scipy.optimize.

brenth#

scipy.optimize.brenth(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 a bracketing interval using Brent’s method with hyperbolic extrapolation.

A variation on the classic Brent routine to find a root of the function f between the arguments a and b that uses hyperbolic extrapolation instead of inverse quadratic extrapolation. Bus & Dekker (1975) guarantee convergence for this method, claiming that the upper bound of function evaluations here is 4 or 5 times that of bisection. f(a) and f(b) cannot have the same signs. Generally, on a par with the brent routine, but not as heavily tested. It is a safe version of the secant method that uses hyperbolic extrapolation. The version here is by Chuck Harris, and implements Algorithm M of [BusAndDekker1975], where further details (convergence properties, additional remarks and such) can be found

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. As with brentq, for nice functions the method will often satisfy the above condition with xtol/2 and rtol/2.
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. As with brentq, for nice functions the method will often satisfy the above condition with xtol/2 and rtol/2.
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

fmin, fmin_powell, fmin_cg, fmin_bfgs, fmin_ncg: multivariate local optimizers
leastsq: nonlinear least squares minimizer
fmin_l_bfgs_b, fmin_tnc, fmin_cobyla: constrained multivariate optimizers
basinhopping, differential_evolution, brute: global optimizers
fminbound, brent, golden, bracket: local scalar minimizers
fsolve: N-D root-finding
brentq, ridder, bisect, newton: 1-D root-finding
fixed_point: scalar fixed-point finder
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 brenth 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

[BusAndDekker1975]

Bus, J. C. P., Dekker, T. J., “Two Efficient Algorithms with Guaranteed Convergence for Finding a Zero of a Function”, ACM Transactions on Mathematical Software, Vol. 1, Issue 4, Dec. 1975, pp. 330-345. Section 3: “Algorithm M”. DOI:10.1145/355656.355659

Examples

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

>>> from scipy import optimize

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

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