550 lines
24 KiB
Python
550 lines
24 KiB
Python
|
import math
|
||
|
|
import numpy as np
|
||
|
|
import scipy._lib._elementwise_iterative_method as eim
|
||
|
|
from scipy._lib._util import _RichResult
|
||
|
|
from scipy._lib._array_api import xp_clip, xp_minimum, xp_sign
|
||
|
|
|
||
|
|
# TODO:
|
||
|
|
# - (maybe?) don't use fancy indexing assignment
|
||
|
|
# - figure out how to replace the new `try`/`except`s
|
||
|
|
|
||
|
|
|
||
|
|
def _chandrupatla(func, a, b, *, args=(), xatol=None, xrtol=None,
|
||
|
|
fatol=None, frtol=0, maxiter=None, callback=None):
|
||
|
|
"""Find the root of an elementwise function using Chandrupatla's algorithm.
|
||
|
|
|
||
|
|
For each element of the output of `func`, `chandrupatla` seeks the scalar
|
||
|
|
root that makes the element 0. This function allows for `a`, `b`, and the
|
||
|
|
output of `func` to be of any broadcastable shapes.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
func : callable
|
||
|
|
The function whose root is desired. The signature must be::
|
||
|
|
|
||
|
|
func(x: ndarray, *args) -> ndarray
|
||
|
|
|
||
|
|
where each element of ``x`` is a finite real and ``args`` is a tuple,
|
||
|
|
which may contain an arbitrary number of components of any type(s).
|
||
|
|
``func`` must be an elementwise function: each element ``func(x)[i]``
|
||
|
|
must equal ``func(x[i])`` for all indices ``i``. `_chandrupatla`
|
||
|
|
seeks an array ``x`` such that ``func(x)`` is an array of zeros.
|
||
|
|
a, b : array_like
|
||
|
|
The lower and upper bounds of the root of the function. Must be
|
||
|
|
broadcastable with one another.
|
||
|
|
args : tuple, optional
|
||
|
|
Additional positional arguments to be passed to `func`.
|
||
|
|
xatol, xrtol, fatol, frtol : float, optional
|
||
|
|
Absolute and relative tolerances on the root and function value.
|
||
|
|
See Notes for details.
|
||
|
|
maxiter : int, optional
|
||
|
|
The maximum number of iterations of the algorithm to perform.
|
||
|
|
The default is the maximum possible number of bisections within
|
||
|
|
the (normal) floating point numbers of the relevant dtype.
|
||
|
|
callback : callable, optional
|
||
|
|
An optional user-supplied function to be called before the first
|
||
|
|
iteration and after each iteration.
|
||
|
|
Called as ``callback(res)``, where ``res`` is a ``_RichResult``
|
||
|
|
similar to that returned by `_chandrupatla` (but containing the current
|
||
|
|
iterate's values of all variables). If `callback` raises a
|
||
|
|
``StopIteration``, the algorithm will terminate immediately and
|
||
|
|
`_chandrupatla` will return a result.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
res : _RichResult
|
||
|
|
An instance of `scipy._lib._util._RichResult` with the following
|
||
|
|
attributes. The descriptions are written as though the values will be
|
||
|
|
scalars; however, if `func` returns an array, the outputs will be
|
||
|
|
arrays of the same shape.
|
||
|
|
|
||
|
|
x : float
|
||
|
|
The root of the function, if the algorithm terminated successfully.
|
||
|
|
nfev : int
|
||
|
|
The number of times the function was called to find the root.
|
||
|
|
nit : int
|
||
|
|
The number of iterations of Chandrupatla's algorithm performed.
|
||
|
|
status : int
|
||
|
|
An integer representing the exit status of the algorithm.
|
||
|
|
``0`` : The algorithm converged to the specified tolerances.
|
||
|
|
``-1`` : The algorithm encountered an invalid bracket.
|
||
|
|
``-2`` : The maximum number of iterations was reached.
|
||
|
|
``-3`` : A non-finite value was encountered.
|
||
|
|
``-4`` : Iteration was terminated by `callback`.
|
||
|
|
``1`` : The algorithm is proceeding normally (in `callback` only).
|
||
|
|
success : bool
|
||
|
|
``True`` when the algorithm terminated successfully (status ``0``).
|
||
|
|
fun : float
|
||
|
|
The value of `func` evaluated at `x`.
