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)[source]#

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

Parameters:

ffunction: Python function returning a number. f must be continuous, and f(a) and f(b) must have opposite signs.
ascalar: One end of the bracketing interval [a,b].
bscalar: The other end of the bracketing interval [a,b].
xtolnumber, 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.
rtolnumber, 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.
maxiterint, optional: If convergence is not achieved in maxiter iterations, an error is raised. Must be >= 0.
argstuple, optional: Containing extra arguments for the function f. f is called by apply(f, (x)+args).
full_outputbool, 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.
dispbool, optional: If True, raise RuntimeError if the algorithm didn’t converge. Otherwise, the convergence status is recorded in any RootResults return object.

Returns:

rootfloat: Root of f between a and b.
rRootResults (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