# !/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 Based on matlab functions derivest.m gradest.m hessdiag.m, hessian.m and jacobianest.m version 1.0 released 12/27/2006 by John D'Errico (e-mail: woodchips@rochester.rr.com) Also based on the python functions approx_fprime, approx_fprime_cs, approx_hess_cs, approx_hess1, approx_hess2 and approx_hess3 in the statsmodels.tools.numdiff module released in 2014 written by Josef Perktold. """ from __future__ import division, print_function import numpy as np from collections import namedtuple from numpy import linalg import warnings # 2016-04-19 PAK use relative imports from .multicomplex import bicomplex from .extrapolation import Richardson, dea3, convolve # CRUFT: nanmedian introduced in numpy 1.9, 2014-09-07. try: from numpy import nanmedian except ImportError: import numpy numpy.nanmedian = numpy.median numpy.nanpercentile = numpy.percentile numpy.nanmin = numpy.min numpy.nanargmin = numpy.argmin # CRUFT: factorial moved from scipy.misc to scipy.special try: from scipy.special import factorial except ImportError: from scipy.misc import factorial __all__ = ('dea3', 'Derivative', 'Jacobian', 'Gradient', 'Hessian', 'Hessdiag', 'MinStepGenerator', 'MaxStepGenerator', 'Richardson') # NOTE: we only do double precision internally so far _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 _CENTRAL_WEIGHTS_AND_POINTS = { (1, 3): (np.array([-1, 0, 1]) / 2.0, np.arange(-1, 2)), (1, 5): (np.array([1, -8, 0, 8, -1]) / 12.0, np.arange(-2, 3)), (1, 7): (np.array([-1, 9, -45, 0, 45, -9, 1]) / 60.0, np.arange(-3, 4)), (1, 9): (np.array([3, -32, 168, -672, 0, 672, -168, 32, -3]) / 840.0, np.arange(-4, 5)), (2, 3): (np.array([1, -2.0, 1]), np.arange(-1, 2)), (2, 5): (np.array([-1, 16, -30, 16, -1]) / 12.0, np.arange(-2, 3)), (2, 7): (np.array([2, -27, 270, -490, 270, -27, 2]) / 180.0, np.arange(-3, 4)), (2, 9): (np.array([-9, 128, -1008, 8064, -14350, 8064, -1008, 128, -9]) / 5040.0, np.arange(-4, 5))} def fornberg_weights_all(x, x0, M=1): """ Return finite difference weights_and_points for derivatives of all orders. Parameters ---------- x : vector, length n x-coordinates for grid points x0 : scalar location where approximations are to be accurate m : scalar integer highest derivative that we want to find weights_and_points for Returns ------- C : array, shape n x m+1 contains coefficients for the j'th derivative in column j (0 <= j <= m) See also: --------- fornberg_weights Reference --------- B. Fornberg (1998) "Calculation of weights_and_points in finite difference formulas", SIAM Review 40, pp. 685-691. http://www.scholarpedia.org/article/Finite_difference_method """ N = len(x) if M >= N: raise ValueError('length(x) must be larger than m') c1, c4 = 1, x[0] - x0 C = np.zeros((N, M + 1)) C[0, 0] = 1 for n in range(1, N): m = np.arange(0, min(n, M) + 1) c2, c5, c4 = 1, c4, x[n] - x0 for v in range(n): c3 = x[n] - x[v] c2, c6, c7 = c2 * c3, m * C[v, m-1], C[v, m] C[v, m] = (c4 * c7 - c6) / c3 C[n, m] = c1 * (c6 - c5 * c7) / c2 c1 = c2 return C def fornberg_weights(x, x0, m=1): """ Return weights for finite difference approximation of the m'th derivative U^m(x0), evaluated at x0, based on n values of U at x[0], x[1],... x[n-1]: U^m(x0) = sum weights[i] * U(x[i]) Parameters ---------- x : vector abscissas used for the evaluation for the derivative at x0. x0 : scalar location where approximations are to be accurate m : integer order of derivative. Note for m=0 this can be used to evaluate the interpolating polynomial itself. Notes ----- The x values can be arbitrarily spaced but must be distinct and len(x) > m. The Fornberg algorithm is much more stable numerically than regular vandermonde systems for large values of n. See also -------- fornberg_weights_all """ return fornberg_weights_all(x, x0, m)[:, -1] def _make_exact(h): """Make sure h is an exact representable number This is important when calculating numerical derivatives and is accomplished by adding 1 and then subtracting 1.. """ return (h + 1.0) - 1.0 def default_scale(method='forward', n=1, order=2): # is_odd = (n % 2) == 1 high_order = int(n > 1 or order >= 4) order2 = max(order // 2-1, 0) n4 = n // 4 return (dict(multicomplex=1.35, complex=1.35).get(method, 2.5) + int((n - 1)) * dict(multicomplex=0, complex=0.0).get(method, 1.3) + order2 * dict(central=3, forward=2, backward=2).get(method, 0) + # is_odd * dict(complex=2.65*int(n//2)).get(method, 0) + (n % 4 == 1) * high_order * dict(complex=3.65 + n4 * (5 + 1.5**n4) ).get(method, 0) + (n % 4 == 3) * dict(complex=3.65*2 + n4 * (5 + 2.1**n4) ).get(method, 0) + (n % 4 == 2) * dict(complex=3.65 + n4 * (5 + 1.7**n4) ).get(method, 