scipy.optimize.ridder¶

scipy.optimize.ridder(f, a, b, args=(), xtol=2e-12, rtol=8.8817841970012523e-16, maxiter=100, full_output=False, disp=True)[source]¶

Find a root of a function in an interval.

Parameters:

Parameters:	f : function Python function returning a number. f must be continuous, and f(a) and f(b) must have opposite signs. a : number One end of the bracketing interval [a,b]. b : number The other end of the bracketing interval [a,b]. xtol : number, optional The computed root `x0` will satisfy `np.allclose(x, x0, atol=xtol, rtol=rtol)`, where `x` is the exact root. The parameter must be nonnegative. rtol : number, optional The computed root `x0` will satisfy `np.allclose(x, x0, atol=xtol, rtol=rtol)`, where `x` is the exact root. The parameter cannot be smaller than its default value of `4np.finfo(float).eps`. maxiter* : number, 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.
Returns:	x0 : float Zero 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.

f : function

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

a : number

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

b : number

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

xtol : number, optional

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

rtol : number, optional

The computed root x0 will satisfy np.allclose(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 : number, 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.

Returns:

x0 : float

Zero 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

fixed_point: scalar fixed-point finder

Notes

Uses [Ridders1979] method to find a zero of the function f between the arguments a and b. Ridders’ method is faster than bisection, but not generally as fast as the Brent rountines. [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.

References

[Ridders1979]

(1, 2, 3) 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