scipy.optimize.

bisect#

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

Find root of a function within an interval using bisection.

Basic bisection routine to find a root of the function f between the arguments a and b. f(a) and f(b) cannot have the same signs. Slow but sure.

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 a 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
fixed_point: scalar fixed-point finder
fsolve: n-dimensional root-finding
elementwise.find_root: efficient elementwise 1-D root-finder

Notes

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 bisect 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.

Examples

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

>>> from scipy import optimize

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

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