|
||
|
|
xl, xr : float
|
||
|
|
The lower and upper ends of the bracket.
|
||
|
|
fl, fr : float
|
||
|
|
The function value at the lower and upper ends of the bracket.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Implemented based on Chandrupatla's original paper [1]_.
|
||
|
|
|
||
|
|
If ``xl`` and ``xr`` are the left and right ends of the bracket,
|
||
|
|
``xmin = xl if abs(func(xl)) <= abs(func(xr)) else xr``,
|
||
|
|
and ``fmin0 = min(func(a), func(b))``, then the algorithm is considered to
|
||
|
|
have converged when ``abs(xr - xl) < xatol + abs(xmin) * xrtol`` or
|
||
|
|
``fun(xmin) <= fatol + abs(fmin0) * frtol``. This is equivalent to the
|
||
|
|
termination condition described in [1]_ with ``xrtol = 4e-10``,
|
||
|
|
``xatol = 1e-5``, and ``fatol = frtol = 0``. The default values are
|
||
|
|
``xatol = 4*tiny``, ``xrtol = 4*eps``, ``frtol = 0``, and ``fatol = tiny``,
|
||
|
|
where ``eps`` and ``tiny`` are the precision and smallest normal number
|
||
|
|
of the result ``dtype`` of function inputs and outputs.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
|
||
|
|
.. [1] Chandrupatla, Tirupathi R.
|
||
|
|
"A new hybrid quadratic/bisection algorithm for finding the zero of a
|
||
|
|
nonlinear function without using derivatives".
|
||
|
|
Advances in Engineering Software, 28(3), 145-149.
|
||
|
|
https://doi.org/10.1016/s0965-9978(96)00051-8
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
brentq, brenth, ridder, bisect, newton
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from scipy import optimize
|
||
|
|
>>> def f(x, c):
|
||
|
|
... return x**3 - 2*x - c
|
||
|
|
>>> c = 5
|
||
|
|
>>> res = optimize._chandrupatla._chandrupatla(f, 0, 3, args=(c,))
|
||
|
|
>>> res.x
|
||
|
|
2.0945514818937463
|
||
|
|
|
||
|
|
>>> c = [3, 4, 5]
|
||
|
|
>>> res = optimize._chandrupatla._chandrupatla(f, 0, 3, args=(c,))
|
||
|
|
>>> res.x
|
||
|
|
array([1.8932892 , 2. , 2.09455148])
|
||
|
|
|
||
|
|
"""
|
||
|
|
res = _chandrupatla_iv(func, args, xatol, xrtol,
|
||
|
|
fatol, frtol, maxiter, callback)
|
||
|
|
func, args, xatol, xrtol, fatol, frtol, maxiter, callback = res
|
||
|
|
|
||
|
|
# Initialization
|
||
|
|
temp = eim._initialize(func, (a, b), args)
|
||
|
|
func, xs, fs, args, shape, dtype, xp = temp
|
||
|
|
x1, x2 = xs
|
||
|
|
f1, f2 = fs
|
||
|
|
status = xp.full_like(x1, eim._EINPROGRESS, dtype=xp.int32) # in progress
|
||
|
|
nit, nfev = 0, 2 # two function evaluations performed above
|
||
|
|
finfo = xp.finfo(dtype)
|
||
|
|
xatol = 4*finfo.smallest_normal if xatol is None else xatol
|
||
|
|
xrtol = 4*finfo.eps if xrtol is None else xrtol
|
||
|
|
fatol = finfo.smallest_normal if fatol is None else fatol
