# scipy.optimize.brentq¶

scipy.optimize.brentq(f, a, b, args=(), xtol=9.9999999999999998e-13, rtol=4.4408920985006262e-16, maxiter=100, full_output=False, disp=True)

Find a root of a function in given interval.

Return float, a zero of f between a and b. f must be a continuous function, and [a,b] must be a sign changing interval.

Description: Uses the classic Brent (1973) 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-Deker-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. 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 routine converges when a root is known to lie within xtol of the value return. Should be >= 0. The routine modifies this to take into account the relative precision of doubles. maxiter : number, optional if convergence is not achieved in maxiter iterations, and 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 : {True, bool} optional If True, raise RuntimeError if the algorithm didn’t converge. 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.

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

Notes

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

 [Brent1973] Brent, R. P., Algorithms for Minimization Without Derivatives. Englewood Cliffs, NJ: Prentice-Hall, 1973. Ch. 3-4.
 [PressEtal1992] 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.”

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