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scipy.optimize.elementwise.

# find_root

scipy.optimize.elementwise.find_root(f, init, /, *, args=(), kwargs=None, 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`.

    kwargs dict of str:array_like, optional

      Additional keyword arguments to be passed to f. See 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.

      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 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).

      xfloat array

        The root of the function, if the algorithm terminated successfully.

      f_xfloat array

        The value of f evaluated at x.

      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.

      brackettuple of float arrays

        The lower and upper endpoints of the final bracket.

      f_brackettuple of float arrays

        The value of f evaluated at the lower and upper endpoints of the bracket.

  See also

  `bracket_root`

  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.

  Array API Standard Support

  `find_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.

  References

  [1] (1,2,3)

  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. DOI:10.1016/s0965-9978(96)00051-8.

  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])