|
||
|
|
frtol = frtol * xp_minimum(xp.abs(f1), xp.abs(f2))
|
||
|
|
maxiter = (math.log2(finfo.max) - math.log2(finfo.smallest_normal)
|
||
|
|
if maxiter is None else maxiter)
|
||
|
|
work = _RichResult(x1=x1, f1=f1, x2=x2, f2=f2, x3=None, f3=None, t=0.5,
|
||
|
|
xatol=xatol, xrtol=xrtol, fatol=fatol, frtol=frtol,
|
||
|
|
nit=nit, nfev=nfev, status=status)
|
||
|
|
res_work_pairs = [('status', 'status'), ('x', 'xmin'), ('fun', 'fmin'),
|
||
|
|
('nit', 'nit'), ('nfev', 'nfev'), ('xl', 'x1'),
|
||
|
|
('fl', 'f1'), ('xr', 'x2'), ('fr', 'f2')]
|
||
|
|
|
||
|
|
def pre_func_eval(work):
|
||
|
|
# [1] Figure 1 (first box)
|
||
|
|
x = work.x1 + work.t * (work.x2 - work.x1)
|
||
|
|
return x
|
||
|
|
|
||
|
|
def post_func_eval(x, f, work):
|
||
|
|
# [1] Figure 1 (first diamond and boxes)
|
||
|
|
# Note: y/n are reversed in figure; compare to BASIC in appendix
|
||
|
|
work.x3, work.f3 = (xp.asarray(work.x2, copy=True),
|
||
|
|
xp.asarray(work.f2, copy=True))
|
||
|
|
j = xp.sign(f) == xp.sign(work.f1)
|
||
|
|
nj = ~j
|
||
|
|
work.x3[j], work.f3[j] = work.x1[j], work.f1[j]
|
||
|
|
work.x2[nj], work.f2[nj] = work.x1[nj], work.f1[nj]
|
||
|
|
work.x1, work.f1 = x, f
|
||
|
|
|
||
|
|
def check_termination(work):
|
||
|
|
# [1] Figure 1 (second diamond)
|
||
|
|
# Check for all terminal conditions and record statuses.
|
||
|
|
|
||
|
|
# See [1] Section 4 (first two sentences)
|
||
|
|
i = xp.abs(work.f1) < xp.abs(work.f2)
|
||
|
|
work.xmin = xp.where(i, work.x1, work.x2)
|
||
|
|
work.fmin = xp.where(i, work.f1, work.f2)
|
||
|
|
stop = xp.zeros_like(work.x1, dtype=xp.bool) # termination condition met
|
||
|
|
|
||
|
|
# If function value tolerance is met, report successful convergence,
|
||
|
|
# regardless of other conditions. Note that `frtol` has been redefined
|
||
|
|
# as `frtol = frtol * minimum(f1, f2)`, where `f1` and `f2` are the
|
||
|
|
# function evaluated at the original ends of the bracket.
|
||
|
|
i = xp.abs(work.fmin) <= work.fatol + work.frtol
|
||
|
|
work.status[i] = eim._ECONVERGED
|
||
|
|
stop[i] = True
|
||
|
|
|
||
|
|
# If the bracket is no longer valid, report failure (unless a function
|
||
|
|
# tolerance is met, as detected above).
|
||
|
|
i = (xp_sign(work.f1) == xp_sign(work.f2)) & ~stop
|
||
|
|
NaN = xp.asarray(xp.nan, dtype=work.xmin.dtype)
|
||
|
|
work.xmin[i], work.fmin[i], work.status[i] = NaN, NaN, eim._ESIGNERR
|
||
|
|
stop[i] = True
|
||
|
|
|
||
|
|
# If the abscissae are non-finite or either function value is NaN,
|
||
|
|
# report failure.
|
||
|
|
x_nonfinite = ~(xp.isfinite(work.x1) & xp.isfinite(work.x2))
|
||
|
|
f_nan = xp.isnan(work.f1) & xp.isnan(work.f2)
