The following functions compute the full Levin u-transform of a series with its error estimate. The error estimate is computed by propagating rounding errors from each term through to the final extrapolation.
These functions are intended for summing analytic series where each term
is known to high accuracy, and the rounding errors are assumed to
originate from finite precision. They are taken to be relative errors of
order GSL_DBL_EPSILON for each term.
The calculation of the error in the extrapolated value is an O(N^2) process, which is expensive in time and memory. A faster but less reliable method which estimates the error from the convergence of the extrapolated value is described in the next section. For the method described here a full table of intermediate values and derivatives through to O(N) must be computed and stored, but this does give a reliable error estimate.
This function allocates a workspace for a Levin u-transform of n terms. The size of the workspace is O(2n^2 + 3n).
This function frees the memory associated with the workspace w.
This function takes the terms of a series in array of size array_size and computes the extrapolated limit of the series using a Levin u-transform. Additional working space must be provided in w. The extrapolated sum is stored in sum_accel, with an estimate of the absolute error stored in abserr. The actual term-by-term sum is returned in
w->sum_plain. The algorithm calculates the truncation error (the difference between two successive extrapolations) and round-off error (propagated from the individual terms) to choose an optimal number of terms for the extrapolation. All the terms of the series passed in through array should be non-zero.