Source code for sherpa.utils

#
#  Copyright (C) 2007, 2015, 2016, 2018 - 2024
#  Smithsonian Astrophysical Observatory
#
#
#  This program is free software; you can redistribute it and/or modify
#  it under the terms of the GNU General Public License as published by
#  the Free Software Foundation; either version 3 of the License, or
#  (at your option) any later version.
#
#  This program is distributed in the hope that it will be useful,
#  but WITHOUT ANY WARRANTY; without even the implied warranty of
#  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
#  GNU General Public License for more details.
#
#  You should have received a copy of the GNU General Public License along
#  with this program; if not, write to the Free Software Foundation, Inc.,
#  51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
#

"""Objects and utilities used by multiple Sherpa subpackages.

Code in this module should not be considered stable, as it may be
moved, changed, or removed.

"""

import inspect
import logging
import operator
import os
import pydoc
import string
import sys
from types import FunctionType, MethodType
from typing import Any, Callable, Generic, Optional, TypeVar
import warnings

import numpy as np

# Note: _utils.gsl_fcmp and _utils.ndtri are not exported from
#       this module; is this intentional?
from sherpa.utils._utils import hist1d, hist2d  # type: ignore
from sherpa.utils import _utils  # type: ignore
from sherpa.utils import _psf    # type: ignore
from sherpa.utils.err import IOErr
from sherpa.utils.random import poisson_noise

# We re-export the symbols from parallel but this will be removed at
# some point.
#
from sherpa.utils.parallel import multi as _multi, ncpus as _ncpus, \
    parallel_map, parallel_map_funcs, run_tasks


warning = logging.getLogger("sherpa").warning
debug = logging.getLogger("sherpa").debug

__all__ = ('NoNewAttributesAfterInit',
           '_guess_ampl_scale', 'apache_muller', 'bisection', 'bool_cast',
           'calc_ftest', 'calc_mlr', 'calc_total_error', 'create_expr',
           'create_expr_integrated',
           'dataspace1d', 'dataspace2d', 'demuller',
           'erf', 'export_method', 'extract_kernel',
           'filter_bins', 'gamma', 'get_fwhm',
           'get_keyword_defaults', 'get_keyword_names', 'get_midpoint',
           'get_num_args', 'get_peak', 'get_position', 'get_valley',
           'guess_amplitude', 'guess_amplitude2d', 'guess_amplitude_at_ref',
           'guess_bounds', 'guess_fwhm', 'guess_position', 'guess_radius',
           'guess_reference', 'histogram1d', 'histogram2d', 'igam', 'igamc',
           'incbet', 'interpolate', 'is_binary_file', 'Knuth_close',
           'lgam', 'linear_interp', 'nearest_interp',
           'neville', 'neville2d',
           'new_muller', 'normalize',
           'pad_bounding_box', 'parallel_map', 'parallel_map_funcs',
           'param_apply_limits', 'parse_expr', 'poisson_noise',
           'print_fields', 'rebin',
           'sao_arange', 'sao_fcmp', 'send_to_pager',
           'set_origin', 'sum_intervals', 'zeroin',
           'multinormal_pdf', 'multit_pdf', 'get_error_estimates', 'quantile',
           )


_guess_ampl_scale = 1.e+3
"""The scaling applied to a value to create its range.

The minimum and maximum values for a range are calculated by
dividing and multiplying the value by ``_guess_ampl_scale``.
"""

###############################################################################
#
# Types
#
###############################################################################



