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 - x0will satisfy- np.allclose(x, x0, atol=xtol, rtol=rtol), where- xis the exact root. The parameter must be positive. For nice functions, Brent’s method will often satisfy the above condition with- xtol/2and- rtol/2. [Brent1973]
- rtolnumber, optional
- The computed root - x0will satisfy- np.allclose(x, x0, atol=xtol, rtol=rtol), where- xis 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/2and- 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- RootResultsobject.
- dispbool, optional
- If True, raise RuntimeError if the algorithm didn’t converge. Otherwise, the convergence status is recorded in any - RootResultsreturn object.
 
- Returns:
- rootfloat
- Root of f between a and b. 
- rRootResults(present iffull_output = True)
- Object containing information about the convergence. In particular, - r.convergedis True if the routine converged.
 
 - See also - fmin,- fmin_powell,- fmin_cg,- fmin_bfgs,- fmin_ncg
- multivariate local optimizers 
- leastsq
- nonlinear least squares minimizer 
- fmin_l_bfgs_b,- fmin_tnc,- fmin_cobyla
- constrained multivariate optimizers 
- basinhopping,- differential_evolution,- brute
- global optimizers 
- fminbound,- brent,- golden,- bracket
- local scalar minimizers 
- fsolve
- N-D root-finding 
- brenth,- ridder,- bisect,- newton
- 1-D root-finding 
- fixed_point
- scalar fixed-point finder 
 - Notes - f must be continuous. f(a) and f(b) must have opposite signs. - 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