Module rgsl::types::series_acceleration [−][src]
Expand description
Series Acceleration
The functions described in this chapter accelerate the convergence of a series using the Levin u-transform. This method takes a small number of terms from the start of a series and uses a systematic approximation to compute an extrapolated value and an estimate of its error. The u-transform works for both convergent and divergent series, including asymptotic series.
Acceleration functions
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.
Acceleration functions without error estimation
The functions described in this section compute the Levin u-transform of series and attempt 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.
References and Further Reading
The algorithms used by these functions are described in the following papers,
T. Fessler, W.F. Ford, D.A. Smith, HURRY: An acceleration algorithm for scalar sequences and series ACM Transactions on Mathematical Software, 9(3):346–354, 1983. and Algorithm 602 9(3):355–357, 1983.
The theory of the u-transform was presented by Levin,
D. Levin, Development of Non-Linear Transformations for Improving Convergence of Sequences, Intern. J. Computer Math. B3:371–388, 1973.
A review paper on the Levin Transform is available online,
Herbert H. H. Homeier, Scalar Levin-Type Sequence Transformations, http://arxiv.org/abs/math/0005209.
Structs
The following functions perform the same calculation without estimating the errors. They require
O(N) storage instead of O(N^2). This may be useful for summing many similar series where the
size of the error has already been estimated reliably and is not expected to change.
Workspace for Levin U Transform with error estimation.