bracket_minimum#
- scipy.optimize.elementwise.bracket_minimum(f, xm0, *, xl0=None, xr0=None, xmin=None, xmax=None, factor=None, args=(), maxiter=1000)[source]#
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 pointsxl < xm < xr
such thatfl >= 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:
- fcallable
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 andargs
is a tuple, which may contain an arbitrary number of arrays that are broadcastable withx
.f must be an elementwise function: each element
f(x)[i]
must equalf(x[i])
for all indicesi
. It must not mutate the arrayx
or the arrays inargs
.- 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, xmaxfloat array_like, optional
Minimum and maximum allowable endpoints of the bracket, inclusive. Must be broadcastable with all other array inputs.
- factorfloat array_like, default: 2
The factor used to grow the bracket. See Notes.
- argstuple 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
.- maxiterint, 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.- successbool array
True
where the algorithm terminated successfully (status0
);False
otherwise.- statusint 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.
- bracket3-tuple of float arrays
The left, middle, and right points of the bracket, if the algorithm terminated successfully.
- f_bracket3-tuple of float arrays
The function value at the left, middle, and right points of the bracket.
- nfevint 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.
- nitint array
The number of iterations of the algorithm that were performed.
Notes
Similar to
scipy.optimize.bracket
, this function seeks to find real pointsxl < xm < xr
such thatf(xl) >= f(xm)
andf(xr) >= f(xm)
, where at least one of the inequalities is strict. Unlikescipy.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 unlikescipy.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 bexr + 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 bexmax - (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.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
andfind_minimum
accept arrays for most arguments. For instance, to find the minimizers and minima for a few values of the parameterc
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.])