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# scipy.optimize.brentq

`scipy.optimize.``brentq`(f, a, b, args=(), xtol=2e-12, rtol=8.8817841970012523e-16, maxiter=100, 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]. 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 : number

  > One end of the bracketing interval \([a, b]\).

  b : number

  > The other end of the bracketing interval \([a, b]\).

  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 will `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 will `xtol/2` and `rtol/2`. [Brent1973]

  maxiter : number, 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.

  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`
    `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.”