def brentq(f, a, b, args=(), xtol=_xtol, rtol=_rtol, maxiter=_iter, full_output=False, disp=True): """ Find a root of a function in given interval. Return float, a zero of `f` between `a` and `b`. `f` must be a continuous function, and [a,b] must be a sign changing interval. Description: Uses the classic Brent (1973) 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-Deker-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. f must be continuous, and f(a) and f(b) must have opposite signs. a : number One end of the bracketing interval [a,b]. b : number The other end of the bracketing interval [a,b]. xtol : number, optional The routine converges when a root is known to lie within xtol of the value return. Should be >= 0. The routine modifies this to take into account the relative precision of doubles. maxiter : number, optional if convergence is not achieved in maxiter iterations, and 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 a RootResults object. disp : bool, optional If True, raise RuntimeError if the algorithm didn't converge. Returns ------- x0 : float Zero of `f` between `a` and `b`. r : RootResults (present if ``full_output = True``) Object containing information about the convergence. In particular, ``r.converged`` is True if the routine converged. See Also -------- multivariate local optimizers `fmin`, `fmin_powell`, `fmin_cg`, `fmin_bfgs`, `fmin_ncg` nonlinear least squares minimizer `leastsq` constrained multivariate optimizers `fmin_l_bfgs_b`, `fmin_tnc`, `fmin_cobyla` global optimizers `anneal`, `brute` local scalar minimizers `fminbound`, `brent`, `golden`, `bracket` n-dimensional root-finding `fsolve` one-dimensional root-finding `brentq`, `brenth`, `ridder`, `bisect`, `newton` scalar fixed-point finder `fixed_point` Notes ----- `f` must be continuous. f(a) and f(b) must have opposite signs. References ---------- .. [Brent1973] 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." """ if type(args) != type(()) : args = (args,) r = _zeros._brentq(f,a,b,xtol,maxiter,args,full_output,disp) return results_c(full_output, r)
def brentq(f, a, b, args=(), xtol=_xtol, rtol=_rtol, maxiter=_iter, full_output=False, disp=False): """Find root of f in [a,b] The classic Brent routine to find a zero of the function f between the arguments a and b. f(a) and f(b) can not have the same signs. Generally the best of the routines here. It is a safe version of the secant method that uses inverse quadratic extrapolation. The version here is a slight modification that uses a different formula in the extrapolation step. A description may be found in Numerical Recipes, but the code here is probably easier to understand. f : Python function returning a number. a : Number, one end of the bracketing interval. b : Number, the other end of the bracketing interval. xtol : Number, the routine converges when a root is known to lie within xtol of the value return. Should be >= 0. The routine modifies this to take into account the relative precision of doubles. maxiter : Number, if convergence is not achieved in maxiter iterations, and error is raised. Must be >= 0. args : tuple containing extra arguments for the function f. f is called by apply(f,(x)+args). 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 RootResults object containing information about the convergence. In particular, r.converged is True if the 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 anneal, brute -- global optimizers fminbound, brent, golden, bracket -- local scalar minimizers fsolve -- n-dimenstional root-finding brentq, brenth, ridder, bisect, newton -- one-dimensional root-finding fixed_point -- scalar fixed-point finder """ if type(args) != type(()) : args = (args,) r = _zeros._brentq(f,a,b,xtol,maxiter,args,full_output,disp) return results_c(full_output, r)
def brentq(f, a, b, args=(), xtol=_xtol, rtol=_rtol, maxiter=_iter, full_output=False, disp=True): """ Find a root of a function in given interval. Return float, a zero of `f` between `a` and `b`. `f` must be a continuous function, and [a,b] must be a sign changing interval. Description: Uses the classic Brent (1973) 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-Deker-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. f must be continuous, and f(a) and f(b) must have opposite signs. a : number One end of the bracketing interval [a,b]. b : number The other end of the bracketing interval [a,b]. xtol : number, optional The routine converges when a root is known to lie within xtol of the value return. Should be >= 0. The routine modifies this to take into account the relative precision of doubles. maxiter : number, optional if convergence is not achieved in maxiter iterations, and 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 a RootResults object. disp : bool, optional If True, raise RuntimeError if the algorithm didn't converge. Returns ------- x0 : float Zero of `f` between `a` and `b`. r : RootResults (present if ``full_output = True``) Object containing information about the convergence. In particular, ``r.converged`` is True if the routine converged. See Also -------- multivariate local optimizers `fmin`, `fmin_powell`, `fmin_cg`, `fmin_bfgs`, `fmin_ncg` nonlinear least squares minimizer `leastsq` constrained multivariate optimizers `fmin_l_bfgs_b`, `fmin_tnc`, `fmin_cobyla` global optimizers `anneal`, `brute` local scalar minimizers `fminbound`, `brent`, `golden`, `bracket` n-dimensional root-finding `fsolve` one-dimensional root-finding `brentq`, `brenth`, `ridder`, `bisect`, `newton` scalar fixed-point finder `fixed_point` Notes ----- `f` must be continuous. f(a) and f(b) must have opposite signs. References ---------- .. [Brent1973] 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." """ if type(args) != type(()): args = (args, ) r = _zeros._brentq(f, a, b, xtol, maxiter, args, full_output, disp) return results_c(full_output, r)
def brentq(f, a, b, args=(), xtol=_xtol, rtol=_rtol, maxiter=_iter, full_output=False, disp=True): """ 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]_. 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 ---------- f : function Python function returning a number. The function :math:`f` must be continuous, and :math:`f(a)` and :math:`f(b)` must have opposite signs. a : scalar One end of the bracketing interval :math:`[a, b]`. b : scalar The other end of the bracketing interval :math:`[a, b]`. xtol : number, optional The computed root ``x0`` will satisfy ``np.allclose(x, x0, atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The parameter must be nonnegative. For nice functions, Brent's method will often satisfy the above condition with ``xtol/2`` and ``rtol/2``. [Brent1973]_ rtol : number, optional The computed root ``x0`` will satisfy ``np.allclose(x, x0, atol=xtol, rtol=rtol)``, where ``x`` is 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/2`` and ``rtol/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 a `RootResults` 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 if ``full_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 `fmin`, `fmin_powell`, `fmin_cg`, `fmin_bfgs`, `fmin_ncg` nonlinear least squares minimizer `leastsq` constrained multivariate optimizers `fmin_l_bfgs_b`, `fmin_tnc`, `fmin_cobyla` global optimizers `basinhopping`, `brute`, `differential_evolution` local scalar minimizers `fminbound`, `brent`, `golden`, `bracket` n-dimensional root-finding `fsolve` one-dimensional root-finding `brenth`, `ridder`, `bisect`, `newton` scalar fixed-point finder `fixed_point` References ---------- .. [Brent1973] 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 """ if not isinstance(args, tuple): args = (args, ) maxiter = operator.index(maxiter) if xtol <= 0: raise ValueError("xtol too small (%g <= 0)" % xtol) if rtol < _rtol: raise ValueError("rtol too small (%g < %g)" % (rtol, _rtol)) r = _zeros._brentq(f, a, b, xtol, rtol, maxiter, args, full_output, disp) return results_c(full_output, r)