__(self, x, *args, **kwds): vals = super(Jacobian, self).__call__(np.atleast_1d(x), *args, **kwds) if vals.ndim == 3: return np.array([np.hstack([np.diag(hj) for hj in hi]) for hi in vals]) return vals
[docs]class Gradient(Jacobian): __doc__ = _cmn_doc % dict( derivative='Gradient', extra_parameter=""" order : int, optional defines the order of the error term in the Taylor approximation used. For 'central' and 'complex' methods, it must be an even number.""", returns=""" Returns ------- grad : array gradient """, extra_note=""" Higher order approximation methods will generally be more accurate, but may also suffer more from numerical problems. First order methods is usually not recommended. If x0 is an nxm array, then f is assumed to be a function of n*m variables. """, example=""" Example ------- >>> import numpy as np >>> import numdifftools as nd >>> fun = lambda x: np.sum(x**2) >>> dfun = nd.Gradient(fun) >>> dfun([1,2,3]) array([ 2., 4., 6.]) # At [x,y] = [1,1], compute the numerical gradient # of the function sin(x-y) + y*exp(x) >>> sin = np.sin; exp = np.exp >>> z = lambda xy: sin(xy[0]-xy[1]) + xy[1]*exp(xy[0]) >>> dz = nd.Gradient(z) >>> grad2 = dz([1, 1]) >>> grad2 array([ 3.71828183, 1.71828183]) # 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=""" See also -------- Derivative, Hessian, Jacobian """) def __call__(self, x, *args, **kwds): return super(Gradient, self).__call__(np.atleast_1d(x).ravel(), *args, **kwds).squeeze()
[docs]class Hessdiag(Derivative): __doc__ = _cmn_doc % dict( derivative='Hessian diagonal', extra_parameter="""order : int, optional defines the order of the error term in the Taylor approximation used. For 'central' and 'complex' methods, it must be an even number.""", returns=""" Returns ------- hessdiag : array hessian diagonal """, extra_note=""" Higher order approximation methods will generally be more accurate, but may also suffer more from numerical problems. First order methods is usually not recommended. """, example=""" Example ------- >>> import numpy as np >>> import numdifftools as nd >>> fun = lambda x : x[0] + x[1]**2 + x[2]**3 >>> Hfun = nd.Hessdiag(fun, full_output=True) >>> hd, info = Hfun([1,2,3]) >>> np.allclose(hd, [ 0., 2., 18.]) True >>> info.error_estimate < 1e-11 array([ True, True, True], dtype=bool) """, see_also=""" See also -------- Derivative, Hessian, Jacobian, Gradient """)
[docs] def __init__(self, f, step=None, method='central', order=2, full_output=False, **step_options): super(Hessdiag, self).__init__(f, step=step, method=method, n=2, order=order, full_output=full_output, **step_options)
n = property(fget=lambda cls: 2, fset=lambda cls, n: cls._set_derivative()) @staticmethod def _central2(f, fx, x, h, *args, **kwds): """Eq. 8""" n = len(x) increments = np.identity(n) * h partials = [(f(x + 2 * hi, *args, **kwds) + f(x - 2 * hi, *args, **kwds) + 2 * fx - 2 * f(x + hi, *args, **kwds) - 2 * f(x - hi, *args, **kwds)) / 4.0 for hi in increments] return np.array(partials) @staticmethod def _central_even(f, fx, x, h, *args, **kwds): """Eq. 9""" n = len(x) increments = np.identity(n) * h partials = [(f(x + hi, *args, **kwds) + f(x - hi, *args, **kwds)) / 2.0 - fx for hi in increments] return np.array(partials) @staticmethod def _backward(f, fx, x, h, *args, **kwds): n = len(x) increments = np.identity(n) * h partials = [fx - f(x - hi, *args, **kwds) for hi in increments] return np.array(partials) @staticmethod def _forward(f, fx, x, h, *args, **kwds): n = len(x) increments = np.identity(n) * h partials = [f(x + hi, *args, **kwds) - fx for hi in increments] return np.array(partials) @staticmethod def _multicomplex2(f, fx, x, h, *args, **kwds): n = len(x) increments = np.identity(n) * h partials = [f(Bicomplex(x + 1j * hi, hi), *args, **kwds).imag12 for hi in increments] return np.array(partials) @staticmethod def _complex_even(f, fx, x, h, *args, **kwargs): n = len(x) increments = np.identity(n) * h * (1j + 1) / np.sqrt(2) partials = [(f(x + hi, *args, **kwargs) + f(x - hi, *args, **kwargs)).imag for hi in increments] return np.array(partials) def __call__(self, x, *args, **kwds): return super(Hessdiag, self).__call__(np.atleast_1d(x), *args, **kwds)
