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from scipy.optimize._bracket import _bracket_root, _bracket_minimum
from scipy.optimize._chandrupatla import _chandrupatla, _chandrupatla_minimize
from scipy._lib._util import _RichResult
def find_root(f, init, /, *, args=(), tolerances=None, maxiter=None, callback=None):
"""Find the root of a monotonic, real-valued function of a real variable.
For each element of the output of `f`, `find_root` seeks the scalar
root that makes the element 0. This function currently uses Chandrupatla's
bracketing algorithm [1]_ and therefore requires argument `init` to
provide a bracket around the root: the function values at the two endpoints
must have opposite signs.
Provided a valid bracket, `find_root` is guaranteed to converge to a solution
that satisfies the provided `tolerances` if the function is continuous within
the bracket.
This function works elementwise when `init` and `args` contain (broadcastable)
arrays.
Parameters
----------
f : callable
The function whose root is desired. The signature must be::
f(x: array, *args) -> array
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``.
`f` must be an elementwise function: each element ``f(x)[i]``
must equal ``f(x[i])`` for all indices ``i``. It must not mutate the
array ``x`` or the arrays in ``args``.
`find_root` seeks an array ``x`` such that ``f(x)`` is an array of zeros.
init : 2-tuple of float array_like
The lower and upper endpoints of a bracket surrounding the desired root.
A bracket is valid if arrays ``xl, xr = init`` satisfy ``xl < xr`` and
``sign(f(xl)) == -sign(f(xr))`` elementwise. Arrays be broadcastable with
one another and `args`.
args : tuple of array_like, optional
Additional positional array arguments to be passed to `f`. Arrays
must be broadcastable with one another and the arrays of `init`.
If the callable for which the root is desired requires arguments that are
not broadcastable with `x`, wrap that callable with `f` such that `f`
accepts only `x` and broadcastable ``*args``.
tolerances : dictionary of floats, optional
Absolute and relative tolerances on the root and function value.
Valid keys of the dictionary are:
- ``xatol`` - absolute tolerance on the root
- ``xrtol`` - relative tolerance on the root
- ``fatol`` - absolute tolerance on the function value
- ``frtol`` - relative tolerance on the function value
See Notes for default values and explicit termination conditions.
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 `find_root` (but containing the current
iterate's values of all variables). If `callback` raises a
``StopIteration``, the algorithm will terminate immediately and
`find_root` will return a result. `callback` must not mutate
`res` or its attributes.
Returns
-------
res : _RichResult
An object similar to an instance of `scipy.optimize.OptimizeResult` with the
following attributes. The descriptions are written as though the values will
be scalars; however, if `f` returns an array, the outputs will be
arrays of the same shape.
success : bool array
``True`` where the algorithm terminated successfully (status ``0``);
``False`` otherwise.
status : int array
An integer representing the exit status of the algorithm.
- ``0`` : The algorithm converged to the specified tolerances.
- ``-1`` : The initial bracket was invalid.
- ``-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 array
The root of the function, if the algorithm terminated successfully.
f_x : float array
The value of `f` evaluated at `x`.
nfev : int array
The number of abscissae at which `f` was evaluated to find the root.
This is distinct from the number of times `f` is *called* because the
the function may evaluated at multiple points in a single call.
nit : int array
The number of iterations of the algorithm that were performed.
bracket : tuple of float arrays
The lower and upper endpoints of the final bracket.
f_bracket : tuple of float arrays
The value of `f` evaluated at the lower and upper endpoints of the
bracket.
Notes
-----
Implemented based on Chandrupatla's original paper [1]_.
Let:
- ``a, b = init`` be the left and right endpoints of the initial bracket,
- ``xl`` and ``xr`` be the left and right endpoints of the final bracket,
- ``xmin = xl if abs(f(xl)) <= abs(f(xr)) else xr`` be the final bracket
endpoint with the smaller function value, and
- ``fmin0 = min(f(a), f(b))`` be the minimum of the two values of the
function evaluated at the initial bracket endpoints.
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``.
However, the default values of the `tolerances` dictionary 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
--------
bracket_root
Examples
--------
Suppose we wish to find the root of the following function.
>>> def f(x, c=5):
... return x**3 - 2*x - c
First, we must find a valid bracket. The function is not monotonic,
but `bracket_root` may be able to provide a bracket.
