5.1.3.3. numdifftools.extrapolation.dea3¶

dea3(v_0, v_1, v_2, symmetric=False)[source]¶

Extrapolate a slowly convergent sequence using Shanks transformations.

Parameters

v_0, v_1, v_2array-like: 3 values of a convergent sequence to extrapolate

Returns

resultarray-like: extrapolated value
abserrarray-like: absolute error estimate

See also

Dea

Notes

DEA3 attempts to extrapolate nonlinearly by Shanks transformations to a better estimate of the sequence’s limiting value based on only three values. The epsilon algorithm of P. Wynn, see [Rc8bfc08f7c28-1], is used to perform the non-linear Shanks transformations. The routine is a vectorized translation of the DQELG function found in the QUADPACK fortran library for LIMEXP=3, see [Rc8bfc08f7c28-2] and [Rc8bfc08f7c28-3].

References

1: Wynn, P. (1956) “On a Device for Computing the em(Sn) Transformation”, Mathematical Tables and Other Aids to Computation, 10, 91-96.
2: R. Piessens, E. De Doncker-Kapenga and C. W. Uberhuber (1983), “QUADPACK: a subroutine package for automatic integration”, Springer, ISBN: 3-540-12553-1, 1983.
3: http://www.netlib.org/quadpack/
4: https://mathworld.wolfram.com/WynnsEpsilonMethod.html

Examples

# integrate sin(x) from 0 to pi/2

>>> import numpy as np
>>> import numdifftools as nd
>>> Ei= np.zeros(3)
>>> linfun = lambda i : np.linspace(0, np.pi/2., 2**(i+5)+1)
>>> for k in np.arange(3):
...    x = linfun(k)
...    Ei[k] = np.trapz(np.sin(x),x)
>>> [En, err] = nd.dea3(Ei[0], Ei[1], Ei[2])
>>> truErr = np.abs(En-1.)
>>> np.all(truErr < err)
True
>>> np.allclose(En, 1)
True
>>> np.all(np.abs(Ei-1)<1e-3)
True