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scipy.optimize.

# brenth

scipy.optimize.brenth(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 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 `x0` will satisfy `np.isclose(x, x0, atol=xtol, rtol=rtol)`, where `x` is the exact root. The parameter must be positive. As with `brentq`, for nice functions the method will often satisfy the above condition with `xtol/2` and `rtol/2`.

    rtolnumber, 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`. As with `brentq`, for nice functions the method will often satisfy the above condition with `xtol/2` and `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 `RootResults` object.

    dispbool, optional

      If True, raise RuntimeError if the algorithm didn’t converge. Otherwise, the convergence status is recorded in any `RootResults` return object.

  Returns:
    rootfloat

      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

  `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 `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 `brenth` 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

  [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