>>> from scipy.optimize import elementwise
>>> res_bracket = elementwise.bracket_root(f, 0)
>>> res_bracket.success
True
>>> res_bracket.bracket
(2.0, 4.0)
Indeed, the values of the function at the bracket endpoints have
opposite signs.
>>> res_bracket.f_bracket
(-1.0, 51.0)
Once we have a valid bracket, `find_root` can be used to provide
a precise root.
>>> res_root = elementwise.find_root(f, res_bracket.bracket)
>>> res_root.x
2.0945514815423265
The final bracket is only a few ULPs wide, so the error between
this value and the true root cannot be much smaller within values
that are representable in double precision arithmetic.
>>> import numpy as np
>>> xl, xr = res_root.bracket
>>> (xr - xl) / np.spacing(xl)
2.0
>>> res_root.f_bracket
(-8.881784197001252e-16, 9.769962616701378e-15)
`bracket_root` and `find_root` accept arrays for most arguments.
For instance, to find the root for a few values of the parameter ``c``
at once:
>>> c = np.asarray([3, 4, 5])
>>> res_bracket = elementwise.bracket_root(f, 0, args=(c,))
>>> res_bracket.bracket
(array([1., 1., 2.]), array([2., 2., 4.]))
>>> res_root = elementwise.find_root(f, res_bracket.bracket, args=(c,))
>>> res_root.x
array([1.8932892 , 2. , 2.09455148])
"""
def reformat_result(res_in):
res_out = _RichResult()
res_out.status = res_in.status
res_out.success = res_in.success
res_out.x = res_in.x
res_out.f_x = res_in.fun
res_out.nfev = res_in.nfev
res_out.nit = res_in.nit
res_out.bracket = (res_in.xl, res_in.xr)
res_out.f_bracket = (res_in.fl, res_in.fr)
res_out._order_keys = ['success', 'status', 'x', 'f_x',
'nfev', 'nit', 'bracket', 'f_bracket']
return res_out
xl, xr = init
default_tolerances = dict(xatol=None, xrtol=None, fatol=None, frtol=0)
tolerances = {} if tolerances is None else tolerances
default_tolerances.update(tolerances)
tolerances = default_tolerances
if callable(callback):
def _callback(res):
return callback(reformat_result(res))
else:
_callback = callback
res = _chandrupatla(f, xl, xr, args=args, **tolerances,
maxiter=maxiter, callback=_callback)
return reformat_result(res)
def find_minimum(f, init, /, *, args=(), tolerances=None, maxiter=100, callback=None):
"""Find the minimum of an unimodal, real-valued function of a real variable.
For each element of the output of `f`, `find_minimum` seeks the scalar minimizer
that minimizes the element. This function currently uses Chandrupatla's
bracketing minimization algorithm [1]_ and therefore requires argument `init`
to provide a three-point minimization bracket: ``x1 < x2 < x3`` such that
``func(x1) >= func(x2) <= func(x3)``, where one of the inequalities is strict.
Provided a valid bracket, `find_minimum` is guaranteed to converge to a local
minimum that satisfies the provided `tolerances` if the function is continuous
within the bracket.
This function works elementwise when `init` and `args` contain (broadcastable)
arrays.
Parameters
----------
f : callable
The function whose minimizer is desired. The signature must be::
f(x: array, *args) -> array
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``.
`f` must be an elementwise function: each element ``f(x)[i]``
must equal ``f(x[i])`` for all indices ``i``. It must not mutate the
array ``x`` or the arrays in ``args``.
`find_minimum` seeks an array ``x`` such that ``f(x)`` is an array of
local minima.
init : 3-tuple of float array_like
The abscissae of a standard scalar minimization bracket. A bracket is
valid if arrays ``x1, x2, x3 = init`` satisfy ``x1 < x2 < x3`` and
``func(x1) >= func(x2) <= func(x3)``, where one of the inequalities
is strict. Arrays must be broadcastable with one another and the arrays
of `args`.
args : tuple of array_like, optional
Additional positional array arguments to be passed to `f`. Arrays
must be broadcastable with one another and the arrays of `init`.
If the callable for which the root is desired requires arguments that are
not broadcastable with `x`, wrap that callable with `f` such that `f`
accepts only `x` and broadcastable ``*args``.
tolerances : dictionary of floats, optional
Absolute and relative tolerances on the root and function value.
