scipy.optimize.brentq#

scipy.optimize.brentq(f, a, b, args=(), xtol=2e-12, rtol=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.

Uses the classic Brent’s method to find a root 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]. A third 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:

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

Notes

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

Related functions fall into several classes:

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

References

[Brent1973] (1,2,3)

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.”

Examples

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

>>> from scipy import optimize

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

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