###############################################################################


[docs] class NoNewAttributesAfterInit(): """ Prevents attribute deletion and setting of new attributes after __init__ has been called. Derived classes must call NoNewAttributesAfterInit.__init__ after all other initialization. """ __initialized = False # Use name mangling def __init__(self): self.__initialized = True def __delattr__(self, name): if self.__initialized and hasattr(self, name): raise AttributeError(f"'{type(self).__name__}' object attribute '{name}' " "cannot be deleted") object.__delattr__(self, name) def __setattr__(self, name, val): if self.__initialized and (not hasattr(self, name)): raise AttributeError(f"'{type(self).__name__}' object has no attribute '{name}'") if self.__initialized and hasattr(self, name): cname = callable(getattr(self, name)) cval = callable(val) if cname and not cval: raise AttributeError(f"'{type(self).__name__}' object attribute '{name}' " "cannot be replaced with a non-callable attribute") if not cname and cval: raise AttributeError(f"'{type(self).__name__}' object attribute '{name}' " "cannot be replaced with a callable attribute") object.__setattr__(self, name, val)
############################################################################### # # Compiled Utilities: _utils # ###############################################################################
[docs] def calc_ftest(dof1, stat1, dof2, stat2): """Compare two models using the F test. The F-test is a model comparison test; that is, it is a test used to select from two competing models which best describes a particular data set. A model comparison test statistic, T, is created from the best-fit statistics of each fit; as with all statistics, it is sampled from a probability distribution p(T). The test significance is defined as the integral of p(T) from the observed value of T to infinity. The significance quantifies the probability that one would select the more complex model when in fact the null hypothesis is correct. See also `calc_mlr`. Parameters ---------- dof1 : int or sequence of int degrees of freedom of the simple model stat1 : number or sequence of number best-fit chi-square statistic value of the simple model dof2 : int or sequence of int degrees of freedom of the complex model stat2 : number or sequence of number best-fit chi-square statistic value of the complex model Returns ------- sig : number or ndarray The significance, or p-value. A standard threshold for selecting the more complex model is significance < 0.05 (the '95% criterion' of statistics). See Also -------- calc_mlr, incbet Notes ----- The F test uses the ratio of the reduced chi2, which follows the F-distribution, (stat1/dof1) / (stat2/dof2). The incomplete Beta function is used to calculate the integral of the tail of the F-distribution. The F test should only be used when: - the simpler of the two models is nested within the other; that is, one can obtain the simpler model by setting the extra parameters of the more complex model (often to zero or one); - the extra parameters have values sampled from normal distributions under the null hypothesis (i.e., if one samples many datasets given the null hypothesis and fits these data with the more complex model, the distributions of values for the extra parameters must be Gaussian); - those normal distributions are not truncated by parameter space boundaries; - the best-fit statistics are sampled from the chi-square distribution. See Protassov et al. 2002 [1]_ for more discussion. References ---------- .. [1] Protassov et al., Statistics, Handle with Care: Detecting Multiple Model Components with the Likelihood Ratio Test, Astrophysical Journal, vol 571, pages 545-559, 2002, http://adsabs.harvard.edu/abs/2002ApJ...571..545P Examples -------- >>> calc_ftest(11, 16.3, 10, 10.2) 0.03452352914891555 >>> calc_ftest([11, 11], [16.3, 16.3], [10, 9], [10.2, 10.5]) array([0.03452353, 0.13819987]) """ return _utils.calc_ftest(dof1, stat1, dof2, stat2)
[docs] def calc_mlr(delta_dof, delta_stat): """Compare two models using the Maximum Likelihood Ratio test. The Maximum Likelihood Ratio (MLR) test is a model comparison test; that is, it is a test used to select from two competing models which best describes a particular data set. A model comparison test statistic, T, is created from the best-fit statistics of each fit; as with all statistics, it is sampled from a probability distribution p(T). The test significance is defined as the integral of p(T) from the observed value of T to infinity. The significance quantifies the probability that one would select the more complex model when in fact the null hypothesis is correct. See also `calc_ftest`. Parameters ---------- delta_dof : int or sequence of int change in the number of degrees of freedom delta_stat : number or sequence of number change in the best-fit statistic value Returns ------- sig : number or ndarray The significance, or p-value. A standard threshold for selecting the more complex model is significance < 0.05 (the '95% criterion' of statistics). See Also -------- calc_ftest Notes ----- The MLR test should only be used when: - the simpler of the two models is nested within the other; that is, one can obtain the simpler model by setting the extra parameters of the more complex model (often to zero or one); - the extra parameters have values sampled from normal distributions under the null hypothesis (i.e., if one samples many datasets given the null hypothesis and fits these data with the more complex model, the distributions of values for the extra parameters must be Gaussian); - those normal distributions are not truncated by parameter space boundaries; - the best-fit statistics for each fit are sampled from the chi-square distribution. See Protassov et al. 2002 [1]_ for more discussion. References ---------- .. [1] Protassov et al., Statistics, Handle with Care: Detecting Multiple Model Components with the Likelihood Ratio Test, Astrophysical Journal, vol 571, pages 545-559, 2002, http://adsabs.harvard.edu/abs/2002ApJ...571..545P Examples -------- In this example, the more-complex model has 2 extra degrees of freedom and a statistic value that is larger by 3.7. The MLR test does not provide any evidence that the complex model is a better fit to the data than the simple model since the result is much larger than 0. >>> calc_mlr(2, 3.7) 0.15723716631362761 """ return _utils.calc_mlr(delta_dof, delta_stat)
[docs] def erf(x): """Calculate the error function. Parameters ---------- x : scalar or array Returns ------- val : scalar or array The error function of the input. See Also -------- gamma Examples -------- >>> erf(0) 0.0 >>> erf([1.0, 2.3]) array([ 0.84270079, 0.99885682]) """ return _utils.erf(x)
[docs] def igamc(a, x): """Calculate the complement of the regularized incomplete Gamma function (upper). The function is defined using the regularized incomplete Gamma function - igam(a,x) - and the Gamma function - gamma(a) - as:: igamc(a,x) = 1 - igam(a,x) = 1 / gamma(a) Int_x^Inf e^(-t) t^(a-1) dt Parameters ---------- a : scalar or array a > 0 x : scalar or array x > 0 Returns ------- val : scalar or array The incomplete Gamma function of the input. See Also -------- gamma, igam Notes ----- In this implementation, which is provided by the Cephes Math Library [1]_, both arguments must be positive. The integral is evaluated by either a power series or continued fraction expansion, depending on the relative values of a and x. Using IEEE arithmetic, the relative errors are ======== ====== ======== ======= ======= domain domain # trials peak rms ======== ====== ======== ======= ======= 0.5,100 0,100 200000 1.9e-14 1.7e-15 0.01,0.5 0,100 200000 1.4e-13 1.6e-15 ======== ====== ======== ======= ======= References ---------- .. [1] Cephes Math Library Release 2.0: April, 1987. Copyright 1985, 1987 by Stephen L. Moshier. Direct inquiries to 30 Frost Street, Cambridge, MA 02140. Examples -------- >>> igamc(1, 2) 0.1353352832366127 >>> igamc([1,1], [2,3]) array([ 0.13533528, 0.04978707]) """ return _utils.igamc(a, x)
[docs] def igam(a, x): """Calculate the regularized incomplete Gamma function (lower). The function is defined using the complete Gamma function - gamma(a) - as:: igam(a,x) = 1 / gamma(a) Int_0^x e^(-t) t^(a^-1) dt Parameters ---------- a : scalar or array a > 0 x : scalar or array x > 0 Returns ------- val : scalar or array The incomplete Gamma function of the input. See Also -------- gamma, igamc Notes ----- In this implementation, which is provided by the Cephes Math Library [1]_, both arguments must be positive. The integral is evaluated by either a power series or continued fraction expansion, depending on the relative values of a and x. Using IEEE arithmetic, the relative errors are ====== ======== ======= ======= domain # trials peak rms ====== ======== ======= ======= 0,30 200000 3.6e-14 2.9e-15 0,100 300000 9.9e-14 1.5e-14 ====== ======== ======= ======= References ---------- .. [1] Cephes Math Library Release 2.0: April, 1987. Copyright 1985, 1987 by Stephen L. Moshier. Direct inquiries to 30 Frost Street, Cambridge, MA 02140. Examples -------- >>> igam(1, 2) 0.8646647167633873 >>> igam([1,1], [2,3]) array([ 0.86466472, 0.95021293]) """ return _utils.igam(a, x)
[docs] def incbet(a, b, x): """Calculate the incomplete Beta function. The function is defined as:: sqrt(a+b)/(sqrt(a) sqrt(b)) Int_0^x t^(a-1) (1-t)^(b-1) dt and the integral from x to 1 can be obtained using the relation:: 1 - incbet(a, b, x) = incbet(b, a, 1-x) Parameters ---------- a : scalar or array a > 0 b : scalar or array b > 0 x : scalar or array 0 <= x <= 1 Returns ------- val : scalar or array The incomplete beta function calculated from the inputs. See Also -------- calc_ftest Notes ----- In this implementation, which is provided by the Cephes Math Library [1]_, the integral is evaluated by a continued fraction expansion or, when b*x is small, by a power series. Using IEEE arithmetic, the relative errors are (tested uniformly distributed random points (a,b,x) with a and b in 'domain' and x between 0 and 1): ======== ======== ======= ======= domain # trials peak rms ======== ======== ======= ======= 0,5 10000 6.9e-15 4.5e-16 0,85 250000 2.2e-13 1.7e-14 0,1000 30000 5.3e-12 6.3e-13 0,1000 250000 9.3e-11 7.1e-12 0,100000 10000 8.7e-10 4.8e-11 ======== ======== ======= ======= Outputs smaller than the IEEE gradual underflow threshold were excluded from these statistics. References ---------- .. [1] Cephes Math Library Release 2.0: April, 1987. Copyright 1985, 1987 by Stephen L. Moshier. Direct inquiries to 30 Frost Street, Cambridge, MA 02140. Examples -------- >>> incbet(0.3, 0.6, 0.5) 0.68786273145845922 >>> incbet([0.3,0.3], [0.6,0.7], [0.5,0.4]) array([ 0.68786273, 0.67356524]) """ return _utils.incbet(a, b, x)
[docs] def gamma(z): """Calculate the Gamma function. Parameters ---------- z : scalar or array -171 <= z <- 171.6 Returns ------- val : scalar or array The gamma function of the input. See Also -------- igam, lgam Notes ----- This implementation is provided by the Cephes Math Library [1]_. Arguments ``|x| >= 34`` are reduced by recurrence and the function approximated by a rational function of degree 6/7 in the interval (2,3). Large arguments are handled by Stirling's formula. Large negative arguments are made positive using a reflection formula. Relative errors are ======== ======== ======= ======= domain # trials peak rms ======== ======== ======= ======= -170,33 20000 2.3e-15 3.3e-16 -33,33 20000 9.4e-16 2.2e-16 33,171.6 20000 2.3e-15 3.2e-16 ======== ======== ======= ======= Errors for arguments outside the test range will be larger owing to amplification by the exponential function. References ---------- .. [1] Cephes Math Library Release 2.0: April, 1987. Copyright 1985, 1987 by Stephen L. Moshier. Direct inquiries to 30 Frost Street, Cambridge, MA 02140. Examples -------- >>> gamma(2.3) 1.1667119051981603 >>> gamma([2.3,1.9]) array([ 1.16671191, 0.96176583]) """ return _utils.gamma(z)
[docs] def lgam(z): """Calculate the log (base e) of the Gamma function. Parameters ---------- z : scalar or array 0 <= z <= 2.556348e305 Returns ------- val : scalar or array The log of the Gamma function of the input. See Also -------- gamma, igam Notes ----- This implementation is provided by the Cephes Math Library [1]_. For arguments greater than 13, the logarithm of the Gamma function is approximated by the logarithmic version of Stirling's formula using a polynomial approximation of degree 4. Arguments between -33 and +33 are reduced by recurrence to the interval [2,3] of a rational approximation. The cosecant reflection formula is employed for arguments less than -33. Relative errors are =============== ======== ======= ======= domain # trials peak rms =============== ======== ======= ======= 0,3 28000 5.4e-16 1.1e-16 2.718,2.556e305 40000 3.5e-16 8.3e-17 =============== ======== ======= ======= The error criterion was relative when the function magnitude was greater than one but absolute when it was less than one. The following test used the relative error criterion, though at certain points the relative error could be much higher than indicated. ======= ======== ======= ======= domain # trials peak rms ======= ======== ======= ======= -200,-4 10000 4.8e-16 1.3e-16 ======= ======== ======= ======= References ---------- .. [1] Cephes Math Library Release 2.0: April, 1987. Copyright 1985, 1987 by Stephen L. Moshier. Direct inquiries to 30 Frost Street, Cambridge, MA 02140. Examples -------- >>> lgam(104.56) 380.21387239435785 >>> lgam([104.56,2823.4]) array([ 380.21387239, 19607.42734396]) """ return _utils.lgam(z)
