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.allclose(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.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. 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

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