0) + (n % 4 == 0) * dict(complex=(n//4) * (10 + 1.5*int(n > 10)) ).get(method, 0)) def valarray(shape, value=np.NaN, typecode=None): """Return an array of all value.""" if typecode is None: typecode = bool out = np.ones(shape, dtype=typecode) * value if not isinstance(out, np.ndarray): out = np.asarray(out) return out def nom_step(x=None): """Return nominal step.""" if x is None: return 1.0 return np.maximum(np.log1p(np.abs(x)), 1.0) def _default_base_step(x, scale, epsilon=None): if epsilon is None: h = EPS ** (1. / scale) * nom_step(x) else: h = valarray(x.shape, value=epsilon) return h class MinStepGenerator(object): """ Generates a sequence of steps where steps = base_step * step_ratio ** (np.arange(num_steps) + offset) Parameters ---------- base_step : float, array-like, optional Defines the base step, if None, then base_step is set to EPS**(1/scale)*max(log(1+|x|), 1) where x is supplied at runtime through the __call__ method. step_ratio : real scalar, optional, default 2 Ratio between sequential steps generated. Note: Ratio > 1 If None then step_ratio is 2 for n=1 otherwise step_ratio is 1.6 num_steps : scalar integer, optional, default n + order - 1 + num_extrap defines number of steps generated. It should be larger than n + order - 1 offset : real scalar, optional, default 0 offset to the base step scale : real scalar, optional scale used in base step. If not None it will override the default computed with the default_scale function. """ def __init__(self, base_step=None, step_ratio=2, num_steps=None, offset=0, scale=None, num_extrap=0, use_exact_steps=True, check_num_steps=True): self.base_step = base_step self.num_steps = num_steps self.step_ratio = step_ratio self.offset = offset self.scale = scale self.check_num_steps = check_num_steps self.use_exact_steps = use_exact_steps self.num_extrap = num_extrap def __repr__(self): class_name = self.__class__.__name__ kwds = ['%s=%s' % (name, str(getattr(self, name))) for name in self.__dict__.keys()] return """%s(%s)""" % (class_name, ','.join(kwds)) def _default_scale(self, method, n, order): scale = self.scale if scale is None: scale = default_scale(method, n, order) return scale def _default_base_step(self, xi, method, n, order=2): scale = self._default_scale(method, n, order) base_step = _default_base_step(xi, scale, self.base_step) if self.use_exact_steps: base_step = _make_exact(base_step) return base_step def _min_num_steps(self, method, n, order): num_steps = n + order - 1 if method in ['central', 'central2', 'complex', 'multicomplex']: step = 2 if method == 'complex': step = 4 if n > 2 or order >= 4 else 2 num_steps = (n + order-1) // step return max(int(num_steps), 1) def _default_num_steps(self, method, n, order): min_num_steps = self._min_num_steps(method, n, order) if self.num_steps is not None: num_steps = int(self.num_steps) if self.check_num_steps: num_steps = max(num_steps, min_num_steps) return num_steps return min_num_steps + int(self.num_extrap) def _default_step_ratio(self, n): if self.step_ratio is None: step_ratio = {1: 2.0}.get(n, 1.6) else: step_ratio = float(self.step_ratio) if self.use_exact_steps: step_ratio = _make_exact(step_ratio) return step_ratio def __call__(self, x, method='central', n=1, order=2): xi = np.asarray(x) base_step = self._default_base_step(xi, method, n, order) step_ratio = self._default_step_ratio(n) num_steps = self._default_num_steps(method, n, order) offset = self.offset for i in range(num_steps-1, -1, -1): h = (base_step * step_ratio**(i + offset)) if (np.abs(h) > 0).all(): yield h class MinMaxStepGenerator(object): """ Generates a sequence of steps where steps = logspace(log10(step_min), log10(step_max), num_steps) Parameters ---------- step_min : float, array-like, optional Defines the minimim step. Default value is: EPS**(1/scale)*max(log(1+|x|), 1) where x and scale are supplied at runtime through the __call__ method. step_max : real scalar, optional maximum step generated. Default value is: exp(log(step_min) * scale / (scale + 1.5)) num_steps : scalar integer, optional defines number of steps generated. scale : real scalar, optional scale used in base step. If set to a value it will override the scale supplied at runtime. """ def __init__(self, step_min=None, step_max=None, num_steps=10, scale=None, num_extrap=0): self.step_min = step_min self.num_steps = num_steps self.step_max = step_max self.scale = scale self.num_extrap = num_extrap def __repr__(self): class_name = self.__class__.__name__ kwds = ['%s=%s' % (name, str(getattr(self, name))) for name in self.