[docs] def sao_arange(start, stop, step=None): """Create a range of values between start and stop. See also `numpy.arange` and `numpy.linspace`. Parameters ---------- start, stop : float The start and stop points. step : float or None, optional If not given the step size defaults to 1.0. Returns ------- vals : NumPy array The values start, start + step, ... The last point is the first position where start + n * step >= stop, which means that it can include a point > stop. Examples -------- >>> sao_arange(1, 3) array([ 1., 2., 3.]) >>> sao_arange(1, 3, 0.6) array([ 1. , 1.6, 2.2, 2.8, 3.4]) """ if step is None: return _utils.sao_arange(start, stop) return _utils.sao_arange(start, stop, step)
[docs] def sao_fcmp(x, y, tol): """Compare y to x, using an absolute tolerance. Parameters ---------- x : number or array_like The expected value, or values. y : number or array_like The value, or values, to check. If x is an array, then y must be an array of the same size. If x is a scalar then y can be a scalar or an array. tol : number The absolute tolerance used for comparison. Returns ------- flags : int or array_like 0, 1, or -1 for each value in second. If the values match, then 0, otherwise -1 if the expected value (x) is less than the comparison value (y) or +1 if x is larger than y. See Also -------- Knuth_close Examples -------- >>> sao_fcmp(1, 1.01, 0.01) 0 >>> sao_fcmp(1, [0.9, 1, 1.1], 0.01) array([ 1, 0, -1], dtype=int32) >>> sao_fcmp([1.2, 2.3], [1.22, 2.29], 0.01) array([-1, 0], dtype=int32) """ return _utils.sao_fcmp(x, y, tol)
[docs] def sum_intervals(src, indx0, indx1): """Sum up data within one or more pairs of indexes. Parameters ---------- src : sequence of floats The data to be summed. indx0, indx1 : scalar or sequence of int The pair of indexes over which to sum the src array. The sizes of indx0 and indx1 must match, and each element of indx1 must be at least as large as the corresponding element in indx0. Returns ------- val : scalar or array The sum of the src over the given interval ranges. Notes ----- It is assumed that all indexes are valid. That is, they are in the range [0, length of src). This condition is not checked for. Examples -------- >>> sum_intervals([1.1, 2.2, 3.3, 4.4], 1, 2) 5.5 >>> sum_intervals([1.1, -2.2, 3.3, 4.4], [1, 0], [3, 0]) array([ 5.5, 1.1]) """ return _utils.sum_intervals(src, indx0, indx1)
[docs] def rebin(y0, x0lo, x0hi, x1lo, x1hi): """Rebin a histogram. Parameters ---------- y0 : sequence of numbers The Y values of the histogram to rebin. x0lo, x0hi : sequence of numbers The lower and upper edges of the X values to rebin. They must match the size of `y0`. x1lo, x1hi : sequence of numbers The lower and upper edges of the X values of the output histogram. Returns ------- yout : NumPy array of numbers The re-binned Y values (same size as `x1lo`). """ return _utils.rebin(y0, x0lo, x0hi, x1lo, x1hi)
[docs] def neville(xout, xin, yin): """Polynomial one-dimensional interpolation using Neville's method. The scheme used for interpolation (Neville's method) is described at [1]_. Parameters ---------- xout : array_like The positions at which to interpolate. xin : array_like The x values of the data to be interpolated. This must be sorted so that it is monotonically increasing. yin : array_like The y values of the data to interpolate (must be the same size as ``xin``). Returns ------- yout : NumPy array of numbers The interpolated y values (same size as ``xout``). See Also -------- interpolate, linear_interp, nearest_interp References ---------- .. [1] http://en.wikipedia.org/wiki/Neville%27s_algorithm Examples -------- >>> import numpy as np >>> x = [1.2, 3.4, 4.5, 5.2] >>> y = [12.2, 14.4, 16.8, 15.5] >>> xgrid = np.linspace(2, 5, 5) >>> ygrid = neville(xgrid, x, y) """ return _utils.neville(xout, xin, yin)
############################################################################### # # Compiled Utilities: _psf # ###############################################################################
[docs] def extract_kernel(kernel, dims_kern, dims_new, center, xlo, xhi, widths, radial): """Extract the kernel. Parameters ---------- kernel dims_kern dims_new center xlo xhi widths radial : int Set to 1 if using a radial profile, 0 otherwise. Returns ------- out, dims, frac, lo, hi """ return _psf.extract_kernel(kernel, dims_kern, dims_new, center, xlo, xhi, widths, radial)
[docs] def normalize(xs): """Normalize an array. Parameters ---------- xs : sequence The values to normalize. This must be a 1D array. Returns ------- ns : ndarray The values of xs / sum of xs. """ return _psf.normalize(xs)
[docs] def set_origin(dims, maxindex=None): """Return the position of the origin of the kernel. Parameters ---------- dims : number or sequence The dimensions of the kernel. This should be a scalar or a one- or two-element sequence. maxindex : None or int, optional If given, then use this location - which is the index into the flattened array - as the center, otherwise use the center of the grid. Returns ------- cs : ndarray or number The coordinates of the center, matching the input dims format. Examples -------- >>> set_origin(12) 5 >>> set_origin([12]) array([5]) >>> set_origin([12], 4) array([4]) >>> set_origin([12, 13]) array([5, 6]) >>> set_origin([12, 13], 42) array([6, 3]) """ if maxindex is None: return _psf.set_origin(dims) return _psf.set_origin(dims, maxindex)
[docs] def pad_bounding_box(kernel, mask): """Expand the kernel to match the mask. Parameters ---------- kernel : numeric sequence The data to copy. The data is a 1D array. mask : int sequence The mask determines the size of the output and where to place the kernel values. It is expected that the number of non-zero mask elements matches the size of the `kernel` parameter. Returns ------- nkernel : ndarray The output is the same size as the mask, and initialized to zero everywhere. Cells where the mask is non-zero are copied from the kernel. Examples -------- >>> pad_bounding_box([1, 2, 3, 4], [1, 1, 0, 1, 1, 0, 0, 0, 0]) array([ 1., 2., 0., 3., 4., 0., 0., 0., 0.]) """ return _psf.pad_bounding_box(kernel, mask)
############################################################################### # # Utilities # ############################################################################### # at what precisions do we assume equality in energy grids? eps = np.finfo(np.float32).eps
[docs] def filter_bins(mins, maxes, axislist, integrated=False): """What mask represents the given set of filters? The ranges are treated as inclusive at both ends if integrated is False, the default, otherwise the lower limit is inclusive but the upper limit is exclusive. Parameters ---------- mins : sequence of values The minimum value of the valid range (elements may be None). maxes : sequence of values The maximum value of the valid range (elements may be None). axislist: sequence of arrays The axis to apply the range to. There must be the same number of elements in mins, maxes, and axislist. The number of elements of each element of axislist must also agree (the cell values do not need to match). integrated : bool, optional Is the data integrated (we have low and high bin edges)? The default is False. When True it is expected that axislist contains a even number of rows, where the odd values are the low edges and the even values the upper edges, and that the mins and maxes only ever contain a single value, given in (None, hi) and (lo, None) ordering. Returns ------- mask : ndarray or None A mask indicating whether the values are included (True) or excluded (False). If any of the input sequences are empty then None will be returned. Examples -------- Calculate those points in xs which are in the range 1.5 <= x <= 4. >>> xs = [1, 2, 3, 4, 5] >>> filter_bins([1.5], [4], [xs]) array([False, True, True, True, False]) Repeat the above calculation by combining filters for x >= 1.5 and x <= 4 (note that the grid must be repeated for each filter): >>> filter_bins([1.5, None], [None, 4], [xs, xs]) array([False, True, True, True, False]) For integrated data sets the lower and upper edges should be sent separately with the max and min limits, along with setting the integrated flag. The following selects the bins that cover the range 2 to 4 and 1.5 to 3.5: >>> xlo = [1, 2, 3, 4, 5] >>> xhi = [2, 3, 4, 5, 6] >>> filter_bins([None, 2], [4, None], [xlo, xhi], integrated=True) array([False, True, True, False, False]) >>> filter_bins([None, 1.5], [3.5, None], [xlo, xhi], integrated=True) array([ True, True, True, False, False]) """ mask = None def locheck(lo, axis): if integrated: return sao_fcmp(lo, axis, eps) < 0 return sao_fcmp(lo, axis, eps) <= 0 def hicheck(hi, axis): if integrated: return sao_fcmp(hi, axis, eps) > 0 return sao_fcmp(hi, axis, eps) >= 0 for lo, hi, axis in zip(mins, maxes, axislist): if (lo is None) and (hi is None): continue axis = np.asarray(axis) axismask = np.ones(axis.size, dtype=bool) if lo is not None: axismask &= locheck(lo, axis) if hi is not None: axismask &= hicheck(hi, axis) if mask is None: mask = axismask else: mask &= axismask return mask
[docs] def bool_cast(val): """Convert a string to a boolean. Parameters ---------- val : bool, str or sequence The input value to decode. Returns ------- flag : bool or ndarray True or False if val is considered to be a true or false term. If val is a sequence then the return value is an ndarray of the same size. Notes ----- The string is compared in a case-insensitive manner to the following: 'true', 'on', 'yes', '1', 't', and 'y' for `True` and 'false', 'off', 'no', '0', 'f', and 'n' for `False`. If there is no match to the above then the default conversion provided by the `bool` routine is used. """ if type(val) in (tuple, list, np.ndarray): return np.asarray([bool_cast(item) for item in val], bool) if type(val) == str: # since built in bool() only returns false for empty strings vlo = val.lower() if vlo in ('false', 'off', 'no', '0', 'f', 'n'): return False if vlo in ('true', 'on', 'yes', '1', 't', 'y'): return True raise TypeError(f"unknown boolean value: '{val}'") # use built in bool cast return bool(val)
# Telling the type system that the signature of meth is "the same" as # the signature of the return value probably needs Python 3.10 for # ParamSpec and then maybe Python 3.12 for the generic support. #
[docs] def export_method(meth: Callable, name: Optional[str] = None, modname: Optional[str] = None ) -> Callable: """ Given a bound instance method, return a simple function that wraps it. The only difference between the interface of the original method and the generated function is that the latter doesn't include 'self' in its argument list. This means that when the wrapper function is called with an incorrect number of arguments, the error message does not include 'self' in argument counts. The only reason to generate such a wrapper is to hide from a user the fact that they're using an instance method rather than a simple function. If meth is not an instance method, it is returned unchanged. If name is None, the generated function will have the same name as the input method. Otherwise, name must be a string containing the desired name of the new method. If modname is not None, it must be a string and will be used as the module name for the generated function. Note that the caller is responsible for assigning the returned function to an appropriate name in the calling scope. """ if not isinstance(meth, MethodType): return meth # Most of the functionality here can be provided by # functools.wraps (and in fact would add more functionality, such # as the ability to handle keyword-only or positional-only # arguments). It would also likely require less maintenance. # However, it does not handle two important issues: # # a) when an error is raised it is reported as from Session.<name> # rather than <name> # # Attempts to change the __qualname__ field have not been # successful. The wrapped function can include a check for an # exception, manually removing any leading "Session." text # from the message, but it is not particularly good code. # # b) Error messages related to the number of arguments include the # self argument (i.e. are 1 more than the user is told to # expect), which is one of the main reasons for this routine. # # The only time name is not None appears to be in the tests, so # can this feature be removed? # if name is None: name = meth.__name__ if name == meth.__name__: old_name = f'_old_{name}' else: old_name = meth.__name__ defaults = meth.__defaults__ doc = meth.__doc__ def tostr(p): if p.kind == p.VAR_KEYWORD: return f"**{p.name}" if p.kind == p.VAR_POSITIONAL: return f"*{p.name}" return p.name # Ideally this would also identify when to add "/" or "*" # to indicate positional-only or keyword-only arguments. # sig = inspect.signature(meth) argspec = ",".join([tostr(p) for p in sig.parameters.values()]) # Create a wrapper function with no default arguments g: dict[str, Any] = {old_name: meth} # The only time modname is None appears to be the test so can we # make it a required argument? # if modname is not None: g['__name__'] = modname fdef = f'def {name}({argspec}): return {old_name}({argspec})' exec(fdef, g) # Create another new function from the one we just made, this time # adding the default arguments, doc string, and any annotations # from the original method. # # Why does this not change the __defaults__ field of new_meth # rather than creating a copy of it? # new_meth = g[name] new_meth = FunctionType(new_meth.__code__, new_meth.__globals__, new_meth.__name__, defaults, new_meth.__closure__) new_meth.__doc__ = doc new_meth.__annotations__ = meth.__annotations__ return new_meth