[docs]class Hessian(Hessdiag): __doc__ = _cmn_doc % dict( derivative='Hessian', extra_parameter="", returns=""" Returns ------- hess : ndarray array of partial second derivatives, Hessian """, extra_note=""" Computes the Hessian according to method as: 'forward' :eq:`7`, 'central' :eq:`9` and 'complex' :eq:`10`: .. math:: \quad ((f(x + d_j e_j + d_k e_k) - f(x + d_j e_j))) / (d_j d_k) :label: 7 .. math:: \quad ((f(x + d_j e_j + d_k e_k) - f(x + d_j e_j - d_k e_k)) - (f(x - d_j e_j + d_k e_k) - f(x - d_j e_j - d_k e_k)) / (4 d_j d_k) :label: 9 .. math:: imag(f(x + i d_j e_j + d_k e_k) - f(x + i d_j e_j - d_k e_k)) / (2 d_j d_k) :label: 10 where :math:`e_j` is a vector with element :math:`j` is one and the rest are zero and :math:`d_j` is a scalar spacing :math:`steps_j`. """, example=""" Example ------- >>> import numpy as np >>> import numdifftools as nd # Rosenbrock function, minimized at [1,1] >>> rosen = lambda x : (1.-x[0])**2 + 105*(x[1]-x[0]**2)**2 >>> Hfun = nd.Hessian(rosen) >>> h = Hfun([1, 1]) >>> h array([[ 842., -420.], [-420., 210.]]) # cos(x-y), at (0,0) >>> cos = np.cos >>> fun = lambda xy : cos(xy[0]-xy[1]) >>> Hfun2 = nd.Hessian(fun) >>> h2 = Hfun2([0, 0]) >>> h2 array([[-1., 1.], [ 1., -1.]])""", see_also=""" See also -------- Derivative, Hessian """) order = property(fget=lambda cls: dict(backward=1, forward=1, complex=2).get(cls.method, 2), fset=lambda cls, order: None) def _apply_fd_rule(self, step_ratio, sequence, steps): """ Return derivative estimates of f at x0 for a sequence of stepsizes h Here the difference rule is already applied. Just return result. """ return self._vstack(sequence, steps) @staticmethod def _complex_high_order(): return False @staticmethod def _complex_even(f, fx, x, h, *args, **kwargs): """ Calculate Hessian with complex-step derivative approximation The stepsize is the same for the complex and the finite difference part """ n = len(x) ee = np.diag(h) hess = 2. * np.outer(h, h) for i in range(n): for j in range(i, n): hess[i, j] = (f(x + 1j * ee[i] + ee[j], *args, **kwargs) - f(x + 1j * ee[i] - ee[j], *args, **kwargs) ).imag / hess[j, i] hess[j, i] = hess[i, j] return hess @staticmethod def _multicomplex2(f, fx, x, h, *args, **kwargs): """Calculate Hessian with Bicomplex-step derivative approximation""" n = len(x) ee = np.diag(h) hess = np.outer(h, h) for i in range(n): for j in range(i, n): zph = Bicomplex(x + 1j * ee[i, :], ee[j, :]) hess[i, j] = (f(zph, *args, **kwargs)).imag12 / hess[j, i] hess[j, i] = hess[i, j] return hess @staticmethod def _central_even(f, fx, x, h, *args, **kwargs): """Eq 9.""" n = len(x) ee = np.diag(h) hess = np.outer(h, h) for i in range(n): hess[i, i] = (f(x + 2 * ee[i, :], *args, **kwargs) - 2 * fx + f(x - 2 * ee[i, :], *args, **kwargs) ) / (4. * hess[i, i]) for j in range(i + 1, n): hess[i, j] = (f(x + ee[i, :] + ee[j, :], *args, **kwargs) - f(x + ee[i, :] - ee[j, :], *args, **kwargs) - f(x - ee[i, :] + ee[j, :], *args, **kwargs) + f(x - ee[i, :] - ee[j, :], *args, **kwargs) ) / (4. * hess[j, i]) hess[j, i] = hess[i, j] return hess @staticmethod def _central2(f, fx, x, h, *args, **kwargs): """Eq. 8""" n = len(x) ee = np.diag(h) dtype = np.result_type(fx) g = np.empty(n, dtype=dtype) gg = np.empty(n, dtype=dtype) for i in range(n): g[i] = f(x + ee[i], *args, **kwargs) gg[i] = f(x - ee[i], *args, **kwargs) hess = np.empty((n, n), dtype=dtype) np.outer(h, h, out=hess) for i in range(n): for j in range(i, n): hess[i, j] = (f(x + ee[i, :] + ee[j, :], *args, **kwargs) - g[i] - g[j] + fx + f(x - ee[i, :] - ee[j, :], *args, **kwargs) - gg[i] - gg[j] + fx) / (2 * hess[j, i]) hess[j, i] = hess[i, j] return hess @staticmethod def _forward(f, fx, x, h, *args, **kwargs): """Eq. 7""" n = len(x) ee = np.diag(h) dtype = np.result_type(fx) g = np.empty(n, dtype=dtype) for i in range(n): g[i] = f(x + ee[i, :], *args, **kwargs) hess = np.empty((n, n), dtype=dtype) np.outer(h, h, out=hess) for i in range(n): for j in range(i, n): hess[i, j] = (f(x + ee[i, :] + ee[j, :], *args, **kwargs) - g[i] - g[j] + fx) / hess[j, i] hess[j, i] = hess[i, j] return hess def _backward(self, f, fx, x, h, *args, **kwargs): return self._forward(f, fx, x, -h, *args, **kwargs)