brenth#
- scipy.optimize.brenth(f, a, b, args=(), xtol=2e-12, rtol=np.float64(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 with hyperbolic extrapolation. - A variation on the classic Brent routine to find a root of the function f between the arguments a and b that uses hyperbolic extrapolation instead of inverse quadratic extrapolation. Bus & Dekker (1975) guarantee convergence for this method, claiming that the upper bound of function evaluations here is 4 or 5 times that of bisection. f(a) and f(b) cannot have the same signs. Generally, on a par with the brent routine, but not as heavily tested. It is a safe version of the secant method that uses hyperbolic extrapolation. The version here is by Chuck Harris, and implements Algorithm M of [BusAndDekker1975], where further details (convergence properties, additional remarks and such) can be found - Parameters:
- ffunction
- Python function returning a number. 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.isclose(x, x0, atol=xtol, rtol=rtol), where- xis the exact root. The parameter must be positive. As with- brentq, for nice functions the method will often satisfy the above condition with- xtol/2and- rtol/2.
- rtolnumber, optional
- The computed root - x0will satisfy- np.isclose(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. As with- brentq, for nice functions the method will often satisfy the above condition with- xtol/2and- rtol/2.
- 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 
- brentq,- ridder,- bisect,- newton
- 1-D root-finding 
- fixed_point
- scalar fixed-point finder 
- elementwise.find_root
- efficient elementwise 1-D root-finder 
 - Notes - As mentioned in the parameter documentation, the computed root - x0will satisfy- np.isclose(x, x0, atol=xtol, rtol=rtol), where- xis the exact root. In equation form, this terminating condition is- abs(x - x0) <= xtol + rtol * abs(x0).- The default value - xtol=2e-12may lead to surprising behavior if one expects- brenthto always compute roots with relative error near machine precision. Care should be taken to select xtol for the use case at hand. Setting- xtol=5e-324, the smallest subnormal number, will ensure the highest level of accuracy. Larger values of xtol may be useful for saving function evaluations when a root is at or near zero in applications where the tiny absolute differences available between floating point numbers near zero are not meaningful.- References [BusAndDekker1975]- Bus, J. C. P., Dekker, T. J., “Two Efficient Algorithms with Guaranteed Convergence for Finding a Zero of a Function”, ACM Transactions on Mathematical Software, Vol. 1, Issue 4, Dec. 1975, pp. 330-345. Section 3: “Algorithm M”. DOI:10.1145/355656.355659 - Examples - >>> def f(x): ... return (x**2 - 1) - >>> from scipy import optimize - >>> root = optimize.brenth(f, -2, 0) >>> root -1.0 - >>> root = optimize.brenth(f, 0, 2) >>> root 1.0