[docs] def get_keyword_names(func, skip=0): """Return the names of the keyword arguments. Parameters ---------- func The function to query. skip : int, optional The number of keyword arguments to skip. Returns ------- names : list of str The names of the keyword arguments. It can be empty. See Also -------- get_keyword_defaults, get_num_args """ # This used to use getargspec but was changed to use inspect # since the former was removed briefly (circa Python 3.6). # sig = inspect.signature(func) kwargs = [p.name for p in sig.parameters.values() if p.kind == p.POSITIONAL_OR_KEYWORD and p.default != p.empty] return kwargs[skip:]
[docs] def get_keyword_defaults(func, skip=0): """Return the keyword arguments and their default values. Note that a similar function `sherpa.plot.backend_utils.get_keyword_defaults` exits, which differs from this one in that it deals with keyword-only arguments, not all arguments. Parameters ---------- func The function to query. skip : int, optional The number of keyword arguments to skip. Returns ------- vals : dict The keys are names of the keyword arguments, the values are the default value for that parameter. It can be empty. See Also -------- get_keyword_names, get_num_args, `sherpa.plot.backend_utils.get_keyword_defaults` """ # This used to use getargspec but was changed to use inspect # since the former was removed briefly (circa Python 3.6). # sig = inspect.signature(func) kwargs = [(p.name, p.default) for p in sig.parameters.values() if p.kind == p.POSITIONAL_OR_KEYWORD and p.default != p.empty] return dict(kwargs[skip:])
[docs] def get_num_args(func): """Return the number of arguments for a function. Parameters ---------- func The function to query. Returns ------- ntotal, npos, nkeyword : int, int, int The total number of arguments, the number of positional arguments, and the number of keyword arguments. See Also -------- get_keyword_defaults, get_keyword_names """ # This used to use getargspec but was changed to use inspect # since the former was removed briefly (circa Python 3.6). # sig = inspect.signature(func) posargs = [True for p in sig.parameters.values() if p.kind == p.POSITIONAL_OR_KEYWORD and p.default == p.empty] kwargs = [True for p in sig.parameters.values() if p.kind == p.POSITIONAL_OR_KEYWORD and p.default != p.empty] npos = len(posargs) nkw = len(kwargs) return (npos + nkw, npos, nkw)
[docs] def create_expr(vals, mask=None, format='%s', delim='-'): """Create a string representation of a filter. Use the mask to convert the input values into a set of comma-separated filters - low value and high value, separated by the delimiter - that represent the data. If the mask is not given then the values must be "channel" values (that is, two values are consecutive if there difference is 1). Parameters ---------- vals : sequence The values that represent the sequence if mask is not None, otherwise the selected channel numbers (in this case integer values). mask : sequence of bool or None, optional The mask setting for the full dataset, without any filtering applied. A value of True indicates the element is included and False means it is excluded. format : str, optional The format used to display each value. delim : str, optional The separator for a range. Raises ------ ValueError If the ``vals`` and ``mask`` sequences do not match: the length of ``vals`` must equal the number of True values in ``mask``. See Also -------- create_expr_integrated, parse_expr Examples -------- >>> create_expr([1, 2, 3, 4]) '1-4' >>> create_expr([1, 2, 4, 5, 7]) '1-2,4-5,7' >>> create_expr([1, 2, 3, 4], [True, True, True, True]) '1-4' >>> create_expr([0.1, 0.2, 0.4, 0.8], [True, True, True, True]) '0.1-0.8' >>> create_expr([0.1, 0.2, 0.4, 0.8], [True, True, True, False, False, True]) '0.1-0.4,0.8' """ if len(vals) == 0: return '' if len(vals) == 1: return format % vals[0] if mask is None: seq = vals else: # Ensure we have a boolean array to make indexing behave sensibly # (NumPy 1.17 or so changed behavior related to this). # mask = np.asarray(mask, dtype=bool) # Ensure that the vals and mask array match: the number of # mask=True elements should equal the number of input values. # if sum(mask) != len(vals): raise ValueError("mask array mismatch with vals") # We only care about the difference between two consecutive # values, so it doesn't matter if index starts at 0 or 1. # index = np.arange(len(mask)) seq = index[mask] exprs = [] start = vals[0] # We follow create_expr_integrated but instead of having separate lo/hi # always we use the same array # startbins = vals[1:] endbins = vals[:-1] diffs = np.diff(seq) idxs, = np.where(diffs != 1) for idx in idxs: exprs.append((start, endbins[idx])) start = startbins[idx] exprs.append((start, vals[-1])) def filt(lo, hi): vstr = format % lo if lo == hi: return vstr return vstr + f"{delim}{format % hi}" return ",".join([filt(*expr) for expr in exprs])
def create_expr_integrated(lovals, hivals, mask=None, format='%s', delim='-', eps=np.finfo(np.float32).eps): """Create a string representation of a filter (integrated). Use the mask to convert the input values into a set of comma-separated filters - low value and high value, separated by the delimiter - that represent the data. Unlike `create_expr` this routine uses the lovals values for the start of the bin and hivals for the end of each bin, and assumes that contiguous bins should be combined. Parameters ---------- lovals, hivals : sequence The lower and upper values of each bin. It is required that they are in ascending order and ``lovals`` < ``hivals``. mask : sequence of bool or None, optional The mask setting for the full dataset, without any filtering applied. A value of True indicates the element is included and False means it is excluded. Note that this is opposite to the numpy convention in numpy masked arrays. format : str, optional The format used to display each value. delim : str, optional The separator for a range. eps : number, optional The tolerance for comparing two numbers with sao_fcmp. Raises ------ ValueError If the ``lovals`` and ``hivals`` sequences do not match. See Also -------- create_expr, parse_expr Examples -------- When there is no mask, or all mask values are True, we just show the full range: >>> create_expr_integrated([1, 2, 3, 4], [2, 3, 4, 5]) '1-5' >>> create_expr_integrated([1, 2, 4, 5, 7], [2, 3, 5, 6, 8]) '1-8' >>> create_expr_integrated([0.1, 0.2, 0.4, 0.8], [0.2, 0.4, 0.8, 1.0]) '0.1-1.0' >>> create_expr_integrated([0.1, 0.2, 0.4, 0.8], [0.2, 0.4, 0.6, 1.0], [True, True, True, True]) '0.1-1.0' If a mask is given then it defines the bins that are grouped together, even if the bins are not contiguous: >>> create_expr_integrated([1, 2, 4], [2, 3, 5], [True, True, False, True]) '1-3,4-5' >>> create_expr_integrated([1, 3, 5], [2, 4, 6], [True, True, False]) '1-4,5-6' More examples of the mask controlling the grouping: >>> create_expr_integrated([0.1, 0.2, 0.6, 0.8], [0.2, 0.4, 0.8, 1.0], [True, True, False, True, True]) '0.1-0.4,0.6-1.0' >>> create_expr_integrated([0.1, 0.2, 0.4, 0.8], [0.2, 0.3, 0.5, 1.0], [True, True, False, True, False, True]) '0.1-0.3,0.4-0.5,0.8-1.0' >>> create_expr_integrated([0.1, 0.2, 0.4, 0.8], [0.2, 0.3, 0.5, 1.0], [False, True, True, False, True, False, True, False]) '0.1-0.3,0.4-0.5,0.8-1.0' An interesting case is that you can add a "break" between contiguous bins (this behavior may be changed): >>> create_expr_integrated([1, 2, 3, 4], [2, 3, 4, 5], [True, False, True, True, True]) '1-2,2-5' """ # Follow create_expr. # if len(lovals) != len(hivals): raise ValueError("hivals array mismatch with lovals") if len(lovals) == 0: return '' # To identify where there's a break we use an array of consecutive # integers that have missing data masked out. # if mask is None: seq = np.arange(len(lovals)) else: mask = np.asarray(mask, dtype=bool) if sum(mask) != len(lovals): raise ValueError("mask array mismatch with lovals") seq = np.arange(len(mask)) seq = seq[mask] out = format % lovals[0] startbins = lovals[1:] endbins = hivals[:-1] diffs = np.diff(seq) idxs, = np.where(diffs != 1) for idx in idxs: out += f"{delim}{format % endbins[idx]},{format % startbins[idx]}" out += f"{delim}{format % hivals[-1]}" return out
[docs] def parse_expr(expr): """Convert a filter expression into its parts. This is intended for parsing a notice or ignore expression given as a string. Parameters ---------- expr : str The filter expression, of the form 'a:b' or a single number, separated by commas, and white space is ignored. The upper or lower limit of a pair may be ignored (e.g. 'a:' or ':b'). Returns ------- filters : list of pairs Each pair gives the lower- and upper-edge of the filter, using ``None`` to represent no limit. See Also -------- create_expr, create_expr_int Notes ----- There is no attempt to validate that the expression contains strictly ordered pairs, or that the pairs do not overlap, or that the lower- and upper-limits are in increasing numerical order. That is, the expression '5:7,:2,4:6,5:3' is allowed. Examples -------- >>> parse_expr('0.5:7') [(0.5, 7.0)] >>> parse_expr('0.5:') [(0.5, None)] >>> parse_expr(':7') [(None, 7.0)] >>> parse_expr(':2, 4 : 5 ,7:8,10:') [(None, 2.0), (4.0, 5.0), (7.0, 8.0), (10.0, None)] >>> parse_expr('4') [(4.0, 4.0)] >>> parse_expr(' ') [(None, None)] """ if expr is None or str(expr).strip() == '': return [(None, None)] res = [] vals = str(expr).strip().split(',') for val in vals: lo, hi = None, None interval = val.strip().split(':') ninterval = len(interval) if ninterval == 1: lo = interval[0] if lo == '': lo = None hi = lo elif ninterval == 2: lo = interval[0] hi = interval[1] if lo == '': lo = None if hi == '': hi = None else: # This check exited but was never hit due to the way the # code was written. It now errors out if a user gives # a:b:c, whereas the old version would have just ignored # the ':c' part. Perhaps we should just keep dropping # it, in case there's existing code that assumes this? # raise TypeError("interval syntax requires a tuple, 'lo:hi'") if lo is not None: try: lo = float(lo) except ValueError: raise TypeError(f"Invalid lower bound '{lo}'") from None if hi is not None: try: hi = float(hi) except ValueError: raise TypeError(f"Invalid upper bound '{hi}'") from None res.append((lo, hi)) return res
[docs] def calc_total_error(staterror=None, syserror=None): """Add statistical and systematic errors in quadrature. Parameters ---------- staterror : array, optional The statistical error, or ``None``. syserror : array, optional The systematic error, or ``None``. Returns ------- error : array or ``None`` The errors, added in quadrature. If both ``staterror`` and ``syserror`` are ``None`` then the return value is ``None``. """ if (staterror is None) and (syserror is None): error = None elif (staterror is not None) and (syserror is None): error = staterror elif (staterror is None) and (syserror is not None): error = syserror else: error = np.sqrt(staterror * staterror + syserror * syserror) return error
[docs] def quantile(sorted_array, f): """Return the quantile element from sorted_array, where f is [0,1] using linear interpolation. Based on the description of the GSL routine gsl_stats_quantile_from_sorted_data - e.g. http://www.gnu.org/software/gsl/manual/html_node/Median-and-Percentiles.html but all errors are my own. sorted_array is assumed to be 1D and sorted. """ sorted_array = np.asarray(sorted_array) if len(sorted_array.shape) != 1: raise RuntimeError("Error: input array is not 1D") n = sorted_array.size q = (n - 1) * f i = int(np.floor(q)) delta = q - i return (1.0 - delta) * sorted_array[i] + delta * sorted_array[i + 1]
[docs] def get_error_estimates(x, sorted=False): """Compute the median and (-1,+1) sigma values for the data. Parameters ---------- x : array of numbers The input values. sorted : bool, optional If ``False``, the default, then ``x`` is assumed to not be sorted. Returns ------- (median, lsig, usig) The median, value that corresponds to -1 sigma, and value that is +1 sigma, for the input distribution. Examples -------- >>> (m, l, h) = get_error_estimates(x) """ xs = np.asarray(x) if not sorted: xs.sort() xs = np.array(xs) sigfrac = 0.682689 median = quantile(xs, 0.5) lval = quantile(xs, (1 - sigfrac) / 2.0) hval = quantile(xs, (1 + sigfrac) / 2.0) return (median, lval, hval)
[docs] def multinormal_pdf(x, mu, sigma): """The PDF of a multivariate-normal. Returns the probability density function (PDF) of a multivariate normal [1]_. Parameters ---------- x : array An array of length k. mu : array An array of length k. sigma : array A matrix of size (k,k). It must be symmetric and positive-definite. See Also -------- multit_pdf References ---------- .. [1] http://en.wikipedia.org/wiki/Multivariate_normal_distribution """ x = np.asarray(x) mu = np.asarray(mu) sigma = np.asarray(sigma) if x.size != mu.size: raise TypeError("x and mu sizes do not match") if mu.size != sigma.diagonal().size: raise TypeError("sigma shape does not match x") if np.min(np.linalg.eigvalsh(sigma)) <= 0: raise ValueError("sigma is not positive definite") if np.max(np.abs(sigma - sigma.T)) >= 1.e-9: raise ValueError("sigma is not symmetric") rank = mu.size coeff = 1.0 / (np.power(2.0 * np.pi, rank / 2.0) * np.sqrt(np.abs(np.linalg.det(sigma)))) xmu = np.asarray(x - mu) invsigma = np.asarray(np.linalg.inv(sigma)) # The matrix multiplication looks backwards, but mu and x # are passed in already transposed. # # mu = [[a,b,c]] # x = [[d,e,f]] # out = coeff * np.exp(-0.5 * ((xmu @ invsigma) @ xmu.T)) return float(out)