|
||
|
|
i = (x_nonfinite | f_nan) & ~stop
|
||
|
|
work.xmin[i], work.fmin[i], work.status[i] = NaN, NaN, eim._EVALUEERR
|
||
|
|
stop[i] = True
|
||
|
|
|
||
|
|
# This is the convergence criterion used in bisect. Chandrupatla's
|
||
|
|
# criterion is equivalent to this except with a factor of 4 on `xrtol`.
|
||
|
|
work.dx = xp.abs(work.x2 - work.x1)
|
||
|
|
work.tol = xp.abs(work.xmin) * work.xrtol + work.xatol
|
||
|
|
i = work.dx < work.tol
|
||
|
|
work.status[i] = eim._ECONVERGED
|
||
|
|
stop[i] = True
|
||
|
|
|
||
|
|
return stop
|
||
|
|
|
||
|
|
def post_termination_check(work):
|
||
|
|
# [1] Figure 1 (third diamond and boxes / Equation 1)
|
||
|
|
xi1 = (work.x1 - work.x2) / (work.x3 - work.x2)
|
||
|
|
phi1 = (work.f1 - work.f2) / (work.f3 - work.f2)
|
||
|
|
alpha = (work.x3 - work.x1) / (work.x2 - work.x1)
|
||
|
|
j = ((1 - xp.sqrt(1 - xi1)) < phi1) & (phi1 < xp.sqrt(xi1))
|
||
|
|
|
||
|
|
f1j, f2j, f3j, alphaj = work.f1[j], work.f2[j], work.f3[j], alpha[j]
|
||
|
|
t = xp.full_like(alpha, 0.5)
|
||
|
|
t[j] = (f1j / (f1j - f2j) * f3j / (f3j - f2j)
|
||
|
|
- alphaj * f1j / (f3j - f1j) * f2j / (f2j - f3j))
|
||
|
|
|
||
|
|
# [1] Figure 1 (last box; see also BASIC in appendix with comment
|
||
|
|
# "Adjust T Away from the Interval Boundary")
|
||
|
|
tl = 0.5 * work.tol / work.dx
|
||
|
|
work.t = xp_clip(t, tl, 1 - tl)
|
||
|
|
|
||
|
|
def customize_result(res, shape):
|
||
|
|
xl, xr, fl, fr = res['xl'], res['xr'], res['fl'], res['fr']
|
||
|
|
i = res['xl'] < res['xr']
|
||
|
|
res['xl'] = xp.where(i, xl, xr)
|
||
|
|
res['xr'] = xp.where(i, xr, xl)
|
||
|
|
res['fl'] = xp.where(i, fl, fr)
|
||
|
|
res['fr'] = xp.where(i, fr, fl)
|
||
|
|
return shape
|
||
|
|
|
||
|
|
return eim._loop(work, callback, shape, maxiter, func, args, dtype,
|
||
|
|
pre_func_eval, post_func_eval, check_termination,
|
||
|
|
post_termination_check, customize_result, res_work_pairs,
|
||
|
|
xp=xp)
|
||
|
|
|
||
|
|
|
||
|
|
def _chandrupatla_iv(func, args, xatol, xrtol,
|
||
|
|
fatol, frtol, maxiter, callback):
|
||
|
|
# Input validation for `_chandrupatla`
|
||
|
|
|
||
|
|
if not callable(func):
|
||
|
|
raise ValueError('`func` must be callable.')
|
||
|
|
|
||
|
|
if not np.iterable(args):
|
||
|
|
args = (args,)
|
||
|
|
|
||
|
|
# tolerances are floats, not arrays; OK to use NumPy
|
||
|
|
tols = np.asarray([xatol if xatol is not None else 1,
|
||
|
|
xrtol if xrtol is not None else 1,
|
||
|
|
fatol if fatol is not None else 1,
|
||
|
|
frtol if frtol is not None else 1])
|
||
|
|
if (not np.issubdtype(tols.dtype, np.number) or np.any(tols < 0)
|
||
|
|
or np.any(np.isnan(tols)) or tols.shape != (4,)):
|
||
|
|
raise ValueError('Tolerances must be non-negative scalars.')
|
||
|
|
|
||
|
|
if maxiter is not None:
|
||
|
|
maxiter_int = int(maxiter)
|
||
|
|
if maxiter != maxiter_int or maxiter < 0:
|
||
|
|
raise ValueError('`maxiter` must be a non-negative integer.')
|
||
|
|
|
||
|
|
if callback is not None and not callable(callback):
|
||
|
|
raise ValueError('`callback` must be callable.')
|
||
|
|
|
||
|
|
return func, args, xatol, xrtol, fatol, frtol, maxiter, callback
|
||
|
|
|
||
|
|
|
||
|
|
def _chandrupatla_minimize(func, x1, x2, x3, *, args=(), xatol=None,
|
||
|
|
xrtol=None, fatol=None, frtol=None, maxiter=100,
|
||
|
|
callback=None):
|
||
|
|
"""Find the minimizer of an elementwise function.
