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 satisfynp.allclose(x, x0, atol=xtol, rtol=rtol)
, wherex
is the exact root. The parameter must be positive. 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:
- rootfloat
Root 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
- N-D root-finding
- 1-D root-finding
- scalar fixed-point finder
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