[docs] def multit_pdf(x, mu, sigma, dof): """The PDF of a multivariate student-t. Returns the probability density function (PDF) of a multivariate student-t distribution [1]_. Parameters ---------- x : array An array of length k. mu : array An array of length k. sigma : array A matrix of size (k,k). It must be symmetric and positive-definite. dof : int See Also -------- multinormal_pdf References ---------- .. [1] http://en.wikipedia.org/wiki/Multivariate_Student_distribution """ n = float(dof) x = np.asarray(x) mu = np.asarray(mu) sigma = np.asarray(sigma) if x.size != mu.size: raise TypeError("x and mu sizes do not match") if mu.size != sigma.diagonal().size: raise TypeError("sigma shape does not match x") if np.min(np.linalg.eigvalsh(sigma)) <= 0: raise ValueError("sigma is not positive definite") if np.max(np.abs(sigma - sigma.T)) >= 1.e-9: raise ValueError("sigma is not symmetric") rank = mu.size npr = float(n + rank) coeff = (gamma(npr / 2.0) / (gamma(n / 2.0) * np.power(n, rank / 2.0) * np.power(np.pi, rank / 2.0) * np.sqrt(np.abs(np.linalg.det(sigma))))) xmu = np.asarray(x - mu) invsigma = np.asarray(np.linalg.inv(sigma)) # The matrix multiplication looks backwards, but mu and x # are passed in already transposed. # # mu = [[a,b,c]] # x = [[d,e,f]] # term = 1.0 + 1.0 / n * ((xmu @ invsigma) @ xmu.T) out = coeff * np.power(term, -npr / 2.0) return float(out)
[docs] def dataspace1d(start, stop, step=1, numbins=None): """ Populates an integrated grid if numbins is None (default) -> numpy.arange(start,stop,step) if numbins is not None -> numpy.linspace(start, stop, numbins) """ if start >= stop: raise TypeError(f"input should be start < stop, found start={start} stop={stop}") if numbins is None: if step <= 0: raise TypeError(f"input should be step > 0, found step={step}") if step >= (stop - start): raise TypeError( f"input has produced less than 2 bins, found start={start} stop={stop} step={step}") # xx = np.arange(start, stop, step, dtype=float) # xx = sao_arange(start, stop, step) xx = None if numbins is not None: if numbins <= 1: raise TypeError( f"input should be numbins > 1, found numbins={numbins}") xx = np.linspace(start, stop, numbins + 1) else: xx = sao_arange(start, stop, step) xlo = np.array(xx[:-1]) xhi = np.array(xx[1:]) y = np.zeros(len(xlo), dtype=float) return xlo, xhi, y
[docs] def dataspace2d(dim): """ Populates a blank image dataset """ if not np.iterable(dim): raise TypeError("dim must be an array of dimensions") if len(dim) < 2: raise TypeError("dimensions for dataspace2d must be > 1") if dim[0] < 1 or dim[1] < 1: raise TypeError(f"dimensions should be > 0, found dim0 {dim[0]} dim1 {dim[1]}") x0 = np.arange(dim[0], dtype=float) + 1.0 x1 = np.arange(dim[1], dtype=float) + 1.0 x0, x1 = np.meshgrid(x0, x1) shape = tuple(x0.shape) x0 = x0.ravel() x1 = x1.ravel() y = np.zeros(np.prod(dim)) return x0, x1, y, shape
[docs] def histogram1d(x, x_lo, x_hi): """Create a 1D histogram from a sequence of samples. See the `numpy.histogram` routine for a version with more options. .. versionchanged:: 4.15.0 The x_lo and x_hi arguments are no-longer changed (sorted) by this routine. Parameters ---------- x : sequence of numbers The array of samples x_lo : sequence of numbers The lower-edges of each bin. x_hi : sequence of numbers The upper-edges of each bin, which must be the same size as ``x_lo``. Returns ------- y : NumPy array The number of samples in each histogram bin defined by the ``x_lo`` and ``x_hi`` arrays. Examples -------- A simple example, calculating the histogram of 1000 values randomly distributed over [0, 1). >>> import numpy as np >>> rng = np.random.default_rng() >>> x = rng.random(1000) >>> edges = np.linspace(0, 1, 11) >>> xlo = edges[:-1] >>> xhi = edges[1:] >>> y = histogram1d(x, xlo, xhi) Given a list of samples, bin them up so that they can be used as the dependent axis (the value to be fitted) in a Sherpa data set: >>> dataspace1d(1, 10, 1) >>> lo, hi = get_indep() >>> n = histogram1d([2, 3, 2, 8, 5, 2], lo, hi) >>> set_dep(n) """ x_lo = np.asarray(x_lo).copy() x_hi = np.asarray(x_hi).copy() x_lo.sort() x_hi.sort() return hist1d(np.asarray(x), x_lo, x_hi)
[docs] def histogram2d(x, y, x_grid, y_grid): """Create 2D histogram from a sequence of samples. See the `numpy.histogram2d` routine for a version with more options. .. versionchanged:: 4.15.0 The x_grid and y_grid arguments are no-longer changed (sorted) by this routine. Parameters ---------- x : sequence of numbers The array of samples (X coordinate) y : sequence of numbers The array of samples (Y coordinate), which must have the same size as the ``x`` sequence. x_grid : sequence of numbers The X bin edges. y_grid : sequence of numbers The Y bin edges. Returns ------- y : NumPy array The number of samples in each histogram bin defined by the ``x_grid`` and ``y_grid`` arrays. Examples -------- Given a list of coordinates (``xvals``, ``yvals``), bin them up so that they match the 5 by 10 pixel image data space. In this case the X grid is [1, 2, ..., 5] and the Y grid is [1, 2, .., 10]. >>> dataspace2d([5, 10]) >>> (xgrid, ygrid) = get_axes() >>> n = histogram2d(xvals, yvals, xgrid, ygrid) >>> set_dep(n) """ x_grid = np.asarray(x_grid).copy() y_grid = np.asarray(y_grid).copy() x_grid.sort() y_grid.sort() vals = hist2d(np.asarray(x), np.asarray(y), x_grid, y_grid) return vals.reshape((len(x_grid), len(y_grid)))
def interp_util(xout, xin, yin): lenxin = len(xin) i1 = np.searchsorted(xin, xout) i1[i1 == 0] = 1 i1[i1 == lenxin] = lenxin - 1 # if 0 == i1: # i1 = 1 # if lenxin == i1: # i1 = lenxin - 1 x0 = xin[i1 - 1] x1 = xin[i1] y0 = yin[i1 - 1] y1 = yin[i1] return x0, x1, y0, y1
[docs] def linear_interp(xout, xin, yin): """Linear one-dimensional interpolation. Parameters ---------- xout : array_like The positions at which to interpolate. xin : array_like The x values of the data to interpolate. This must be sorted so that it is monotonically increasing. yin : array_like The y values of the data to interpolate (must be the same size as ``xin``). Returns ------- yout : NumPy array of numbers The interpolated y values (same size as ``xout``). See Also -------- interpolate, nearest_interp, neville Examples -------- >>> import numpy as np >>> x = np.asarray([1.2, 3.4, 4.5, 5.2]) >>> y = np.asarray([12.2, 14.4, 16.8, 15.5]) >>> xgrid = np.linspace(2, 5, 5) >>> ygrid = linear_interp(xgrid, x, y) """ x0, x1, y0, y1 = interp_util(xout, xin, yin) val = (xout - x0) / (x1 - x0) * (y1 - y0) + y0 if np.isnan(val).any(): # to handle the case where two adjacent elements of xout are equal return nearest_interp(xout, xin, yin) return val
[docs] def nearest_interp(xout, xin, yin): """Nearest-neighbor one-dimensional interpolation. Parameters ---------- xout : array_like The positions at which to interpolate. xin : array_like The x values of the data to interpolate. This must be sorted so that it is monotonically increasing. yin : array_like The y values of the data to interpolate (must be the same size as ``xin``). Returns ------- yout : NumPy array of numbers The interpolated y values (same size as ``xout``). See Also -------- interpolate, linear_interp, neville Examples -------- >>> import numpy as np >>> x = np.asarray([1.2, 3.4, 4.5, 5.2]) >>> y = np.asarray([12.2, 14.4, 16.8, 15.5]) >>> xgrid = np.linspace(2, 5, 5) >>> ygrid = nearest_interp(xgrid, x, y) """ x0, x1, y0, y1 = interp_util(xout, xin, yin) return np.where((np.abs(xout - x0) < np.abs(xout - x1)), y0, y1)
[docs] def interpolate(xout, xin, yin, function=linear_interp): """One-dimensional interpolation. Parameters ---------- xout : array_like The positions at which to interpolate. xin : array_like The x values of the data to interpolate. This must be sorted so that it is monotonically increasing. yin : array_like The y values of the data to interpolate (must be the same size as ``xin``). function : func, optional The function to perform the interpolation. It accepts the arguments (xout, xin, yin) and returns the interpolated values. The default is to use linear interpolation. Returns ------- yout : array_like The interpolated y values (same size as ``xout``). See Also -------- linear_interp, nearest_interp, neville Examples -------- Use linear interpolation to calculate the Y values for the ``xgrid`` array: >>> import numpy as np >>> x = np.asarray([1.2, 3.4, 4.5, 5.2]) >>> y = np.asarray([12.2, 14.4, 16.8, 15.5]) >>> xgrid = np.linspace(2, 5, 5) >>> ygrid = interpolate(xgrid, x, y) Use Neville's algorithm for the interpolation: >>> ygrid = interpolate(xgrid, x, y, neville) """ if not callable(function): raise TypeError(f"input function '{repr(function)}' is not callable") return function(xout, xin, yin)
[docs] def is_binary_file(filename): """Estimate if a file is a binary file. Parameters ---------- filename : str The name of the file. Returns ------- flag : bool Returns True if a non-printable character is found in the first 1024 bytes of the file. Notes ----- For this function, "binary" means the file contains a non-ASCII character. """ # Originally "binary" was defined as a character not being in # string.printable. With Python 3, we can also use UnicodeDecodeError # as an indicator of a "binary" file, but the check against # string.printable is kept in, since this is more restrictive # than UnicodeDecodeError. # with open(filename, 'r') as fd: try: lines = fd.readlines(1024) except UnicodeDecodeError: return True if len(lines) == 0: return False # Are there any non-printable characters in the buffer? for line in lines: for char in line: if char not in string.printable: return True return False
[docs] def get_midpoint(a): """Estimate the middle of the data. Parameters ---------- a : array_like The data points. Returns ------- ans : scalar The middle of the data. See Also -------- get_peak, guess_bounds Notes ----- The estimate is based on the range of the data, and not the distribution of the points. """ # return np.abs(a.max() - a.min())/2. + a.min() return np.abs(a.max() + a.min()) / 2.0
[docs] def get_peak(y, x, xhi=None): """Estimate the peak position of the data. Parameters ---------- y, x : array_like The data points. xhi : None or array_like, optional If given then the x array is taken to be the low-edge of each bin. Returns ------- ans : scalar The X position of the peak. See Also -------- get_fwhm, get_midpoint, get_valley Notes ----- If there are multiple peaks of the same height then the location of the first peak is used. """ return x[y.argmax()]
[docs] def get_valley(y, x, xhi=None): """Estimate the position of the minimum of the data. Parameters ---------- y, x : array_like The data points. xhi : None or array_like, optional If given then the x array is taken to be the low-edge of each bin. Returns ------- ans : scalar The X position of the minimum. See Also -------- get_fwhm, get_peak Notes ----- If there are multiple minima with the same value then the location of the first minimum is used. """ return x[y.argmin()]
[docs] def get_fwhm(y, x, xhi=None): """Estimate the width of the data. This is only valid for positive data values (``y``). Parameters ---------- y, x : array_like The data points. The x array must be in ascending order. xhi : None or array_like, optional If given then the x array is taken to be the low-edge of each bin. This is unused. Returns ------- ans : scalar An estimate of the full-width half-maximum of the peak. If the data is negative, or no edge is found then half the X range is returned. See Also -------- get_peak, get_valley, guess_fwhm Notes ----- If there are multiple peaks of the same height then the first peak is used. The approach is to find the maximum position and then extend out to the first bins which fall below half the height. The difference of the two points is used. If only one side falls below the value then twice this separation is used. If the half-height is not reached then the value is set to be half the width of the x array. In all cases the upper-edge of the x arrays is ignored, if given. """ # Pick half the width of the X array, purely as a guess. # The x array is required to be ordered, so we can just # take the first and last points. # guess_fwhm = (x[-1] - x[0]) / 2 y_argmax = y.argmax() if y[y_argmax] <= 0: return guess_fwhm half_max_val = y[y_argmax] / 2.0 x_max = x[y_argmax] # Where do the values fall below the half-height? The assumption # is that the arrays are not so large that evaluating the whole # array, rather than just looping out from the maximum location, # is not an expensive operation. # flags = (y - half_max_val) < 0 # Find the distances from these points to the # maximum location. # dist = x[flags] - x_max # We want the maximum value of the negative distances, # and the minimum value of the positive distances. # There's no guarantee either exist. # try: ldist = -1 * max(dist[dist < 0]) except ValueError: ldist = None try: rdist = min(dist[dist > 0]) except ValueError: rdist = None # If we have both HWHM values then sum them and use that, # otherwise if we have one then double it. # if ldist is not None and rdist is not None: return ldist + rdist if ldist is not None: return 2 * ldist if rdist is not None: return 2 * rdist # No value, so use the guess. # return guess_fwhm
[docs] def guess_fwhm(y, x, xhi=None, scale=1000): """Estimate the value and valid range for the FWHM of the data. Parameters ---------- y, x : array_like The data points. xhi : None or array_like, optional If given then the x array is taken to be the low-edge of each bin. scale : number, optional The scaling factor applied to the value to calculate the minimum (divide) and maximum (multiply) value for the FWHM range. Returns ------- ans : dict The keys are 'val', 'min', and 'max', which give the full-width half-maximum and its range. See Also -------- get_fwhm Notes ----- If there are multiple peaks of the same height then the first peak is used. """ fwhm = get_fwhm(y, x, xhi) return {'val': fwhm, 'min': fwhm / scale, 'max': fwhm * scale}