|
||
|
|
|
||
|
|
For each element of the output of `func`, `_chandrupatla_minimize` seeks
|
||
|
|
the scalar minimizer that minimizes the element. This function allows for
|
||
|
|
`x1`, `x2`, `x3`, and the elements of `args` to be arrays of any
|
||
|
|
broadcastable shapes.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
func : callable
|
||
|
|
The function whose minimizer is desired. The signature must be::
|
||
|
|
|
||
|
|
func(x: ndarray, *args) -> ndarray
|
||
|
|
|
||
|
|
where each element of ``x`` is a finite real and ``args`` is a tuple,
|
||
|
|
which may contain an arbitrary number of arrays that are broadcastable
|
||
|
|
with `x`. ``func`` must be an elementwise function: each element
|
||
|
|
``func(x)[i]`` must equal ``func(x[i])`` for all indices ``i``.
|
||
|
|
`_chandrupatla` seeks an array ``x`` such that ``func(x)`` is an array
|
||
|
|
of minima.
|
||
|
|
x1, x2, x3 : array_like
|
||
|
|
The abscissae of a standard scalar minimization bracket. A bracket is
|
||
|
|
valid if ``x1 < x2 < x3`` and ``func(x1) > func(x2) <= func(x3)``.
|
||
|
|
Must be broadcastable with one another and `args`.
|
||
|
|
args : tuple, optional
|
||
|
|
Additional positional arguments to be passed to `func`. Must be arrays
|
||
|
|
broadcastable with `x1`, `x2`, and `x3`. If the callable to be
|
||
|
|
differentiated requires arguments that are not broadcastable with `x`,
|
||
|
|
wrap that callable with `func` such that `func` accepts only `x` and
|
||
|
|
broadcastable arrays.
|
||
|
|
xatol, xrtol, fatol, frtol : float, optional
|
||
|
|
Absolute and relative tolerances on the minimizer and function value.
|
||
|
|
See Notes for details.
|
||
|
|
maxiter : int, optional
|
||
|
|
The maximum number of iterations of the algorithm to perform.
|
||
|
|
callback : callable, optional
|
||
|
|
An optional user-supplied function to be called before the first
|
||
|
|
iteration and after each iteration.
|
||
|
|
Called as ``callback(res)``, where ``res`` is a ``_RichResult``
|
||
|
|
similar to that returned by `_chandrupatla_minimize` (but containing
|
||
|
|
the current iterate's values of all variables). If `callback` raises a
|
||
|
|
``StopIteration``, the algorithm will terminate immediately and
|
||
|
|
`_chandrupatla_minimize` will return a result.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
res : _RichResult
|
||
|
|
An instance of `scipy._lib._util._RichResult` with the following
|
||
|
|
attributes. (The descriptions are written as though the values will be
|
||
|
|
scalars; however, if `func` returns an array, the outputs will be
|
||
|
|
arrays of the same shape.)
|
||
|
|
|
||
|
|
success : bool
|
||
|
|
``True`` when the algorithm terminated successfully (status ``0``).
|
||
|
|
status : int
|
||
|
|
An integer representing the exit status of the algorithm.
|
||
|
|
``0`` : The algorithm converged to the specified tolerances.
|
||
|
|
``-1`` : The algorithm encountered an invalid bracket.
|
||
|
|
``-2`` : The maximum number of iterations was reached.
|
||
|
|
``-3`` : A non-finite value was encountered.
|
||
|
|
``-4`` : Iteration was terminated by `callback`.
|
||
|
|
``1`` : The algorithm is proceeding normally (in `callback` only).
|
||
|
|
x : float
|
||
|
|
The minimizer of the function, if the algorithm terminated
|
||
|
|
successfully.
|
||
|
|
fun : float
|
||
|
|
The value of `func` evaluated at `x`.
|
||
|
|
nfev : int
|
||
|
|
The number of points at which `func` was evaluated.
|
||
|
|
nit : int
|
||
|
|
The number of iterations of the algorithm that were performed.
|
||
|
|
xl, xm, xr : float
|
||
|
|
The final three-point bracket.
|
||
|
|
fl, fm, fr : float
|
||
|
|
The function value at the bracket points.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Implemented based on Chandrupatla's original paper [1]_.
