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 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 : scalar
One end of the bracketing interval \([a, b]\).
- b : scalar
The other end of the bracketing interval \([a, b]\).
- xtol : number, 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]- rtol : number, 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]- maxiter : int, 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 aRootResults
object.- disp : bool, optional
If True, raise RuntimeError if the algorithm didn’t converge. Otherwise the convergence status is recorded in any
RootResults
return object.
Returns: - x0 : float
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.
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.” 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