Valid keys of the dictionary are:
- ``xatol`` - absolute tolerance on the root
- ``xrtol`` - relative tolerance on the root
- ``fatol`` - absolute tolerance on the function value
- ``frtol`` - relative tolerance on the function value
See Notes for default values and explicit termination conditions.
maxiter : int, default: 100
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 `find_minimum` (but containing the current
iterate's values of all variables). If `callback` raises a
``StopIteration``, the algorithm will terminate immediately and
`find_root` will return a result. `callback` must not mutate
`res` or its attributes.
Returns
-------
res : _RichResult
An object similar to an instance of `scipy.optimize.OptimizeResult` with the
following attributes. The descriptions are written as though the values will
be scalars; however, if `f` returns an array, the outputs will be
arrays of the same shape.
success : bool array
``True`` where the algorithm terminated successfully (status ``0``);
``False`` otherwise.
status : int array
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 array
The minimizer of the function, if the algorithm terminated successfully.
f_x : float array
The value of `f` evaluated at `x`.
nfev : int array
The number of abscissae at which `f` was evaluated to find the root.
This is distinct from the number of times `f` is *called* because the
the function may evaluated at multiple points in a single call.
nit : int array
The number of iterations of the algorithm that were performed.
bracket : tuple of float arrays
The final three-point bracket.
f_bracket : tuple of float arrays
The value of `f` evaluated at the bracket points.
Notes
-----
Implemented based on Chandrupatla's original paper [1]_.
If ``xl < xm < xr`` are the points of the bracket and ``fl >= fm <= fr``
(where one of the inequalities is strict) are the values of `f` evaluated
at those points, then the algorithm is considered to have converged when:
- ``xr - xl <= abs(xm)*xrtol + xatol`` or
- ``(fl - 2*fm + fr)/2 <= abs(fm)*frtol + fatol``.
Note that first of these differs from the termination conditions described
in [1]_.
The default value 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
--------
bracket_minimum
Examples
--------
Suppose we wish to minimize the following function.
>>> def f(x, c=1):
... return (x - c)**2 + 2
First, we must find a valid bracket. The function is unimodal,
so `bracket_minium` will easily find a bracket.
>>> from scipy.optimize import elementwise
>>> res_bracket = elementwise.bracket_minimum(f, 0)
>>> res_bracket.success
True
>>> res_bracket.bracket
(0.0, 0.5, 1.5)
Indeed, the bracket points are ordered and the function value
at the middle bracket point is less than at the surrounding
points.
>>> xl, xm, xr = res_bracket.bracket
>>> fl, fm, fr = res_bracket.f_bracket
>>> (xl < xm < xr) and (fl > fm <= fr)
True
Once we have a valid bracket, `find_minimum` can be used to provide
an estimate of the minimizer.
>>> res_minimum = elementwise.find_minimum(f, res_bracket.bracket)
>>> res_minimum.x
1.0000000149011612
The function value changes by only a few ULPs within the bracket, so
the minimizer cannot be determined much more precisely by evaluating
the function alone (i.e. we would need its derivative to do better).
>>> import numpy as np
>>> fl, fm, fr = res_minimum.f_bracket
>>> (fl - fm) / np.spacing(fm), (fr - fm) / np.spacing(fm)
(0.0, 2.0)
Therefore, a precise minimum of the function is given by:
>>> res_minimum.f_x
2.0
`bracket_minimum` and `find_minimum` accept arrays for most arguments.
For instance, to find the minimizers and minima for a few values of the
parameter ``c`` at once:
>>> c = np.asarray([1, 1.5, 2])
>>> res_bracket = elementwise.bracket_minimum(f, 0, args=(c,))
>>> res_bracket.bracket
(array([0. , 0.5, 0.5]), array([0.5, 1.5, 1.5]), array([1.5, 2.5, 2.5]))
>>> res_minimum = elementwise.find_minimum(f, res_bracket.bracket, args=(c,))
>>> res_minimum.x
array([1.00000001, 1.5 , 2. ])
>>> res_minimum.f_x
array([2., 2., 2.])
