bracket_root#
- scipy.optimize.elementwise.bracket_root(f, xl0, xr0=None, *, xmin=None, xmax=None, factor=None, args=(), maxiter=1000)[source]#
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 endpointsxl
andxr
such thatsign(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:
- 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
.- 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, 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 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`.
- bracket2-tuple of float arrays
The lower and upper endpoints of the bracket, if the algorithm terminated successfully.
- f_bracket2-tuple of float arrays
The values of f evaluated at the endpoints of
res.bracket
, respectively.- 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.
See also
Notes
This function generalizes an algorithm found in pieces throughout the
scipy.stats
codebase. The strategy is to iteratively grow the bracket (l, r) untilf(l) < 0 < f(r)
orf(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.
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
andfind_root
accept arrays for most arguments. For instance, to find the root for a few values of the parameterc
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])