SciPy

scipy.optimize.brentq

scipy.optimize.brentq(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 a bracketing interval using Brent’s method.

Uses the classic Brent’s method to find a zero of the function f on the sign changing interval [a , b]. Generally considered the best of the rootfinding routines here. It is a safe version of the secant method that uses inverse quadratic extrapolation. Brent’s method combines root bracketing, interval bisection, and inverse quadratic interpolation. It is sometimes known as the van Wijngaarden-Dekker-Brent method. Brent (1973) claims convergence is guaranteed for functions computable within [a,b].

[Brent1973] provides the classic description of the algorithm. Another description can be found in a recent edition of Numerical Recipes, including [PressEtal1992]. Another description is at http://mathworld.wolfram.com/BrentsMethod.html. It should be easy to understand the algorithm just by reading our code. Our code diverges a bit from standard presentations: we choose a different formula for the extrapolation step.

Parameters:

f : function

Python function returning a number. The function \(f\) 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. For nice functions, Brent’s method will often satisfy the above condition with xtol/2 and rtol/2. [Brent1973]

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. For nice functions, Brent’s method will often satisfy the above condition with xtol/2 and rtol/2. [Brent1973]

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

multivariate
fmin, fmin_powell, fmin_cg, fmin_bfgs, fmin_ncg
nonlinear
leastsq
constrained
fmin_l_bfgs_b, fmin_tnc, fmin_cobyla
global
basinhopping, brute, differential_evolution
local
fminbound, brent, golden, bracket
n-dimensional
fsolve
one-dimensional
brenth, ridder, bisect, newton
scalar
fixed_point

Notes

f must be continuous. f(a) and f(b) must have opposite signs.

References

[Brent1973](1, 2, 3, 4) Brent, R. P., Algorithms for Minimization Without Derivatives. Englewood Cliffs, NJ: Prentice-Hall, 1973. Ch. 3-4.
[PressEtal1992](1, 2) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 352-355, 1992. Section 9.3: “Van Wijngaarden-Dekker-Brent Method.”