"""
def reformat_result(res_in):
res_out = _RichResult()
res_out.status = res_in.status
res_out.success = res_in.success
res_out.x = res_in.x
res_out.f_x = res_in.fun
res_out.nfev = res_in.nfev
res_out.nit = res_in.nit
res_out.bracket = (res_in.xl, res_in.xm, res_in.xr)
res_out.f_bracket = (res_in.fl, res_in.fm, res_in.fr)
res_out._order_keys = ['success', 'status', 'x', 'f_x',
'nfev', 'nit', 'bracket', 'f_bracket']
return res_out
xl, xm, xr = init
default_tolerances = dict(xatol=None, xrtol=None, fatol=None, frtol=None)
tolerances = {} if tolerances is None else tolerances
default_tolerances.update(tolerances)
tolerances = default_tolerances
if callable(callback):
def _callback(res):
return callback(reformat_result(res))
else:
_callback = callback
res = _chandrupatla_minimize(f, xl, xm, xr, args=args, **tolerances,
maxiter=maxiter, callback=_callback)
return reformat_result(res)
def bracket_root(f, xl0, xr0=None, *, xmin=None, xmax=None, factor=None, args=(),
maxiter=1000):
"""Bracket the root of a monotonic, real-valued function of a real variable.
For each element of the output of `f`, `bracket_root` seeks the scalar
bracket endpoints ``xl`` and ``xr`` such that ``sign(f(xl)) == -sign(f(xr))``
elementwise.
The function is guaranteed to find a valid bracket if the function is monotonic,
but it may find a bracket under other conditions.
This function works elementwise when `xl0`, `xr0`, `xmin`, `xmax`, `factor`, and
the elements of `args` are (mutually broadcastable) arrays.
Parameters
----------
f : callable
The function for which the root is to be bracketed. The signature must be::
f(x: array, *args) -> array
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``.
`f` must be an elementwise function: each element ``f(x)[i]``
must equal ``f(x[i])`` for all indices ``i``. It must not mutate the
array ``x`` or the arrays in ``args``.
xl0, xr0: float array_like
Starting guess of bracket, which need not contain a root. If `xr0` is
not provided, ``xr0 = xl0 + 1``. Must be broadcastable with all other
array inputs.
xmin, xmax : float array_like, optional
Minimum and maximum allowable endpoints of the bracket, inclusive. Must
be broadcastable with all other array inputs.
factor : float array_like, default: 2
The factor used to grow the bracket. See Notes.
args : tuple of array_like, optional
Additional positional array arguments to be passed to `f`.
If the callable for which the root is desired requires arguments that are
not broadcastable with `x`, wrap that callable with `f` such that `f`
accepts only `x` and broadcastable ``*args``.
maxiter : int, default: 1000
The maximum number of iterations of the algorithm to perform.
Returns
-------
res : _RichResult
An object similar to an instance of `scipy.optimize.OptimizeResult` with the
following attributes. The descriptions are written as though the values will
be scalars; however, if `f` returns an array, the outputs will be
arrays of the same shape.
success : bool array
``True`` where the algorithm terminated successfully (status ``0``);
``False`` otherwise.
status : int array
An integer representing the exit status of the algorithm.
- ``0`` : The algorithm produced a valid bracket.
- ``-1`` : The bracket expanded to the allowable limits without success.
- ``-2`` : The maximum number of iterations was reached.
- ``-3`` : A non-finite value was encountered.
- ``-4`` : Iteration was terminated by `callback`.
- ``-5``: The initial bracket does not satisfy`xmin <= xl0 < xr0 < xmax`.
bracket : 2-tuple of float arrays
The lower and upper endpoints of the bracket, if the algorithm
terminated successfully.
f_bracket : 2-tuple of float arrays
The values of `f` evaluated at the endpoints of ``res.bracket``,
respectively.
nfev : int array
The number of abscissae at which `f` was evaluated to find the root.
This is distinct from the number of times `f` is *called* because the
the function may evaluated at multiple points in a single call.
nit : int array
The number of iterations of the algorithm that were performed.
Notes
-----
This function generalizes an algorithm found in pieces throughout the
`scipy.stats` codebase. The strategy is to iteratively grow the bracket `(l, r)`
until ``f(l) < 0 < f(r)`` or ``f(r) < 0 < f(l)``. The bracket grows to the left
as follows.
- If `xmin` is not provided, the distance between `xl0` and `l` is iteratively
increased by `factor`.
- If `xmin` is provided, the distance between `xmin` and `l` is iteratively
decreased by `factor`. Note that this also *increases* the bracket size.
Growth of the bracket to the right is analogous.
Growth of the bracket in one direction stops when the endpoint is no longer
finite, the function value at the endpoint is no longer finite, or the
endpoint reaches its limiting value (`xmin` or `xmax`). Iteration terminates
when the bracket stops growing in both directions, the bracket surrounds
the root, or a root is found (by chance).
