10.1 Function list

levin_sum( a, truncate=False, info_dict=None)
Return ( $ A, \epsilon$ ) where $ A$ is the approximated sum of the series (10.1) and $ \epsilon$ is its absolute error estimate.

The calculation of the error in the extrapolated value is an O$ (N^2)$ process, which is expensive in time and memory. 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.

A faster but less reliable method which estimates the error from the convergence of the extrapolated value is employed if truncate is True. This attempts to estimate the error from the ``truncation error'' in the extrapolation, the difference between the final two approximations. Using this method avoids the need to compute an intermediate table of derivatives because the error is estimated from the behavior of the extrapolated value itself. Consequently this algorithm is an O$ (N)$ process and only requires O$ (N)$ terms of storage. If the series converges sufficiently fast then this procedure can be acceptable. It is appropriate to use this method when there is a need to compute many extrapolations of series with similar convergence properties at high-speed. For example, when numerically integrating a function defined by a parameterized series where the parameter varies only slightly. A reliable error estimate should be computed first using the full algorithm described above in order to verify the consistency of the results.

If a dictionary is passed as info_dict, then two entries will be added: info_dict['terms_used'] will be the number of terms used10.1 and info_dict['sum_plain'] will be the sum of these terms without acceleration.



Footnotes

... used10.1
Note that it appears that this is the number of terms beyond the first term that are used. I.e. there are a total of $ \var{terms_used}+1$ terms:

$\displaystyle \var{sum_plain} = \sum_{n=0}^{\var{terms_used}} a_{n}$ (10.2)