[docs] def param_apply_limits(param_limits, par, limits=True, values=True): """Apply the given limits to a parameter. This is primarily used by the ``guess`` routine of a model to set one or more of its parameters to a given value or range. Parameters ---------- param_limits : dict par : sherpa.models.parameter.Parameter instance If the parameter is frozen then nothing is changed. limits : bool, optional The parameter limits are not changed when ``values`` is ``True`` and ``limits`` is ``False``. In all other cases the limits are changed. values : bool, optional When ``True`` the parameter value is changed and the original value is stored (for use by the parameter's ``reset`` method). Examples -------- Create an initial guess for the ``mdl.fwhm`` parameter, changing both the value and the soft limits, based on the ``x`` and ``y`` arrays. >>> vals = guess_fwhm(y, x) >>> param_apply_limits(vals, mdl.fwhm) Change the soft limits for the ``xpos`` and ``ypos`` parameters of the ``src`` model: >>> pos = guess_position(y, x0, x1) >>> param_apply_limits(pos[0], src.xpos, limits=True, values=False) >>> param_apply_limits(pos[1], src.ypos, limits=True, values=False) """ # only guess thawed parameters! if par.frozen: return if limits and values: default_val = par.val par.set(param_limits['val'], param_limits['min'], param_limits['max'], default_min=par.min, default_max=par.max) # set original value as default outside the property interface par._default_val = default_val # set guessed flag to enable user-defined limit reset par._guessed = True elif values: default_val = par.val par.set(param_limits['val']) # set original value as default outside the property interface par._default_val = default_val else: par.set(min=param_limits['min'], max=param_limits['max'], default_min=par.min, default_max=par.max) # set guessed flag to enable user-defined limit reset par._guessed = True
def get_amplitude_position(arr, mean=False): """ Guess model amplitude and position of array returns (xval, xmin, xmax, xpos) """ xpos = xmin = xmax = xval = 0 amax = arr.max() amin = arr.min() if((amax > 0.0 and amin >= 0.0) or (amax > 0.0 and amin < 0.0 and (abs(amin) <= amax))): xpos = arr.argmax() if mean: xpos = np.where(arr == amax) xmax = amax * _guess_ampl_scale xmin = amax / _guess_ampl_scale xval = amax elif((amax > 0.0 and amin < 0.0 and abs(amin) > amax) or (amax == 0.0 and amin < 0.0) or (amax < 0.0)): xpos = arr.argmin() if mean: xpos = np.where(arr == amin) xmax = amin / _guess_ampl_scale xmin = amin * _guess_ampl_scale xval = amin elif (amax == 0.0 and amin == 0.0): xpos = arr.argmax() if mean: xpos = np.where(arr == amax) xmax = 100.0 / _guess_ampl_scale xmin = 0.0 xval = 0.0 return (xval, xmin, xmax, xpos)
[docs] def guess_amplitude(y, x, xhi=None): """ Guess model parameter amplitude (val, min, max) """ val, ymin, ymax, pos = get_amplitude_position(y) amin, amax = None, None if ymin != 0.0 or ymax != 0.0: amin = ymin amax = ymax if xhi is not None: binsize = np.abs(xhi[pos] - x[pos]) if amin is not None: amin /= binsize if amax is not None: amax /= binsize val /= binsize return {'val': val, 'min': amin, 'max': amax}
[docs] def guess_amplitude_at_ref(r, y, x, xhi=None): """ Guess model parameter amplitude (val, min, max) only valid for beta1d, bpl1d, powlaw1d """ t = 1.0 if x[1] > x[0] and r < x[0]: t = np.abs(y[0] + y[1]) / 2.0 elif x[1] > x[0] and r > x[-1]: t = np.abs(y[-1] + y[-2]) / 2.0 elif x[1] < x[0] and r > x[0]: t = np.abs(y[0] + y[1]) / 2.0 elif x[1] < x[0] and r < x[-1]: t = np.abs(y[-1] + y[-2]) / 2.0 else: for i in range(len(x) - 1): if ((r >= x[i] and r < x[i + 1]) or (r >= x[i + 1] and r < x[i])): t = np.abs(y[i] + y[i + 1]) / 2.0 break if t == 0.0: totband = 0.0 dv = 0.0 i = 1 for j in range(len(x) - 1): dv = x[i] - x[i - 1] t += y[i] * dv totband += dv i += 1 t /= totband return {'val': t, 'min': t / _guess_ampl_scale, 'max': t * _guess_ampl_scale}
[docs] def guess_amplitude2d(y, x0lo, x1lo, x0hi=None, x1hi=None): """ Guess 2D model parameter amplitude (val, min, max) """ limits = guess_amplitude(y, x0lo) # if (x0hi is not None and x1hi is not None): # binsize = np.abs((x0hi[0]-x0lo[0])*(x1hi[0]-x1lo[0])) # if limits['min'] is not None: # limits['min'] /= binsize # if limits['max'] is not None: # limits['max'] /= binsize # limits['val'] /= binsize return limits
[docs] def guess_reference(pmin, pmax, x, xhi=None): """ Guess model parameter reference (val, min, max) """ xmin = x.min() xmax = x.max() if xmin >= 1: pmin = 1 if xmax <= 1: pmax = 1 val = 0.0 if xmin < 1.0 and xmax > 1.0: val = 1.0 else: refval = np.floor((xmin + xmax) / 2.0) if refval < pmin or refval > pmax: refval = (xmin + xmax) / 2.0 val = refval return {'val': val, 'min': None, 'max': None}
[docs] def get_position(y, x, xhi=None): """ Get 1D model parameter positions pos (val, min, max) """ pos = get_amplitude_position(y, mean=True) xpos = pos[3] val = np.mean(x[xpos]) xmin = x.min() xmax = x.max() if xhi is not None: xmax = xhi.max() return {'val': val, 'min': xmin, 'max': xmax}
[docs] def guess_position(y, x0lo, x1lo, x0hi=None, x1hi=None): """ Guess 2D model parameter positions xpos, ypos ({val0, min0, max0}, {val1, min1, max1}) """ # pos = int(y.argmax()) # return the average of location of brightest pixels pos = np.where(y == y.max()) x0, x1 = x0lo, x1lo r1 = {'val': np.mean(x0[pos]), 'min': x0.min()} r2 = {'val': np.mean(x1[pos]), 'min': x1.min()} if x0hi is None and x1hi is None: r1['max'] = x0.max() r2['max'] = x1.max() else: r1['max'] = x0hi.max() r2['max'] = x1hi.max() return (r1, r2)
[docs] def guess_bounds(x, xhi=True): """Guess the bounds of a parameter from the independent axis. Parameters ---------- x : array_like The axis values. xhi : bool, optional When ``True``, the return value is two dictionaries, with values set to 1/3 and 2/3 along the axis, otherwise a single dictionary, with a value set to the mid-point is returned. Returns ------- ans : dict or (dict, dict) When ``xhi`` is True then two dictionaries are returned, otherwise one. The keys are 'val', 'min', and 'max'. See Also -------- get_midpoint """ xmin = x.min() xmax = x.max() lo = xmin + (xmax - xmin) / 2.0 if xhi: lo = xmin + (xmax - xmin) / 3.0 hi = xmin + 2.0 * (xmax - xmin) / 3.0 return ({'val': lo, 'min': xmin, 'max': xmax}, {'val': hi, 'min': xmin, 'max': xmax}) return {'val': lo, 'min': xmin, 'max': xmax}
[docs] def guess_radius(x0lo, x1lo, x0hi=None, x1hi=None): """Guess the radius parameter of a 2D model. Parameters ---------- x0lo, x1lo : array_like The independent axes of the grid, in 1D form. x0hi, x1hi : array_like or None, optional The upper bounds of each pixel. Returns ------- ans : dict The keys are 'val', 'min', and 'max', which give the radius and its range, in units of the input grid (``x0lo`` and ``x1lo``). Notes ----- Currently only ``x0lo`` is used, and it is assumed to be arranged so that this axis varies fastest (that is ``x0lo[1] > x0lo[0]``) as well as representing square pixels of the same size. The scaling factor for the minimum and maximum values is taken from the module's `_guess_ampl_scale` variable. """ # TODO: the following was the original code, but # a) x1 isn't used # b) there's no difference between the two branches # So, x0hi/x1hi are currently unused. # # if x0hi is None and x1hi is None: # x0, x1 = x0lo, x1lo # else: # x0, x1 = x0lo, x1lo x0 = x0lo delta = np.apply_along_axis(np.diff, 0, x0)[0] rad = np.abs(10 * delta) return {'val': rad, 'min': rad / _guess_ampl_scale, 'max': rad * _guess_ampl_scale}
################################# Neville2d ###################################
[docs] def neville2d(xinterp, yinterp, x, y, fval): """Polynomial two-dimensional interpolation using Neville's method. The scheme used for interpolation (Neville's method) is described at [1]_, where the interpolation is done first over the Y axis and then the X axis. References ---------- .. [1] http://en.wikipedia.org/wiki/Neville%27s_algorithm """ nrow = fval.shape[0] # ncol = fval.shape[1] tmp = np.zeros(nrow) for row in range(nrow): tmp[row] = neville(yinterp, y, fval[row]) return neville(xinterp, x, tmp)
################################## Hessian #################################### class NumDeriv: def __init__(self, func, fval0): self.nfev, self.func = func_counter(func) self.fval_0 = fval0 class NumDerivCentralOrdinary(NumDeriv): """ Subtract the following Taylor series expansion:: 2 ' h '' 3 f( x +/- h ) = f( x ) +/- h f ( x ) + - f ( x ) + O( h ) 2 gives:: ' 3 f( x + h ) - f( x - h ) = 2 h f ( x ) + O( h ) ' solving for f ( x ):: ' f( x + h ) - f( x - h ) 2 f ( x ) = ----------------------- + O( h ) 2 h In addition to the truncation error of order h^2, there is a round off error due to the finite numerical precision ~ r f( x ):: ' f( x + h ) - f( x - h ) r f( x ) 2 f ( x ) = ----------------------- + --------- + O( h ) 2 h h r 2 Error ~= - + h h minimizing the error by differentiating wrt h, the solve for h:: h ~ r^1/3 """ def __init__(self, func, fval0=None): NumDeriv.__init__(self, func, fval0) def __call__(self, x, h): if 0.0 == h: return np.inf return (self.func(x + h) - self.func(x - h)) / (2.0 * h) class NumDerivFowardPartial(NumDeriv): def __init__(self, func, fval0): NumDeriv.__init__(self, func, fval0) def __call__(self, x, h, *args): if 0.0 == h: h = pow(np.finfo(np.float32).eps, 1.0 / 3.0) ith = args[0] jth = args[1] ei = np.zeros(len(x), float) ej = np.zeros(len(x), float) deltai = h * abs(x[ith]) if 0.0 == deltai: deltai = h ei[ith] = deltai deltaj = h * abs(x[jth]) if 0.0 == deltaj: deltaj = h ej[jth] = deltaj fval = self.fval_0 fval += self.func(x + ei + ej) fval -= self.func(x + ei) fval -= self.func(x + ej) fval /= deltai * deltaj return fval class NumDerivCentralPartial(NumDeriv): """Add the following Taylor series expansion:: 2 ' h '' 3 f( x +/- h ) = f( x ) +/- h f ( x ) + - f ( x ) + O( h ) 2 '' and solve for f ( x ), gives:: '' f( x + h ) + f( x - h ) - 2 f( x ) 2 f ( x ) = ------------------------------------ + O( h ) 2 h In addition to the truncation error of order h^2, there is a round off error due to the finite numerical precision ~ r f( x ):: '' f( x + h ) + f( x - h ) - 2 f( x ) r f( x ) 2 f ( x ) = ------------------------------------ + -------- + O( h ) 2 2 h h r 2 Error ~= - + h 2 h minimizing the error by differentiating wrt h, the solve for h:: h ~ r^1/4 """ def __init__(self, func, fval0): NumDeriv.__init__(self, func, fval0) def __call__(self, x, h, *args): if 0.0 == h: h = pow(np.finfo(np.float32).eps, 1.0 / 3.0) ith = args[0] jth = args[1] ei = np.zeros(len(x), float) if ith == jth: delta = h * abs(x[ith]) if 0.0 == delta: delta = h ei[ith] = delta fval = - 2.0 * self.fval_0 fval += self.func(x + ei) + self.func(x - ei) fval /= delta * delta return fval ej = np.zeros(len(x), float) deltai = h * abs(x[ith]) if 0.0 == deltai: deltai = h ei[ith] = deltai deltaj = h * abs(x[jth]) if 0.0 == deltaj: deltaj = h ej[jth] = deltaj fval = self.func(x + ei + ej) fval -= self.func(x + ei - ej) fval -= self.func(x - ei + ej) fval += self.func(x - ei - ej) fval /= (4.0 * deltai * deltaj) return fval class NoRichardsonExtrapolation: def __init__(self, sequence, verbose=False): self.sequence = sequence self.verbose = verbose def __call__(self, x, t, tol, maxiter, h, *args): self.sequence(x, h, *args) class RichardsonExtrapolation(NoRichardsonExtrapolation): """From Wikipedia, the free encyclopedia In numerical analysis, Richardson extrapolation is a sequence acceleration method, used to improve the rate of convergence of a sequence. It is named after Lewis Fry Richardson, who introduced the technique in the early 20th century.[1][2] In the words of Birkhoff and Rota, '... its usefulness for practical computations can hardly be overestimated.' 1. Richardson, L. F. (1911). \"The approximate arithmetical solution by finite differences of physical problems including differential equations, with an application to the stresses in a masonry dam \". Philosophical Transactions of the Royal Society of London, Series A 210. 