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scipy.optimize.

# toms748

scipy.optimize.toms748(f, a, b, args=(), k=1, xtol=2e-12, rtol=np.float64(8.881784197001252e-16), maxiter=100, full_output=False, disp=True)

  Find a root using TOMS Algorithm 748 method.

  Implements the Algorithm 748 method of Alefeld, Potro and Shi to find a root of the function f on the interval `[a , b]`, where f(a) and f(b) must have opposite signs.

  It uses a mixture of inverse cubic interpolation and “Newton-quadratic” steps. [APS1995].

  Parameters:
    f function

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

    a scalar,

      lower boundary of the search interval

    b scalar,

      upper boundary of the search interval

    args tuple, optional

      containing extra arguments for the function f. f is called by `f(x, *args)`.

    k int, optional

      The number of Newton quadratic steps to perform each iteration. `k>=1`.

    xtol scalar, 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.

    rtol scalar, optional

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

    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 the `RootResults` return object.

  Returns:
    root float

      Approximate root of f

    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

  `brentq`, `brenth`, `ridder`, `bisect`, `newton`
  `fsolve`

    find roots in N dimensions.

  `elementwise.find_root`

    efficient elementwise 1-D root-finder

  Notes

  f must be continuous. Algorithm 748 with `k=2` is asymptotically the most efficient algorithm known for finding roots of a four times continuously differentiable function. In contrast with Brent’s algorithm, which may only decrease the length of the enclosing bracket on the last step, Algorithm 748 decreases it each iteration with the same asymptotic efficiency as it finds the root.

  For easy statement of efficiency indices, assume that f has 4 continuous deriviatives. For `k=1`, the convergence order is at least 2.7, and with about asymptotically 2 function evaluations per iteration, the efficiency index is approximately 1.65. For `k=2`, the order is about 4.6 with asymptotically 3 function evaluations per iteration, and the efficiency index 1.66. For higher values of k, the efficiency index approaches the kth root of `(3k-2)`, hence `k=1` or `k=2` are usually appropriate.

  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 `toms748` 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

  [APS1995]

  Alefeld, G. E. and Potra, F. A. and Shi, Yixun, Algorithm 748: Enclosing Zeros of Continuous Functions, ACM Trans. Math. Softw. Volume 221(1995) doi = {10.1145/210089.210111}

  Examples

      >>> def f(x):
      ...     return (x**3 - 1)  # only one real root at x = 1

      >>> from scipy import optimize
      >>> root, results = optimize.toms748(f, 0, 2, full_output=True)
      >>> root
      1.0
      >>> results
            converged: True
                 flag: converged
       function_calls: 11
           iterations: 5
                 root: 1.0
               method: toms748