scipy.optimize.brentq¶
- scipy.optimize.brentq(f, a, b, args=(), xtol=1e-12, rtol=4.4408920985006262e-16, maxiter=100, full_output=False, disp=True)[source]¶
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-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. 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.
rtol : number, optional
The routine converges when a root is known to lie within rtol times the value returned of the value returned. Should be >= 0. Defaults to np.finfo(float).eps * 2.
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
- brentq, 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) 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.”