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)[source]#

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:

ffunction: Python function returning a scalar. The function \(f\) must be continuous, and \(f(a)\) and \(f(b)\) have opposite signs.
ascalar,: lower boundary of the search interval
bscalar,: upper boundary of the search interval
argstuple, optional: containing extra arguments for the function f. f is called by f(x, *args).
kint, optional: The number of Newton quadratic steps to perform each iteration. k>=1.
xtolscalar, 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.
rtolscalar, optional: The computed root x0 will satisfy np.isclose(x, x0, atol=xtol, rtol=rtol), where x is the exact root.
maxiterint, optional: If convergence is not achieved in maxiter iterations, an error is raised. Must be >= 0.
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 the RootResults return object.

Returns:

rootfloat: Approximate root of f
rRootResults (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