__dict__.keys()] return """%s(%s)""" % (class_name, ','.join(kwds)) def __call__(self, x, method='forward', n=1, order=None): if self.scale is not None: scale = self.scale xi = np.asarray(x) step_min, step_max = self.step_min, self.step_max delta = _default_base_step(xi, scale, step_min) if step_min is None: step_min = (10 * EPS)**(1. / scale) if step_max is None: step_max = np.exp(np.log(step_min) * scale / (scale + 1.5)) steps = np.logspace(0, np.log10(step_max) - np.log10(step_min), self.num_steps)[::-1] for step in steps: h = _make_exact(delta * step) if (np.abs(h) > 0).all(): yield h class MaxStepGenerator(MinStepGenerator): """ Generates a sequence of steps where steps = base_step * step_ratio ** (-np.arange(num_steps) + offset) base_step = step_max * step_nom Parameters ---------- max_step : float, array-like, optional default 2 Defines the maximum step step_ratio : real scalar, optional, default 2 Ratio between sequential steps generated. Note: Ratio > 1 num_steps : scalar integer, optional, default n + order - 1 + num_extrap defines number of steps generated. It should be larger than n + order - 1 step_nom : default maximum(log1p(abs(x)), 1) Nominal step. offset : real scalar, optional, default 0 offset to the base step: max_step * nom_step """ def __init__(self, step_max=2.0, step_ratio=2.0, num_steps=15, step_nom=None, offset=0, num_extrap=0, use_exact_steps=False, check_num_steps=True): self.base_step = None self.step_max = step_max self.step_ratio = step_ratio self.num_steps = num_steps self.step_nom = step_nom self.offset = offset self.num_extrap = num_extrap self.check_num_steps = check_num_steps self.use_exact_steps = use_exact_steps def _default_step_nom(self, x): if self.step_nom is None: return nom_step(x) return valarray(x.shape, value=self.step_nom) def _default_base_step(self, xi, method, n): base_step = self.base_step if base_step is None: base_step = self.step_max * self._default_step_nom(xi) if self.use_exact_steps: base_step = _make_exact(base_step) return base_step def __call__(self, x, method='forward', n=1, order=None): xi = np.asarray(x) offset = self.offset base_step = self._default_base_step(xi, method, n) step_ratio = self._default_step_ratio(n) num_steps = self._default_num_steps(method, n, order) for i in range(num_steps): h = base_step * step_ratio**(-i + offset) if (np.abs(h) > 0).all(): yield h _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) if step or method in in ['complex', 'multicomplex'], otherwise MaxStepGenerator(step_ratio=None, num_extrap=14) 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. 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 """ class _Derivative(object): info = namedtuple('info', ['error_estimate', 'final_step', 'index']) def __init__(self, f, step=None, method='central', order=2, n=1, full_output=False): self.f = f self.n = n self.order = order self.method = method self.full_output = full_output self.richardson_terms = 2 self.step = self._make_generator(step) def _make_generator(self, step): if hasattr(step, '__call__'): return step if step is None and self.method not in ['complex', 'multicomplex']: return MaxStepGenerator(step_ratio=None, num_extrap=14) return MinStepGenerator(base_step=step, step_ratio=None, num_extrap=0) def _get_arg_min(self, errors): shape = errors.shape try: arg_mins = np.nanargmin(errors, axis=0) min_errors = np.nanmin(errors, axis=0) except ValueError as msg: warnings.warn(str(msg)) ix = np.arange(shape[1]) return ix for i, min_error in enumerate(min_errors): idx = np.flatnonzero(errors[:, i] == min_error) arg_mins[i] = idx[idx.size // 2] ix = np.ravel_multi_index((arg_mins, np.arange(shape[1])), shape) return ix def _add_error_to_outliers(self, der, trim_fact=10): try: median = np.nanmedian(der, axis=0) p75 = np.nanpercentile(der, 75, axis=0) p25 = np.nanpercentile(der, 25, axis=0) iqr = np.abs(p75-p25) except ValueError as msg: warnings.warn(str(msg)) return 0 * der a_median = np.abs(median) outliers = (((abs(der) < (a_median / trim_fact)) + (abs(der) > (a_median * trim_fact))) * (a_median > 1e-8) + ((der < p25-1.5*iqr) + (p75+1.5*iqr < der))) errors = outliers * np.abs(der - median) return errors def _get_best_estimate(self, der, errors, steps, shape): errors += self._add_error_to_outliers(der) ix = self._get_arg_min(errors) final_step = steps.flat[ix].reshape(shape) err = errors.flat[ix].reshape(shape) return der.flat[ix].reshape(shape), self.info(err, final_step, ix) @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 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.n % 2 == 0 else 2 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_extrapolate = Richardson(step_ratio=step_ratio, step=step, order=order, num_terms=num_terms) def _wynn_extrapolate(self, der, steps): der, errors = dea3(der[0:-2], der[1:-1], der[2:], symmetric=False) return der, errors, steps[2:] def _extrapolate(self, results, steps, shape): der, errors, steps = self._richardson_extrapolate(results, steps) if len(der) > 2: # der, errors, steps = self._richardson_extrapolate(results, steps) der, errors, steps = self._wynn_extrapolate(der, steps) der, info = self._get_best_estimate(der, errors, steps, shape) return der, info def _get_function_name(self): name = '_%s' % self.method even_derivative_order = self._is_even_derivative() if even_derivative_order and self.method in ('central', 'complex'): name = name + '_even' if self.method in ('complex') and self._is_fourth_derivative(): name = name + '_higher' else: if self._complex_high_order() and self._is_odd_derivative(): name = name + '_odd' if self._is_third_derivative(): name = name + '_higher' elif self.method == 'multicomplex' and self.n > 1: if self.n == 2: name = name + '2' else: raise ValueError('Multicomplex method only support first ' 'and second order derivatives.') 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 return [step for step in self.step(xi, method, n, order)] def _is_odd_derivative(self): return self.n % 2 == 1 def _is_even_derivative(self): return self.n % 2 == 0 def _is_third_derivative(self): return self.n % 4 == 3 def _is_fourth_derivative(self): return self.n % 4 == 0 def _eval_first_condition(self): even_derivative = self._is_even_derivative() return ((even_derivative and self.method in ('central', 'central2')) or self.method in ['forward', 'backward'] or self.method == 'complex' and self._is_fourth_derivative()) def _eval_first(self, f, x, *args, **kwds): if self._eval_first_condition(): return f(x, *args, **kwds) return 0.0 def _vstack(self, sequence, steps): # sequence = np.atleast_2d(sequence) original_shape = np.shape(sequence[0]) f_del = np.vstack([list(np.ravel(r)) for r in sequence]) h = np.vstack([list(np.ravel(np.ones(original_shape)*step)) for step in steps]) if f_del.size != h.size: raise ValueError('fun did not return data of correct size ' + '(it must be vectorized)') return f_del, h, original_shape def _compute_step_ratio(self, steps): if len(steps) < 2: return 1 return np.unique(steps[0]/steps[1]).mean() 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 class Derivative(_Derivative): __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 >>> from bumps 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 """) """ Find the n-th derivative of a function at a point. Given a function, use a difference formula with spacing `dx` to compute the `n`-th derivative at `x0`. Parameters ---------- f : function Input function. x0 : float The point at which `n`-th derivative is found. dx : float, optional Spacing. method : Method of estimation. Valid options are: 'central', 'forward' or 'backward'. (Default 'central') n : int, optional (Default 1) Order of the derivative. order : int, optional (Default 2) defining order of basic method used. For 'central' methods, it must be an even number eg. [2,4]. Notes ----- Decreasing the step size too small can result in round-off error. Note on order: higher order methods will generally be more accurate, but may also suffer more from numerical problems. First order methods would usually not be recommended. 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. Central difference methods are almost as accurate and has no restriction on type of function, but sometimes one can only allow evaluation in forward or backward direction. """ @property def n(self): return self._n @n.setter def n(self, n): self._n = n if n == 0: self._derivative = self._derivative_zero_order else: self._derivative = self._derivative_nonzero_order @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) nterms : scalar, integer number of terms """ try: step = [1, 2, 2, 4, 4, 4, 4][parity] except Exception as e: msg = '%s. Parity must be 0, 1, 2, 3, 4, 5 or 6! (%d)' % (str(e), parity) raise ValueError(msg) 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/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))) def _flip_fd_rule(self): n = self.n return ((self._is_even_derivative() and (self.method == 'backward')) or (self.method == 'complex' and (n % 8 in [3, 4, 5, 6]))) 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 = 0 if (method.startswith('central') or (method.startswith('complex') and self.n == 1 and method_order < 4)): parity = (order % 2) + 1 elif self.method == 'complex': if self._is_odd_derivative(): parity = 6 if self._is_third_derivative() else 5 else: parity = 4 if self._is_fourth_derivative() else 3 