2. Richardson, L. F. (1927). \" The deferred approach to the limit \". Philosophical Transactions of the Royal Society of London, Series A 226:""" def __call__(self, x, t, tol, maxiter, h, *args): richardson = np.zeros((maxiter, maxiter), dtype=np.float64) richardson[0, 0] = self.sequence(x, h, *args) t_sqr = t * t for ii in range(1, maxiter): h /= t richardson[ii, 0] = self.sequence(x, h, *args) ii_1 = ii - 1 for jj in range(1, ii + 1): # jjp1 = jj + 1 -- this variable is not used jj_1 = jj - 1 factor = pow(t_sqr, jj) factor_1 = factor - 1 richardson[ii, jj] = (factor * richardson[ii, jj_1] - richardson[ii_1, jj_1]) / factor_1 arg_jj = richardson[ii, jj] arg_jj -= richardson[ii, jj_1] arg_ii = richardson[ii, jj] arg_ii -= richardson[ii_1, jj_1] if Knuth_close(richardson[ii, ii], richardson[ii_1, ii_1], tol): if self.verbose: print_low_triangle(richardson, jj) return richardson[ii, ii] if self.verbose: print_low_triangle(richardson, maxiter - 1) return richardson[maxiter - 1, maxiter - 1] def hessian(func, par, extrapolation, algorithm, maxiter, h, tol, t): num_dif = algorithm(func, func(par)) deriv = extrapolation(num_dif) npar = len(par) Hessian = np.zeros((npar, npar), dtype=np.float64) for ii in range(npar): for jj in range(ii + 1): answer = deriv(par, t, tol, maxiter, h, ii, jj) Hessian[ii, jj] = answer / 2.0 Hessian[jj, ii] = Hessian[ii, jj] return Hessian, num_dif.nfev[0] def print_low_triangle(matrix, num): # print matrix for ii in range(num): print(matrix[ii, 0], end=' ') for jj in range(1, ii + 1): print(matrix[ii, jj], end=' ') print() def symmetric_to_low_triangle(matrix, num): low_triangle = [] for ii in range(num): for jj in range(ii + 1): low_triangle.append(matrix[ii, jj]) # print_low_triangle( matrix, num ) # print low_triangle return low_triangle ############################### Root of all evil ############################## def printf(format, *args): """Format args with the first argument as format string, and write. Return the last arg, or format itself if there are no args.""" sys.stdout.write(str(format) % args) # WARNING: where is if_ meant to be defined? return if_(args, args[-1], format) # With ParamSpec, added in Python 3.10, we might be able to annotate # this so that we can match the arguments that `func` uses are the # same as are sent to the __call__ method, although it might be easier # to do after the generics changes added in Python 3.12. # T = TypeVar("T") class FuncCounter(Generic[T]): """Store the number of times the function is called. .. versionadded:: 4.17.0 """ __slots__ = ("nfev", "func") def __init__(self, func: Callable[..., T]) -> None: self.nfev = 0 self.func = func def __call__(self, *args) -> T: self.nfev += 1 return self.func(*args) def func_counter(func): """DEPRECATED. .. deprecated:: 4.17.0 Use the `FuncCounter` class instead. """ # This is FutureWarning rather than DeprecationWarning to make # sure users see the message. # warnings.warn("func_counter is deprecated in 4.17.0: use FuncCounter instead", FutureWarning) def func_counter_wrapper(x, *args): nfev[0] += 1 return func(x, *args) return nfev, func_counter_wrapper def is_in(arg, seq): """DEPRECATED. .. deprecated:: 4.17.0 Use the Python `in` operator instead. """ # This is FutureWarning rather than DeprecationWarning to make # sure users see the message. # warnings.warn("is_in is deprecated in 4.17.0: use Python's in instead", FutureWarning) for x in seq: if arg == x: return True return False def is_iterable(arg): return isinstance(arg, (list, tuple, np.ndarray)) or np.iterable(arg) def is_sequence(start, mid, end): return start < mid < end
[docs] def Knuth_close(x, y, tol, myop=operator.__or__): """Check whether two floating-point numbers are close together. See Also -------- sao_fcmp Notes ----- The following text was taken verbatim from [1]_: In most cases it is unreasonable to use an operator==(...) for a floating-point values equality check. The simple solution like ``abs(f1-f2) <= e`` does not work for very small or very big values. This floating-point comparison algorithm is based on the more confident solution presented by D. E. Knuth in 'The art of computer programming (vol II)'. For a given floating point values u and v and a tolerance e:: | u - v | <= e * |u| and | u - v | <= e * |v| (1) defines a "very close with tolerance e" relationship between u and v:: | u - v | <= e * |u| or | u - v | <= e * |v| (2) defines a "close enough with tolerance e" relationship between u and v. Both relationships are commutative but are not transitive. The relationship defined by inequations (1) is stronger that the relationship defined by inequations (2) (i.e. (1) => (2) ). References ---------- .. [1] http://www.boost.org/doc/libs/1_35_0/libs/test/doc/components/test_tools/floating_point_comparison.html#Introduction """ diff = abs(x - y) if 0.0 == x or 0.0 == y: return diff <= tol return myop(diff <= tol * abs(x), diff <= tol * abs(y))
def safe_div(num, denom): dbl_max = sys.float_info.max dbl_min = sys.float_info.min # avoid overflow if denom < 1 and num > denom * dbl_max: return dbl_max # avoid underflow if 0.0 == num or denom > 1 and num < denom * dbl_min: return 0 return num / denom def Knuth_boost_close(x, y, tol, myop=operator.__or__): """ The following text was taken verbatim from: http://www.boost.org/doc/libs/1_35_0/libs/test/doc/components/test_tools/floating_point_comparison.html#Introduction In most cases it is unreasonable to use an operator==(...) for a floating-point values equality check. The simple solution like abs(f1-f2) <= e does not work for very small or very big values. This floating-point comparison algorithm is based on the more confident solution presented by D. E. Knuth in 'The art of computer programming (vol II)'. For a given floating point values u and v and a tolerance e: | u - v | <= e * |u| and | u - v | <= e * |v| (1) defines a "very close with tolerance e" relationship between u and v | u - v | <= e * |u| or | u - v | <= e * |v| (2) defines a "close enough with tolerance e" relationship between u and v. Both relationships are commutative but are not transitive. The relationship defined by inequations (1) is stronger that the relationship defined by inequations (2) (i.e. (1) => (2) ). Because of the multiplication in the right side of inequations, that could cause an unwanted underflow condition, the implementation is using modified version of the inequations (1) and (2) where all underflow, overflow conditions could be guarded safely: | u - v | / |u| <= e and | u - v | / |v| <= e (1`) | u - v | / |u| <= e or | u - v | / |v| <= e (2`)""" diff = abs(x - y) if 0.0 == x or 0.0 == y: return diff <= tol diff_x = safe_div(diff, x) diff_y = safe_div(diff, y) return myop(diff_x <= tol, diff_y <= tol) def list_to_open_interval(arg): if not np.iterable(arg): return arg return f'({arg[0]:e}, {arg[1]:e})' class OutOfBoundErr(Exception): """Indicate an out-of-bounds exception in the error analysis""" # Should this just move to sherpa.estmethods? pass class QuadEquaRealRoot: """ solve for the real roots of the quadratic equation: a * x^2 + b * x + c = 0""" def __call__(self, a, b, c): if 0.0 == a: # # 0 * x^2 + b * x + c = 0 # if 0.0 != b: # # 0 * x^2 + b * x + c = 0 # the folowing still works even if c == 0 # answer = - c / b return [answer, answer] # # 0 * x^2 + 0 * x + c = 0 # # a == 0, b == 0, so if c == 0 then all numbers work so # returning nan is not right. However if c != 0 then no # roots exist. # return [None, None] if 0.0 == b: # # a * x^2 + 0 * x + c = 0 # if 0.0 == c: # a * x^2 + 0 * x + 0 = 0 return [0.0, 0.0] # a * x^2 + 0 * x + c = 0 if np.sign(a) == np.sign(c): return [None, None] answer = np.sqrt(c / a) return [-answer, answer] if 0.0 == c: # # a * x^2 + b * x + 0 = 0 # return [0.0, - b / a] discriminant = b * b - 4.0 * a * c debug("disc=%s", discriminant) sqrt_disc = np.sqrt(discriminant) t = - (b + np.sign(b) * sqrt_disc) / 2.0 return [c / t, t / a]
[docs] def bisection(fcn, xa, xb, fa=None, fb=None, args=(), maxfev=48, tol=1.0e-6): """A basic root finding algorithm that uses standard bisection Bisection is a relatively slow method for root finding, but it guaranteed to work for a continuous function with a root in a bracketed interval; in other words the function must undergo a sign change between the bracketing values. See https://en.wikipedia.org/wiki/Bisection_method for a description of the bisection method. Parameters ---------- fcn : callable The function with a root. The function signature is ``fcn(x, *args)``. xa : float Lower limit of the bracketing interval xb : float Upper limit of the bracketing interval fa : float or None Function value at ``xa``. This parameter is optional and can be passed to save time in cases where ``fcn(xa, *args)`` is already known and function evaluation takes a long time. If `None`, it will be calculated. fb : float or None Function value at ``xb``. This parameter is optional and can be passed to save time in cases where ``fcn(xb, *args)`` is already known and function evaluation takes a long time. If `None`, it will be calculated. args : tuple Additional parameters that will be passed through to ``fcn``. maxfev : int Maximal number of function evaluations tol : float The root finding algorithm stops if a value x with ``abs(fcn(x)) < tol`` is found. Returns ------- out : list The output has the form of a list: ``[[x, fcn(x)], [x1, fcn(x1)], [x2, fcn(x2)], nfev]`` where ``x`` is the location of the root, and ``x1`` and ``x2`` are the previous steps. The function value for those steps is returned as well. ``nfev`` is the total number of function evaluations. If any of those values is not available, ``None`` will be returned instead. """ myfcn = FuncCounter(fcn) try: if fa is None: fa = myfcn(xa, *args) if abs(fa) <= tol: return [[xa, fa], [[xa, fa], [xa, fa]], myfcn.nfev] if fb is None: fb = myfcn(xb, *args) if abs(fb) <= tol: return [[xb, fb], [[xb, fb], [xb, fb]], myfcn.nfev] if np.sign(fa) == np.sign(fb): # TODO: is this a useful message for the user? warning('%s: %s fa * fb < 0 is not met', __name__, fcn.__name__) return [[None, None], [[None, None], [None, None]], myfcn.nfev] while myfcn.nfev < maxfev: if abs(fa) > tol and abs(fb) > tol: xc = (xa + xb) / 2.0 fc = myfcn(xc, *args) if abs(xa - xb) < min(tol * abs(xb), tol / 10.0): return [[xc, fc], [[xa, fa], [xb, fb]], myfcn.nfev] if np.sign(fa) != np.sign(fc): xb, fb = xc, fc else: xa, fa = xc, fc else: if abs(fa) <= tol: return [[xa, fa], [[xa, fa], [xb, fb]], myfcn.nfev] return [[xb, fb], [[xa, fa], [xb, fb]], myfcn.nfev] xc = (xa + xb) / 2.0 fc = myfcn(xc, *args) return [[xc, fc], [[xa, fa], [xb, fb]], myfcn.nfev] except OutOfBoundErr: return [[None, None], [[xa, fa], [xb, fb]], myfcn.nfev]
def quad_coef(x, f): """ p( x ) = f( xc ) + A ( x - xc ) + B ( x - xc ) ( x - xb ) = f( xc ) + A ( x - xc ) + B ( ( x - xc ) ( x - xc ) + ( x - xc ) ( xc - xb ) ) = f( xc ) + ( A + B ( xc - xb ) ) ( x - xc ) + B ( x - xc )^2 = f( xc ) + C ( x - xc ) + B ( x - xc )^2 ; C = A + B ( xc - xb ) = f( xc ) + C x - C xc + B ( x^2 - 2 x xc + xc^2 ) = B x^2 + ( C - 2 * B xc ) x + f( xc ) - C xc + B xc^2 = B x^2 + ( C - 2 * B x[2] ) x + f[ 2 ] + x[2] * ( B x[ 2 ] - C ) """ [B, C] = transformed_quad_coef(x, f) B_x2 = B * x[2] return [B, C - 2 * B_x2, f[2] + x[2] * (B_x2 - C)] def transformed_quad_coef(x, f): """ p( x ) = f( xc ) + A ( x - xc ) + B ( x - xc ) ( x - xb ) where A and B are the divided differences:: f( xc ) - f( xb ) A = ----------------- xc - xb 1 ( f( xc ) - f( xb ) f( xb ) - f( xa ) ) B = ------- ( ----------------- - ----------------- ) xc - xa ( xc - xb xb - xa ) p( x ) = f( xc ) + A ( x - xc ) + B ( x - xc ) ( x - xb ) = f( xc ) + A ( x - xc ) + B ( ( x - xc ) ( x - xc ) + ( x - xc ) ( xc - xb ) ) = f( xc ) + ( A + B ( xc - xb ) ) ( x - xc ) + B ( x - xc )^2 = f( xc ) + C ( x - xc ) + B ( x - xc )^2 where C = A + B ( xc - xb ) The root of p( x ), using the quadratic formula:: 1 ( ) x - xc = --- ( - C +/- sqrt( C^2 - 4 f( xc ) B ) ) 2 B ( ) Rationalize the numerator to avoid subtractive cancellation:: 2 f( xc ) x - xc = ------------------------------- C +/- sqrt( C^2 - 4 f( xc ) B ) The sign should be chosen to maximize the denominator. Therefore, the next point in the iteration is:: 2 f( xc ) x = xc - -------------------------------------- C + sgn( C ) sqrt( C^2 - 4 f( xc ) B ) { -1, x < 0 where sgn(x) = { { 1, x >= 0 """ xa, xb, xc = x[0], x[1], x[2] fa, fb, fc = f[0], f[1], f[2] xc_xb = xc - xb fc_fb = fc - fb A = fc_fb / xc_xb fb_fa = fb - fa xb_xa = xb - xa xc_xa = xc - xa B = (A - fb_fa / xb_xa) / xc_xa C = A + B * xc_xb return [B, C]
[docs] def demuller(fcn, xa, xb, xc, fa=None, fb=None, fc=None, args=(), maxfev=32, tol=1.0e-6): """A root-finding algorithm using Muller's method. The algorithm is described at https://en.wikipedia.org/wiki/Muller%27s_method. :: p( x ) = f( xc ) + A ( x - xc ) + B ( x - xc ) ( x - xb ) Notes ----- The general case:: 2 f( x ) n x = x - ---------------------------------------- n+1 n C + sgn( C ) sqrt( C^2 - 4 f( x ) B ) n n n n n 1 ( f( x ) - f( x ) f( x ) - f( x ) ) ( n n-1 n-1 n-2 ) B = ------- ( ------------------ - ------------------- ) n x - x ( x - x x - x ) n n-2 ( n n-1 n-1 n-2 ) f( x ) - f( x ) n n-1 A = ----------------- n x - x n n-1 C = A + B ( x - x ) n n n n n-1 The convergence rate for Muller's method can be shown to be the real root of the cubic x - x^3, that is:: p = (a + 4 / a + 1) / 3 a = (19 + 3 sqrt(33))^1/3 In other words: O(h^p) where p is approximately 1.839286755. Parameters ---------- fcn : callable The function with a root. The function signature is ``fcn(x, *args)``. xa, xb, xc : float Muller's method requires three initial values. fa, fb, fc : float or None Function values at ``xa``, ``xb``, and ``xc``. These parameters are optional and can be passed to save time in cases where ``fcn(xa, *args)`` is already known and function evaluation takes a long time. If `None`, they will be calculated. args : tuple Additional parameters that will be passed through to ``fcn``. maxfev : int Maximal number of function evaluations tol : float The root finding algorithm stops if the function value a value x with ``abs(fcn(x)) < tol`` is found. Returns ------- out : list The output has the form of a list: ``[[x, fcn(x)], [x1, fcn(x1)], [x2, fcn(x2)], nfev]`` where ``x`` is the location of the root, and ``x1`` and ``x2`` are the previous steps. The function value for those steps is returned as well. ``nfev`` is the total number of function evaluations. If any of those values is not available, ``None`` will be returned instead. """ def is_nan(arg): if arg != arg: return True if arg is np.nan: return True return np.isnan(arg) myfcn = FuncCounter(fcn) try: if fa is None: fa = myfcn(xa, *args) if abs(fa) <= tol: return [[xa, fa], [[xa, fa], [xa, fa]], myfcn.nfev] if fb is None: fb = myfcn(xb, *args) if abs(fb) <= tol: return [[xb, fb], [[xb, fb], [xb, fb]], myfcn.nfev] if fc is None: fc = myfcn(xc, *args) if abs(fc) <= tol: return [[xc, fc], [[xc, fc], [xc, fc]], myfcn.nfev] while myfcn.nfev < maxfev: [B, C] = transformed_quad_coef([xa, xb, xc], [fa, fb, fc]) discriminant = max(C * C - 4.0 * fc * B, 0.0) if is_nan(B) or is_nan(C) or \ 0.0 == C + np.sign(C) * np.sqrt(discriminant): return [[None, None], [[None, None], [None, None]], myfcn.nfev] xd = xc - 2.0 * fc / (C + np.sign(C) * np.sqrt(discriminant)) fd = myfcn(xd, *args) if abs(fd) <= tol: return [[xd, fd], [[None, None], [None, None]], myfcn.nfev] xa = xb fa = fb xb = xc fb = fc xc = xd fc = fd return [[xd, fd], [[None, None], [None, None]], myfcn.nfev] except ZeroDivisionError: return [[xd, fd], [[None, None], [None, None]], myfcn.nfev]
