scipy.optimize.brentq¶

scipy.optimize.
brentq
(f, a, b, args=(), xtol=2e12, rtol=8.881784197001252e16, 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 WijngaardenDekkerBrent 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 satisfynp.allclose(x, x0, atol=xtol, rtol=rtol)
, wherex
is the exact root. The parameter must be nonnegative. For nice functions, Brent’s method will often satisfy the above condition withxtol/2
andrtol/2
. [Brent1973] rtolnumber, optional
The computed root
x0
will satisfynp.allclose(x, x0, atol=xtol, rtol=rtol)
, wherex
is the exact root. The parameter cannot be smaller than its default value of4*np.finfo(float).eps
. For nice functions, Brent’s method will often satisfy the above condition withxtol/2
andrtol/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 aRootResults
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
 x0float
Zero of f between a and b.
 r
RootResults
(present iffull_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
 nonlinear least squares minimizer
 constrained multivariate optimizers
 global optimizers
 local scalar minimizers
 ND rootfinding
 1D rootfinding
 scalar fixedpoint finder
References
 Brent1973(1,2,3)
Brent, R. P., Algorithms for Minimization Without Derivatives. Englewood Cliffs, NJ: PrenticeHall, 1973. Ch. 34.
 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. 352355, 1992. Section 9.3: “Van WijngaardenDekkerBrent 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