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scipy.optimize.elementwise.

# bracket_root

scipy.optimize.elementwise.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.

      successbool array

        `True` where the algorithm terminated successfully (status `0`); `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

  `find_root`

  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.

  Array API Standard Support

  `bracket_root` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

  Library | CPU | GPU
  --- | --- | ---
  NumPy | ✅ | n/a
  CuPy | n/a | ✅
  PyTorch | ✅ | ✅
  JAX | ⛔ | ⛔
  Dask | ⛔ | n/a

  > See Support for the array API standard for more information.

  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])