If two brackets are found - that is, a bracket is found on both sides in
the same iteration, the smaller of the two is returned.
If roots of the function are found, both `xl` and `xr` are set to the
leftmost root.
See Also
--------
find_root
Examples
--------
Suppose we wish to find the root of the following function.
>>> def f(x, c=5):
... return x**3 - 2*x - c
First, we must find a valid bracket. The function is not monotonic,
but `bracket_root` may be able to provide a bracket.
>>> from scipy.optimize import elementwise
>>> res_bracket = elementwise.bracket_root(f, 0)
>>> res_bracket.success
True
>>> res_bracket.bracket
(2.0, 4.0)
Indeed, the values of the function at the bracket endpoints have
opposite signs.
>>> res_bracket.f_bracket
(-1.0, 51.0)
Once we have a valid bracket, `find_root` can be used to provide
a precise root.
>>> res_root = elementwise.find_root(f, res_bracket.bracket)
>>> res_root.x
2.0945514815423265
`bracket_root` and `find_root` accept arrays for most arguments.
For instance, to find the root for a few values of the parameter ``c``
at once:
>>> import numpy as np
>>> c = np.asarray([3, 4, 5])
>>> res_bracket = elementwise.bracket_root(f, 0, args=(c,))
>>> res_bracket.bracket
(array([1., 1., 2.]), array([2., 2., 4.]))
>>> res_root = elementwise.find_root(f, res_bracket.bracket, args=(c,))
>>> res_root.x
array([1.8932892 , 2. , 2.09455148])
""" # noqa: E501
res = _bracket_root(f, xl0, xr0=xr0, xmin=xmin, xmax=xmax, factor=factor,
args=args, maxiter=maxiter)
res.bracket = res.xl, res.xr
res.f_bracket = res.fl, res.fr
del res.xl
del res.xr
del res.fl
del res.fr
return res
def bracket_minimum(f, xm0, *, xl0=None, xr0=None, xmin=None, xmax=None,
factor=None, args=(), maxiter=1000):
"""Bracket the minimum of a unimodal, real-valued function of a real variable.
For each element of the output of `f`, `bracket_minimum` seeks the scalar
bracket points ``xl < xm < xr`` such that ``fl >= fm <= fr`` where one of the
inequalities is strict.
The function is guaranteed to find a valid bracket if the function is
strongly unimodal, but it may find a bracket under other conditions.
This function works elementwise when `xm0`, `xl0`, `xr0`, `xmin`, `xmax`, `factor`,
and the elements of `args` are (mutually broadcastable) arrays.
Parameters
----------
f : callable
The function for which the root is to be bracketed. The signature must be::
f(x: array, *args) -> array
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``.
`f` must be an elementwise function: each element ``f(x)[i]``
must equal ``f(x[i])`` for all indices ``i``. It must not mutate the
array ``x`` or the arrays in ``args``.
xm0: float array_like
Starting guess for middle point of bracket.
xl0, xr0: float array_like, optional
Starting guesses for left and right endpoints of the bracket. Must
be broadcastable with all other array inputs.
xmin, xmax : float array_like, optional
Minimum and maximum allowable endpoints of the bracket, inclusive. Must
be broadcastable with all other array inputs.
factor : float array_like, default: 2
The factor used to grow the bracket. See Notes.
args : tuple of array_like, optional
Additional positional array arguments to be passed to `f`.
If the callable for which the root is desired requires arguments that are
not broadcastable with `x`, wrap that callable with `f` such that `f`
accepts only `x` and broadcastable ``*args``.
maxiter : int, default: 1000
The maximum number of iterations of the algorithm to perform.
Returns
-------
res : _RichResult
An object similar to an instance of `scipy.optimize.OptimizeResult` with the
following attributes. The descriptions are written as though the values will
be scalars; however, if `f` returns an array, the outputs will be
arrays of the same shape.
success : bool array
``True`` where the algorithm terminated successfully (status ``0``);
``False`` otherwise.
status : int array
An integer representing the exit status of the algorithm.
- ``0`` : The algorithm produced a valid bracket.
- ``-1`` : The bracket expanded to the allowable limits. Assuming
unimodality, this implies the endpoint at the limit is a minimizer.
- ``-2`` : The maximum number of iterations was reached.
- ``-3`` : A non-finite value was encountered.
- ``-4`` : ``None`` shall pass.