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, fd_rule, 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) ne = h.shape[0] if ne < fd_rule.size: raise ValueError('num_steps (%d) must be larger than ' '(%d) n + order - 1 = %d + %d -1' ' (%s)' % (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 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 = self._get_steps(xi) fxi = self._eval_first(f, xi, *args, **kwds) results = [diff(f, fxi, xi, h, *args, **kwds) for h in steps] step_ratio = self._compute_step_ratio(steps) self._set_richardson_rule(step_ratio, self.richardson_terms) fd_rule = self._get_finite_difference_rule(step_ratio) return self._apply_fd_rule(fd_rule, results, steps) @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 class Gradient(Derivative): def __init__(self, f, step=None, method='central', order=2, full_output=False): super(Gradient, self).__init__(f, step=step, method=method, n=1, order=order, full_output=full_output) __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. """, example=""" Example ------- >>> import numpy as np >>> from bumps 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 """) @staticmethod def _central(f, fx, x, h, *args, **kwds): n = len(x) increments = np.identity(n) * h partials = [(f(x + hi, *args, **kwds) - f(x - hi, *args, **kwds)) / 2.0 for hi in increments] return np.array(partials).T @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).T @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).T @staticmethod def _complex(f, fx, x, h, *args, **kwds): # From Guilherme P. de Freitas, numpy mailing list # http://mail.scipy.org/pipermail/numpy-discussion/2010-May/050250.html n = len(x) increments = np.identity(n) * 1j * h partials = [f(x + ih, *args, **kwds).imag for ih in increments] return np.array(partials).T @staticmethod def _complex_odd(f, fx, x, h, *args, **kwds): n = len(x) increments = np.identity(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).T @staticmethod def _multicomplex(f, fx, x, h, *args, **kwds): n = len(x) increments = np.identity(n) * 1j * h partials = [f(bicomplex(x + hi, 0), *args, **kwds).imag for hi in increments] return np.array(partials).T def __call__(self, x, *args, **kwds): return super(Gradient, self).__call__(np.atleast_1d(x), *args, **kwds) class Jacobian(Gradient): __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 ------- >>> from bumps 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 """) class Hessdiag(Derivative): def __init__(self, f, step=None, method='central', order=2, full_output=False): super(Hessdiag, self).__init__(f, step=step, method=method, n=2, order=order, full_output=full_output) __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 >>> from bumps 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 """) @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) class Hessian(_Derivative): def __init__(self, f, step=None, method='central', full_output=False): order = dict(backward=1, forward=1, complex=2).get(method, 2) super(Hessian, self).__init__(f, n=2, step=step, method=method, order=order, full_output=full_output) __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 >>> from bumps 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 """) def _complex_high_order(self): return False def _derivative(self, xi, args, kwds): xi = np.atleast_1d(xi) diff, f = self._get_functions() steps = self._get_steps(xi) fxi = self._eval_first(f, xi, *args, **kwds) results = [diff(f, fxi, xi, h, *args, **kwds) for h in steps] step_ratio = self._compute_step_ratio(steps) self._set_richardson_rule(step_ratio, self.richardson_terms) return self._vstack(results, steps) @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) # h = _default_base_step(x, 3, base_step, n) ee = np.diag(h) hes = 2. * np.outer(h, h) for i in range(n): for j in range(i, n): hes[i, j] = (f(x + 1j * ee[i] + ee[j], *args, **kwargs) - f(x + 1j * ee[i] - ee[j], *args, **kwargs) ).imag / hes[j, i] hes[j, i] = hes[i, j] return hes @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) # h = _default_base_step(x, 4, base_step, n) 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) # NOTE: ridout suggesting using eps**(1/4)*theta # h = _default_base_step(x, 3, base_step, n) 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) def test_docstrings(): import doctest # Whitespace changes between numpy 1.13 and 1.14 will cause the doctests # to fail; when doctests are updated to 1.14 format, this can be removed. try: # CRUFT np.set_printoptions(legacy='1.13') except TypeError: pass doctest.testmod(optionflags=doctest.NORMALIZE_WHITESPACE) if __name__ == '__main__': # pragma : no cover test_docstrings()