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scipy.optimize.

# ridder

scipy.optimize.ridder(f, a, b, args=(), xtol=2e-12, rtol=np.float64(8.881784197001252e-16), maxiter=100, full_output=False, disp=True)

  Find a root of a function in an interval using Ridder’s method.

  Parameters:
    f function

      Python function returning a number. f must be continuous, and f(a) and f(b) must have opposite signs.

    a scalar

      One end of the bracketing interval [a,b].

    b scalar

      The other end of the bracketing interval [a,b].

    xtol number, optional

      The computed root `x0` will satisfy `np.isclose(x, x0, atol=xtol, rtol=rtol)`, where `x` is the exact root. The parameter must be positive.

    rtol number, optional

      The computed root `x0` will satisfy `np.isclose(x, x0, atol=xtol, rtol=rtol)`, where `x` is the exact root. The parameter cannot be smaller than its default value of `4*np.finfo(float).eps`.

    maxiter int, optional

      If convergence is not achieved in maxiter iterations, an error is raised. Must be >= 0.

    args tuple, optional

      Containing extra arguments for the function f. f is called by `apply(f, (x)+args)`.

    full_output bool, optional

      If full_output is False, the root is returned. If full_output is True, the return value is `(x, r)`, where x is the root, and r is a `RootResults` object.

    disp bool, optional

      If True, raise RuntimeError if the algorithm didn’t converge. Otherwise, the convergence status is recorded in any `RootResults` return object.

  Returns:
    root float

      Root of f between a and b.

    r `RootResults` (present if `full_output = True`)

      Object containing information about the convergence. In particular, `r.converged` is True if the routine converged.

  See also

  `brentq`, `brenth`, `bisect`, `newton`

    1-D root-finding

  `fixed_point`

    scalar fixed-point finder

  `elementwise.find_root`

    efficient elementwise 1-D root-finder

  Notes

  Uses [Ridders1979] method to find a root of the function f between the arguments a and b. Ridders’ method is faster than bisection, but not generally as fast as the Brent routines. [Ridders1979] provides the classic description and source of the algorithm. A description can also be found in any recent edition of Numerical Recipes.

  The routine used here diverges slightly from standard presentations in order to be a bit more careful of tolerance.

  As mentioned in the parameter documentation, the computed root `x0` will satisfy `np.isclose(x, x0, atol=xtol, rtol=rtol)`, where `x` is the exact root. In equation form, this terminating condition is `abs(x - x0) <= xtol + rtol * abs(x0)`.

  The default value `xtol=2e-12` may lead to surprising behavior if one expects `ridder` to always compute roots with relative error near machine precision. Care should be taken to select xtol for the use case at hand. Setting `xtol=5e-324`, the smallest subnormal number, will ensure the highest level of accuracy. Larger values of xtol may be useful for saving function evaluations when a root is at or near zero in applications where the tiny absolute differences available between floating point numbers near zero are not meaningful.

  References

  [Ridders1979] (1,2)

  Ridders, C. F. J. “A New Algorithm for Computing a Single Root of a Real Continuous Function.” IEEE Trans. Circuits Systems 26, 979-980, 1979.

  Examples

      >>> def f(x):
      ...     return (x**2 - 1)

      >>> from scipy import optimize

      >>> root = optimize.ridder(f, 0, 2)
      >>> root
      1.0

      >>> root = optimize.ridder(f, -2, 0)
      >>> root
      -1.0