- ``-5`` : The initial bracket does not satisfy
`xmin <= xl0 < xm0 < xr0 <= xmax`.
bracket : 3-tuple of float arrays
The left, middle, and right points of the bracket, if the algorithm
terminated successfully.
f_bracket : 3-tuple of float arrays
The function value at the left, middle, and right points of the bracket.
nfev : int array
The number of abscissae at which `f` was evaluated to find the root.
This is distinct from the number of times `f` is *called* because the
the function may evaluated at multiple points in a single call.
nit : int array
The number of iterations of the algorithm that were performed.
Notes
-----
Similar to `scipy.optimize.bracket`, this function seeks to find real
points ``xl < xm < xr`` such that ``f(xl) >= f(xm)`` and ``f(xr) >= f(xm)``,
where at least one of the inequalities is strict. Unlike `scipy.optimize.bracket`,
this function can operate in a vectorized manner on array input, so long as
the input arrays are broadcastable with each other. Also unlike
`scipy.optimize.bracket`, users may specify minimum and maximum endpoints
for the desired bracket.
Given an initial trio of points ``xl = xl0``, ``xm = xm0``, ``xr = xr0``,
the algorithm checks if these points already give a valid bracket. If not,
a new endpoint, ``w`` is chosen in the "downhill" direction, ``xm`` becomes the new
opposite endpoint, and either `xl` or `xr` becomes the new middle point,
depending on which direction is downhill. The algorithm repeats from here.
The new endpoint `w` is chosen differently depending on whether or not a
boundary `xmin` or `xmax` has been set in the downhill direction. Without
loss of generality, suppose the downhill direction is to the right, so that
``f(xl) > f(xm) > f(xr)``. If there is no boundary to the right, then `w`
is chosen to be ``xr + factor * (xr - xm)`` where `factor` is controlled by
the user (defaults to 2.0) so that step sizes increase in geometric proportion.
If there is a boundary, `xmax` in this case, then `w` is chosen to be
``xmax - (xmax - xr)/factor``, with steps slowing to a stop at
`xmax`. This cautious approach ensures that a minimum near but distinct from
the boundary isn't missed while also detecting whether or not the `xmax` is
a minimizer when `xmax` is reached after a finite number of steps.
See Also
--------
scipy.optimize.bracket
scipy.optimize.elementwise.find_minimum
Examples
--------
Suppose we wish to minimize the following function.
>>> def f(x, c=1):
... return (x - c)**2 + 2
First, we must find a valid bracket. The function is unimodal,
so `bracket_minium` will easily find a bracket.
>>> from scipy.optimize import elementwise
>>> res_bracket = elementwise.bracket_minimum(f, 0)
>>> res_bracket.success
True
>>> res_bracket.bracket
(0.0, 0.5, 1.5)
Indeed, the bracket points are ordered and the function value
at the middle bracket point is less than at the surrounding
points.
>>> xl, xm, xr = res_bracket.bracket
>>> fl, fm, fr = res_bracket.f_bracket
>>> (xl < xm < xr) and (fl > fm <= fr)
True
Once we have a valid bracket, `find_minimum` can be used to provide
an estimate of the minimizer.
>>> res_minimum = elementwise.find_minimum(f, res_bracket.bracket)
>>> res_minimum.x
1.0000000149011612
`bracket_minimum` and `find_minimum` accept arrays for most arguments.
For instance, to find the minimizers and minima for a few values of the
parameter ``c`` at once:
>>> import numpy as np
>>> c = np.asarray([1, 1.5, 2])
>>> res_bracket = elementwise.bracket_minimum(f, 0, args=(c,))
>>> res_bracket.bracket
(array([0. , 0.5, 0.5]), array([0.5, 1.5, 1.5]), array([1.5, 2.5, 2.5]))
>>> res_minimum = elementwise.find_minimum(f, res_bracket.bracket, args=(c,))
>>> res_minimum.x
array([1.00000001, 1.5 , 2. ])
>>> res_minimum.f_x
array([2., 2., 2.])
""" # noqa: E501
res = _bracket_minimum(f, xm0, xl0=xl0, xr0=xr0, xmin=xmin, xmax=xmax,
factor=factor, args=args, maxiter=maxiter)
res.bracket = res.xl, res.xm, res.xr
res.f_bracket = res.fl, res.fm, res.fr
del res.xl
del res.xm
del res.xr
del res.fl
del res.fm
del res.fr
return res
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