[docs] def new_muller(fcn, xa, xb, fa=None, fb=None, args=(), maxfev=32, tol=1.e-6): '''Alternative implementation of Mueller's method for root finding Parameters ---------- fcn : callable The function with a root. The function signature is ``fcn(x, *args)``. xa, xb: float Muller's method requires three initial values. fa, fb: float or None Function values at ``xa`` and ``xb``. These parameters are optional and can be passed to save time in cases where ``fcn(xa, *args)`` is already known and function evaluation takes a long time. If `None`, they will be calculated. args : tuple Additional parameters that will be passed through to ``fcn``. maxfev : int Maximal number of function evaluations tol : float The root finding algorithm stops if the function value a value x with ``abs(fcn(x)) < tol`` is found. Returns ------- out : list The output has the form of a list: ``[[x, fcn(x)], [x1, fcn(x1)], [x2, fcn(x2)], nfev]`` where ``x`` is the location of the root, and ``x1`` and ``x2`` are the previous steps. The function value for those steps is returned as well. ``nfev`` is the total number of function evaluations. If any of those values is not available, ``None`` will be returned instead. ''' myfcn = FuncCounter(fcn) try: if fa is None: fa = myfcn(xa, *args) if abs(fa) <= tol: return [[xa, fa], [[xa, fa], [xa, fa]], myfcn.nfev] if fb is None: fb = myfcn(xb, *args) if abs(fb) <= tol: return [[xb, fb], [[xb, fb], [xb, fb]], myfcn.nfev] if np.sign(fa) == np.sign(fb): warning('%s: %s fa * fb < 0 is not met', __name__, fcn.__name__) return [[None, None], [[None, None], [None, None]], myfcn.nfev] while myfcn.nfev < maxfev: xc = (xa + xb) / 2.0 fc = myfcn(xc, *args) if abs(fc) <= tol: return [[xc, fc], [[xa, fa], [xb, fb]], myfcn.nfev] tran = transformed_quad_coef([xa, xb, xc], [fa, fb, fc]) B = tran[0] C = tran[1] discriminant = max(C * C - 4.0 * fc * B, 0.0) xd = xc - 2.0 * fc / (C + np.sign(C) * np.sqrt(discriminant)) fd = myfcn(xd, *args) if abs(fd) <= tol: return [[xd, fd], [[xa, fa], [xb, fb]], myfcn.nfev] if np.sign(fa) != np.sign(fc): xb, fb = xc, fc continue if np.sign(fd) != np.sign(fc) and xc < xd: xa, fa = xc, fc xb, fb = xd, fd continue if np.sign(fb) != np.sign(fd): xa, fa = xd, fd continue if np.sign(fa) != np.sign(fd): xb, fb = xd, fd continue if np.sign(fc) != np.sign(fd) and xd < xc: xa, fa = xd, fd xb, fb = xc, fc continue if np.sign(fc) != np.sign(fd): xa, fa = xc, fc continue return [[xd, fd], [[xa, fa], [xb, fb]], myfcn.nfev] except (ZeroDivisionError, OutOfBoundErr): return [[xd, fd], [[xa, fa], [xb, fb]], myfcn.nfev]
# # /* # * Licensed to the Apache Software Foundation (ASF) under one or more # * contributor license agreements. See the NOTICE file distributed with # * this work for additional information regarding copyright ownership. # * The ASF licenses this file to You under the Apache License, Version 2.0 # * (the "License"); you may not use this file except in compliance with # * the License. You may obtain a copy of the License at # * # * http://www.apache.org/licenses/LICENSE-2.0 # * # * Unless required by applicable law or agreed to in writing, software # * distributed under the License is distributed on an "AS IS" BASIS, # * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. # * See the License for the specific language governing permissions and # * limitations under the License. # */ #
[docs] def apache_muller(fcn, xa, xb, fa=None, fb=None, args=(), maxfev=32, tol=1.0e-6): '''An alternative implementation of Muller's method for root finding. Unlike the rest of Sherpa, this method is available the Apache Software Foundation (ASF) licence - see code for this method for details. Parameters ---------- fcn : callable The function with a root. The function signature is ``fcn(x, *args)``. xa, xb: float Muller's method requires three initial values. fa, fb: float or None Function values at ``xa`` and ``xb``. These parameters are optional and can be passed to save time in cases where ``fcn(xa, *args)`` is already known and function evaluation takes a long time. If `None`, they will be calculated. args : tuple Additional parameters that will be passed through to ``fcn``. maxfev : int Maximal number of function evaluations tol : float The root finding algorithm stops if the function value a value x with ``abs(fcn(x)) < tol`` is found. Returns ------- out : list The output has the form of a list: ``[[x, fcn(x)], [x1, fcn(x1)], [x2, fcn(x2)], nfev]`` where ``x`` is the location of the root, and ``x1`` and ``x2`` are the previous steps. The function value for those steps is returned as well. ``nfev`` is the total number of function evaluations. If any of those values is not available, ``None`` will be returned instead. ''' myfcn = FuncCounter(fcn) try: if fa is None: fa = myfcn(xa, *args) if abs(fa) <= tol: return [[xa, fa], [[xa, fa], [xa, fa]], myfcn.nfev] if fb is None: fb = myfcn(xb, *args) if abs(fb) <= tol: return [[xb, fb], [[xb, fb], [xb, fb]], myfcn.nfev] if np.sign(fa) == np.sign(fb): warning('%s: %s fa * fb < 0 is not met', __name__, fcn.__name__) return [[None, None], [[None, None], [None, None]], myfcn.nfev] xc = (xa + xb) / 2.0 fc = myfcn(xc, *args) if abs(fc) <= tol: return [[xc, fc], [[xc, fc], [xc, fc]], myfcn.nfev] xbest, fbest = xa, fa if abs(fb) < abs(fa): xbest, fbest = xb, fb if abs(fc) < abs(fbest): xbest, fbest = xc, fc oldx = 1.0e128 while myfcn.nfev < maxfev: tran = transformed_quad_coef([xa, xb, xc], [fa, fb, fc]) B = tran[0] C = tran[1] discriminant = max(C * C - 4.0 * fc * B, 0.0) den = np.sign(C) * np.sqrt(discriminant) xplus = xc - 2.0 * fc / (C + den) if C != den: xminus = xc - 2.0 * fc / (C - den) else: xminus = 1.0e128 if is_sequence(xa, xplus, xb): x = xplus else: x = xminus # print 'xa=', xa, '\tx=', x, '\txb=', xb, '\txc=', xc # fubar = quad_coef( [xa,xb,xc], [fa,fb,fc] ) # quad = QuadEquaRealRoot( ) # print quad( fubar[0], fubar[1], fubar[2] ) # print # sanity check if not is_sequence(xa, x, xb): x = (xa + xb) / 2.0 y = myfcn(x, *args) if abs(y) < abs(fbest): xbest, fbest = x, y tolerance = min(tol * abs(x), tol) if abs(y) <= tol or abs(x - oldx) <= tolerance: return [[x, y], [[xa, fa], [xb, fb]], myfcn.nfev] mybisect = (x < xc and (xc - xa) > 0.95 * (xb - xa)) or \ (x > xc and (xb - xc) > 0.95 * (xb - xa)) or \ (x == xc) if not mybisect: if x > xc: xa = xc fa = fc if x < xc: xb = xc fb = fc xc, fc = x, y oldx = x else: xmid = (xa + xb) / 2.0 fmid = myfcn(xmid, *args) if abs(fmid) < abs(fbest): xbest, fbest = xmid, fmid if abs(fmid) <= tol: return [[xmid, fmid], [[xa, fa], [xb, fb]], myfcn.nfev] if np.sign(fa) + np.sign(fmid) == 0: xb = xmid fb = fmid else: xa = xmid fa = fmid xc = (xa + xb) / 2.0 fc = myfcn(xc, *args) if abs(fc) < abs(fbest): xbest, fbest = xc, fc if abs(fc) <= tol: return [[xc, fc], [[xa, fa], [xb, fb]], myfcn.nfev] oldx = 1.0e128 # # maxfev has exceeded, return the minimum so far # return [[xbest, fbest], [[xa, fa], [xb, fb]], myfcn.nfev] # # Something drastic has happened # except (ZeroDivisionError, OutOfBoundErr): return [[xbest, fbest], [[xa, fa], [xb, fb]], myfcn.nfev]
[docs] def zeroin(fcn, xa, xb, fa=None, fb=None, args=(), maxfev=32, tol=1.0e-2): """Obtain a zero of a function of one variable using Brent's root finder. Return an approximate location for the root with accuracy:: 4*DBL_EPSILON*abs(x) + tol using the algorithm from [1]_. References ---------- .. [1] G.Forsythe, M.Malcolm, C.Moler, Computer methods for mathematical computations. M., Mir, 1980, p.180 of the Russian edition Notes ----- The function makes use of a bisection procedure combined with a linear or quadratic inverse interpolation. At each step the code operates three abscissae - a, b, and c: - b - the last and the best approximation to the root - a - the last but one approximation - c - the last but one or even an earlier approximation such that: 1) ``|f(b)| <= |f(c)|`` 2) f(b) and f(c) have opposite signs, i.e. b and c encompass the root Given these abscissae, the code computes two new approximations, one by the bisection procedure and the other one from interpolation (if a,b, and c are all different the quadratic interpolation is used, linear otherwise). If the approximation obtained by the interpolation looks reasonable (i.e. falls within the current interval [b,c], not too close to the end points of the interval), the point is accepted as a new approximation to the root. Otherwise, the result of the bissection is used. Parameters ---------- fcn : callable The function with a root. The function signature is ``fcn(x, *args)``. xa : float Lower limit of the bracketing interval xb : float Upper limit of the bracketing interval fa : float or None Function value at ``xa``. This parameter is optional and can be passed to save time in cases where ``fcn(xa, *args)`` is already known and function evaluation takes a long time. If `None`, it will be calculated. fb : float or None Function value at ``xb``. This parameter is optional and can be passed to save time in cases where ``fcn(xb, *args)`` is already known and function evaluation takes a long time. If `None`, it will be calculated. args : tuple Additional parameters that will be passed through to ``fcn``. maxfev : int Maximal number of function evaluations tol : float The root finding algorithm stops if a value x with ``abs(fcn(x)) < tol`` is found. Returns ------- out : list The output has the form of a list: ``[[x, fcn(x)], [x1, fcn(x1)], [x2, fcn(x2)], nfev]`` where ``x`` is the location of the root, and ``x1`` and ``x2`` are the previous steps. The function value for those steps is returned as well. ``nfev`` is the total number of function evaluations. If any of those values is not available, ``None`` will be returned instead. """ myfcn = FuncCounter(fcn) try: if fa is None: fa = myfcn(xa, *args) if abs(fa) <= tol: return [[xa, fa], [[xa, fa], [xb, fb]], myfcn.nfev] if fb is None: fb = myfcn(xb, *args) if abs(fb) <= tol: return [[xb, fb], [[xa, fa], [xb, fb]], myfcn.nfev] if np.sign(fa) == np.sign(fb): warning('%s: %s fa * fb < 0 is not met', __name__, fcn.__name__) return [[None, None], [[None, None], [None, None]], myfcn.nfev] xc = xa fc = fa DBL_EPSILON = np.finfo(np.float32).eps while myfcn.nfev < maxfev: prev_step = xb - xa if abs(fc) < abs(fb): xa, fa = xb, fb xb, fb = xc, fc xc, fc = xa, fa tol_act = 2.0 * DBL_EPSILON * abs(xb) + tol / 2.0 new_step = (xc - xb) / 2.0 if abs(fb) <= tol: return [[xb, fb], [[xa, fa], [xb, fb]], myfcn.nfev] if abs(new_step) <= tol_act: if np.sign(fb) != np.sign(fa): tmp = apache_muller(fcn, xa, xb, fa, fb, args=args, maxfev=maxfev - myfcn.nfev, tol=tol) tmp[-1] += myfcn.nfev return tmp if np.sign(fb) != np.sign(fc): tmp = apache_muller(fcn, xb, xc, fb, fc, args=args, maxfev=maxfev - myfcn.nfev, tol=tol) tmp[-1] += myfcn.nfev return tmp return [[xb, fb], [[xa, fa], [xb, fb]], myfcn.nfev] if abs(prev_step) >= tol_act and abs(fa) > abs(fb): cb = xc - xb if xa == xc: t1 = fb / fa p = cb * t1 q = 1.0 - t1 else: t1 = fb / fc t2 = fb / fa q = fa / fc p = t2 * (cb * q * (q - t1) - (xb - xa) * (t1 - 1.0)) q = (q - 1.0) * (t1 - 1.0) * (t2 - 1.0) if p > 0: q = -q else: p = -p if 2 * p < (1.5 * cb * q - abs(tol_act * q)) and \ 2 * p < abs(prev_step * q): new_step = p / q if abs(new_step) < tol_act: if new_step > 0: new_step = tol_act else: new_step = -tol_act xa = xb fa = fb xb += new_step fb = myfcn(xb, *args) if fb > 0 and fc > 0 or fb < 0 and fc < 0: xc = xa fc = fa return [[xb, fb], [[xa, fa], [xc, fc]], myfcn.nfev] except (ZeroDivisionError, OutOfBoundErr): return [[xb, fb], [[xa, fa], [xc, fc]], myfcn.nfev]
def public(f): """Use a decorator to avoid retyping function/class names. * Based on an idea by Duncan Booth: http://groups.google.com/group/comp.lang.python/msg/11cbb03e09611b8a * Improved via a suggestion by Dave Angel: http://groups.google.com/group/comp.lang.python/msg/3d400fb22d8a42e1 See also https://bugs.python.org/issue26632 """ _all = sys.modules[f.__module__].__dict__.setdefault('__all__', []) if f.__name__ not in _all: # Prevent duplicates if run from an IDE. _all.append(f.__name__) return f def send_to_pager(txt: str, filename=None, clobber: bool = False) -> None: """Write out the given string, using pagination if supported. This used to call out to using less/more but now is handled by pydoc.pager Parameters ---------- txt : str The text to display filename : str or StringIO or None, optional If not None, write the output to the given file or filelike object. clobber : bool, optional If filename is a string, then - when clobber is set - refuse to overwrite the file if it already exists. """ if filename is None: pydoc.pager(txt) return # Have we been sent a StringIO-like object? # if hasattr(filename, 'write'): print(txt, file=filename) return # Assume a filename clobber = bool_cast(clobber) if os.path.isfile(filename) and not clobber: raise IOErr('filefound', filename) with open(filename, 'w', encoding="UTF-8") as fh: print(txt, file=fh)