|
||
|
|
|
||
|
|
If ``x1 < x2 < x3`` are the points of the bracket and ``f1 > f2 <= f3``
|
||
|
|
are the values of ``func`` at those points, then the algorithm is
|
||
|
|
considered to have converged when ``x3 - x1 <= abs(x2)*xrtol + xatol``
|
||
|
|
or ``(f1 - 2*f2 + f3)/2 <= abs(f2)*frtol + fatol``. Note that first of
|
||
|
|
these differs from the termination conditions described in [1]_. The
|
||
|
|
default values of `xrtol` is the square root of the precision of the
|
||
|
|
appropriate dtype, and ``xatol = fatol = frtol`` is the smallest normal
|
||
|
|
number of the appropriate dtype.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Chandrupatla, Tirupathi R. (1998).
|
||
|
|
"An efficient quadratic fit-sectioning algorithm for minimization
|
||
|
|
without derivatives".
|
||
|
|
Computer Methods in Applied Mechanics and Engineering, 152 (1-2),
|
||
|
|
211-217. https://doi.org/10.1016/S0045-7825(97)00190-4
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
golden, brent, bounded
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from scipy.optimize._chandrupatla import _chandrupatla_minimize
|
||
|
|
>>> def f(x, args=1):
|
||
|
|
... return (x - args)**2
|
||
|
|
>>> res = _chandrupatla_minimize(f, -5, 0, 5)
|
||
|
|
>>> res.x
|
||
|
|
1.0
|
||
|
|
>>> c = [1, 1.5, 2]
|
||
|
|
>>> res = _chandrupatla_minimize(f, -5, 0, 5, args=(c,))
|
||
|
|
>>> res.x
|
||
|
|
array([1. , 1.5, 2. ])
|
||
|
|
"""
|
||
|
|
res = _chandrupatla_iv(func, args, xatol, xrtol,
|
||
|
|
fatol, frtol, maxiter, callback)
|
||
|
|
func, args, xatol, xrtol, fatol, frtol, maxiter, callback = res
|
||
|
|
|
||
|
|
# Initialization
|
||
|
|
xs = (x1, x2, x3)
|
||
|
|
temp = eim._initialize(func, xs, args)
|
||
|
|
func, xs, fs, args, shape, dtype, xp = temp # line split for PEP8
|
||
|
|
x1, x2, x3 = xs
|
||
|
|
f1, f2, f3 = fs
|
||
|
|
phi = dtype.type(0.5 + 0.5*5**0.5) # golden ratio
|
||
|
|
status = np.full_like(x1, eim._EINPROGRESS, dtype=int) # in progress
|
||
|
|
nit, nfev = 0, 3 # three function evaluations performed above
|
||
|
|
fatol = np.finfo(dtype).tiny if fatol is None else fatol
|
||
|
|
frtol = np.finfo(dtype).tiny if frtol is None else frtol
|
||
|
|
xatol = np.finfo(dtype).tiny if xatol is None else xatol
|
||
|
|
xrtol = np.sqrt(np.finfo(dtype).eps) if xrtol is None else xrtol
|
||
|
|
|
||
|
|
# Ensure that x1 < x2 < x3 initially.
|
||
|
|
xs, fs = np.vstack((x1, x2, x3)), np.vstack((f1, f2, f3))
|
||
|
|
i = np.argsort(xs, axis=0)
|
||
|
|
x1, x2, x3 = np.take_along_axis(xs, i, axis=0)
|
||
|
|
f1, f2, f3 = np.take_along_axis(fs, i, axis=0)
|
||
|
|
q0 = x3.copy() # "At the start, q0 is set at x3..." ([1] after (7))
|
||
|
|
|
||
|
|
work = _RichResult(x1=x1, f1=f1, x2=x2, f2=f2, x3=x3, f3=f3, phi=phi,
|
||
|
|
xatol=xatol, xrtol=xrtol, fatol=fatol, frtol=frtol,
|
||
|
|
nit=nit, nfev=nfev, status=status, q0=q0, args=args)
|
||
|
|
res_work_pairs = [('status', 'status'),
|
||
|
|
('x', 'x2'), ('fun', 'f2'),
|
||
|
|
('nit', 'nit'), ('nfev', 'nfev'),
|
||
|
|
('xl', 'x1'), ('xm', 'x2'), ('xr', 'x3'),
|
||
|
|
('fl', 'f1'), ('fm', 'f2'), ('fr', 'f3')]
|
||
|
|
|
||
|
|
def pre_func_eval(work):
