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scipy.optimize.

# brentq

scipy.optimize.brentq(f, a, b, args=(), xtol=2e-12, rtol=np.float64(8.881784197001252e-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 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:
    f function

      Python function returning a number. The function \(f\) must be continuous, and \(f(a)\) and \(f(b)\) must have opposite signs.

    a scalar

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

    b scalar

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

    args tuple, optional

      Containing extra arguments for the function f. f is called by `apply(f, (x)+args)`.

    xtol number, optional

      The computed root `x0` will satisfy `np.isclose(x, x0, atol=xtol, rtol=rtol)`, where `x` is the exact root. The parameter must be positive. 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.isclose(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.

    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:
    root float

      Root 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

  `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

  `elementwise.find_root`

    efficient elementwise 1-D root-finder

  Notes

  f must be continuous. f(a) and f(b) must have opposite signs.

  As mentioned in the parameter documentation, the computed root `x0` will satisfy `np.isclose(x, x0, atol=xtol, rtol=rtol)`, where `x` is the exact root. In equation form, this terminating condition is `abs(x - x0) <= xtol + rtol * abs(x0)`.

  The default value `xtol=2e-12` may lead to surprising behavior if one expects `brentq` to 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

  [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