|
||
|
|
# `_check_termination` is called first -> `x3 - x2 > x2 - x1`
|
||
|
|
# But let's calculate a few terms that we'll reuse
|
||
|
|
x21 = work.x2 - work.x1
|
||
|
|
x32 = work.x3 - work.x2
|
||
|
|
|
||
|
|
# [1] Section 3. "The quadratic minimum point Q1 is calculated using
|
||
|
|
# the relations developed in the previous section." [1] Section 2 (5/6)
|
||
|
|
A = x21 * (work.f3 - work.f2)
|
||
|
|
B = x32 * (work.f1 - work.f2)
|
||
|
|
C = A / (A + B)
|
||
|
|
# q1 = C * (work.x1 + work.x2) / 2 + (1 - C) * (work.x2 + work.x3) / 2
|
||
|
|
q1 = 0.5 * (C*(work.x1 - work.x3) + work.x2 + work.x3) # much faster
|
||
|
|
# this is an array, so multiplying by 0.5 does not change dtype
|
||
|
|
|
||
|
|
# "If Q1 and Q0 are sufficiently close... Q1 is accepted if it is
|
||
|
|
# sufficiently away from the inside point x2"
|
||
|
|
i = abs(q1 - work.q0) < 0.5 * abs(x21) # [1] (7)
|
||
|
|
xi = q1[i]
|
||
|
|
# Later, after (9), "If the point Q1 is in a +/- xtol neighborhood of
|
||
|
|
# x2, the new point is chosen in the larger interval at a distance
|
||
|
|
# tol away from x2."
|
||
|
|
# See also QBASIC code after "Accept Ql adjust if close to X2".
|
||
|
|
j = abs(q1[i] - work.x2[i]) <= work.xtol[i]
|
||
|
|
xi[j] = work.x2[i][j] + np.sign(x32[i][j]) * work.xtol[i][j]
|
||
|
|
|
||
|
|
# "If condition (7) is not satisfied, golden sectioning of the larger
|
||
|
|
# interval is carried out to introduce the new point."
|
||
|
|
# (For simplicity, we go ahead and calculate it for all points, but we
|
||
|
|
# change the elements for which the condition was satisfied.)
|
||
|
|
x = work.x2 + (2 - work.phi) * x32
|
||
|
|
x[i] = xi
|
||
|
|
|
||
|
|
# "We define Q0 as the value of Q1 at the previous iteration."
|
||
|
|
work.q0 = q1
|
||
|
|
return x
|
||
|
|
|
||
|
|
def post_func_eval(x, f, work):
|
||
|
|
# Standard logic for updating a three-point bracket based on a new
|
||
|
|
# point. In QBASIC code, see "IF SGN(X-X2) = SGN(X3-X2) THEN...".
|
||
|
|
# There is an awful lot of data copying going on here; this would
|
||
|
|
# probably benefit from code optimization or implementation in Pythran.
|
||
|
|
i = np.sign(x - work.x2) == np.sign(work.x3 - work.x2)
|
||
|
|
xi, x1i, x2i, x3i = x[i], work.x1[i], work.x2[i], work.x3[i],
|
||
|
|
fi, f1i, f2i, f3i = f[i], work.f1[i], work.f2[i], work.f3[i]
|
||
|
|
j = fi > f2i
|
||
|
|
x3i[j], f3i[j] = xi[j], fi[j]
|
||
|
|
j = ~j
|
||
|
|
x1i[j], f1i[j], x2i[j], f2i[j] = x2i[j], f2i[j], xi[j], fi[j]
|
||
|
|
|
||
|
|
ni = ~i
|
||
|
|
xni, x1ni, x2ni, x3ni = x[ni], work.x1[ni], work.x2[ni], work.x3[ni],
|
||
|
|
fni, f1ni, f2ni, f3ni = f[ni], work.f1[ni], work.f2[ni], work.f3[ni]
|
||
|
|
j = fni > f2ni
|
||
|
|
x1ni[j], f1ni[j] = xni[j], fni[j]
|
||
|
|
j = ~j
|
||
|
|
x3ni[j], f3ni[j], x2ni[j], f2ni[j] = x2ni[j], f2ni[j], xni[j], fni[j]
|
||
|
|
|
||
|
|
work.x1[i], work.x2[i], work.x3[i] = x1i, x2i, x3i
|
||
|
|
work.f1[i], work.f2[i], work.f3[i] = f1i, f2i, f3i
|
||
|
|
work.x1[ni], work.x2[ni], work.x3[ni] = x1ni, x2ni, x3ni,
|
||
|
|
work.f1[ni], work.f2[ni], work.f3[ni] = f1ni, f2ni, f3ni
|
||
|
|
|
||
|
|
def check_termination(work):
|
||
|
|
# Check for all terminal conditions and record statuses.
|
||
|
|
stop = np.zeros_like(work.x1, dtype=bool) # termination condition met
|
||
|
|
|
||
|
|
# Bracket is invalid; stop and don't return minimizer/minimum
|
||
|
|
i = ((work.f2 > work.f1) | (work.f2 > work.f3))
|
||
|
|
work.x2[i], work.f2[i] = np.nan, np.nan
|
||
|
|
stop[i], work.status[i] = True, eim._ESIGNERR
|
||
|
|
|
||
|
|
# Non-finite values; stop and don't return minimizer/minimum
|
||
|
|
finite = np.isfinite(work.x1+work.x2+work.x3+work.f1+work.f2+work.f3)
|
||
|
|
i = ~(finite | stop)
|
||
|
|
work.x2[i], work.f2[i] = np.nan, np.nan
|
||
|
|
stop[i], work.status[i] = True, eim._EVALUEERR
|
||
|
|
|
||
|
|
# [1] Section 3 "Points 1 and 3 are interchanged if necessary to make
|
||
|
|
# the (x2, x3) the larger interval."
|
||
|
|
# Note: I had used np.choose; this is much faster. This would be a good
|
||
|
|
# place to save e.g. `work.x3 - work.x2` for reuse, but I tried and
|
||
|
|
# didn't notice a speed boost, so let's keep it simple.
|
||
|
|
i = abs(work.x3 - work.x2) < abs(work.x2 - work.x1)
|
||
|
|
temp = work.x1[i]
|
||
|
|
work.x1[i] = work.x3[i]
|
||
|
|
work.x3[i] = temp
|
||
|
|
temp = work.f1[i]
|
||
|
|
work.f1[i] = work.f3[i]
|
||
|
|
work.f3[i] = temp
|
||
|
|
|
||
|
|
# [1] Section 3 (bottom of page 212)
|
||
|
|
# "We set a tolerance value xtol..."
|
||
|
|
work.xtol = abs(work.x2) * work.xrtol + work.xatol # [1] (8)
|
||
|
|
# "The convergence based on interval is achieved when..."
|
||
|
|
# Note: Equality allowed in case of `xtol=0`
|
||
|
|
i = abs(work.x3 - work.x2) <= 2 * work.xtol # [1] (9)
|
||
|
|
|
||
|
|
# "We define ftol using..."
|
||
|
|
ftol = abs(work.f2) * work.frtol + work.fatol # [1] (10)
|
||
|
|
# "The convergence based on function values is achieved when..."
|
||
|
|
# Note 1: modify in place to incorporate tolerance on function value.
|
||
|
|
# Note 2: factor of 2 is not in the text; see QBASIC start of DO loop
|
||
|
|
i |= (work.f1 - 2 * work.f2 + work.f3) <= 2*ftol # [1] (11)
|
||
|
|
i &= ~stop
|
||
|
|
stop[i], work.status[i] = True, eim._ECONVERGED
|
||
|
|
|
||
|
|
return stop
|
||
|
|
|
||
|
|
def post_termination_check(work):
|
||
|
|
pass
|
||
|
|
|
||
|
|
def customize_result(res, shape):
|
||
|
|
xl, xr, fl, fr = res['xl'], res['xr'], res['fl'], res['fr']
|
||
|
|
i = res['xl'] < res['xr']
|
||
|
|
res['xl'] = np.choose(i, (xr, xl))
|
||
|
|
res['xr'] = np.choose(i, (xl, xr))
|
||
|
|
res['fl'] = np.choose(i, (fr, fl))
|
||
|
|
res['fr'] = np.choose(i, (fl, fr))
|
||
|
|
return shape
|
||
|
|
|
||
|
|
return eim._loop(work, callback, shape, maxiter, func, args, dtype,
|
||
|
|
pre_func_eval, post_func_eval, check_termination,
|
||
|
|
post_termination_check, customize_result, res_work_pairs,
|
||
|
|
xp=xp)
|