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 @cindex acceleration of series
 @cindex summation, acceleration
 @cindex series, acceleration
 @cindex u-transform for series
 @cindex Levin u-transform
 @cindex convergence, accelerating a series

 The functions described in this chapter accelerate the convergence of a
 series using the Levin @math{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
 @math{u}-transform works for both convergent and divergent series, including
 asymptotic series.

 These functions are declared in the header file @file{gsl_sum.h}.

 @menu
 * Acceleration functions::
 * Acceleration functions without error estimation::
 * Example of accelerating a series::
 * Series Acceleration References::
 @end menu

 @node Acceleration functions
 @section Acceleration functions

 The following functions compute the full Levin @math{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 @code{GSL_DBL_EPSILON} for each term.

 The calculation of the error in the extrapolated value is an
 @math{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 @math{O(N)} must be computed and stored, but this
 does give a reliable error estimate.

 @deftypefun {gsl_sum_levin_u_workspace *} gsl_sum_levin_u_alloc (size_t @var{n})
 This function allocates a workspace for a Levin @math{u}-transform of @var{n}
 terms.  The size of the workspace is @math{O(2n^2 + 3n)}.
 @end deftypefun

 @deftypefun void gsl_sum_levin_u_free (gsl_sum_levin_u_workspace * @var{w})
 This function frees the memory associated with the workspace @var{w}.
 @end deftypefun

 @deftypefun int gsl_sum_levin_u_accel (const double * @var{array}, size_t @var{array_size}, gsl_sum_levin_u_workspace * @var{w}, double * @var{sum_accel}, double * @var{abserr})
 This function takes the terms of a series in @var{array} of size
 @var{array_size} and computes the extrapolated limit of the series using
 a Levin @math{u}-transform.  Additional working space must be provided in
 @var{w}.  The extrapolated sum is stored in @var{sum_accel}, with an
 estimate of the absolute error stored in @var{abserr}.  The actual
 term-by-term sum is returned in @code{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 @var{array} should be non-zero.
 @end deftypefun


 @node Acceleration functions without error estimation
 @section Acceleration functions without error estimation

 The functions described in this section compute the Levin @math{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 @math{O(N)}
 process and only requires @math{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.

 @deftypefun {gsl_sum_levin_utrunc_workspace *} gsl_sum_levin_utrunc_alloc (size_t @var{n})
 This function allocates a workspace for a Levin @math{u}-transform of @var{n}
 terms, without error estimation.  The size of the workspace is
 @math{O(3n)}.
 @end deftypefun

 @deftypefun void gsl_sum_levin_utrunc_free (gsl_sum_levin_utrunc_workspace * @var{w})
 This function frees the memory associated with the workspace @var{w}.
 @end deftypefun

 @deftypefun int gsl_sum_levin_utrunc_accel (const double * @var{array}, size_t @var{array_size}, gsl_sum_levin_utrunc_workspace * @var{w}, double * @var{sum_accel}, double * @var{abserr_trunc})
 This function takes the terms of a series in @var{array} of size
 @var{array_size} and computes the extrapolated limit of the series using
 a Levin @math{u}-transform.  Additional working space must be provided in
 @var{w}.  The extrapolated sum is stored in @var{sum_accel}.  The actual
 term-by-term sum is returned in @code{w->sum_plain}. The algorithm
 terminates when the difference between two successive extrapolations
 reaches a minimum or is sufficiently small. The difference between these
 two values is used as estimate of the error and is stored in
 @var{abserr_trunc}.  To improve the reliability of the algorithm the
 extrapolated values are replaced by moving averages when calculating the
 truncation error, smoothing out any fluctuations.
 @end deftypefun


 @node Example of accelerating a series
 @section Examples

 The following code calculates an estimate of @math{\zeta(2) = \pi^2 / 6}
 using the series,
 @tex
 \beforedisplay
 $$
 \zeta(2) = 1 + 1/2^2 + 1/3^2 + 1/4^2 + \dots
 $$
 \afterdisplay
 @end tex
 @ifinfo

 @example
 \zeta(2) = 1 + 1/2^2 + 1/3^2 + 1/4^2 + ...
 @end example

 @end ifinfo
 @noindent
 After @var{N} terms the error in the sum is @math{O(1/N)}, making direct
 summation of the series converge slowly.

 @example
 @verbatiminclude examples/sum.c
 @end example

 @noindent
 The output below shows that the Levin @math{u}-transform is able to obtain an
 estimate of the sum to 1 part in
 @c{$10^{10}$}
 @math{10^10} using the first eleven terms of the series.  The
 error estimate returned by the function is also accurate, giving
 the correct number of significant digits.

 @example
 $ ./a.out
 @verbatiminclude examples/sum.out
 @end example

 @noindent
 Note that a direct summation of this series would require
 @c{$10^{10}$}
 @math{10^10} terms to achieve the same precision as the accelerated
 sum does in 13 terms.

 @node Series Acceleration References
 @section References and Further Reading

 The algorithms used by these functions are described in the following papers,

 @itemize @asis
 @item
 T. Fessler, W.F. Ford, D.A. Smith,
 @sc{hurry}: An acceleration algorithm for scalar sequences and series
 @cite{ACM Transactions on Mathematical Software}, 9(3):346--354, 1983.
 and Algorithm 602 9(3):355--357, 1983.
 @end itemize

 @noindent
 The theory of the @math{u}-transform was presented by Levin,

 @itemize @asis
 @item
 D. Levin,
 Development of Non-Linear Transformations for Improving Convergence of
 Sequences, @cite{Intern.@: J.@: Computer Math.} B3:371--388, 1973.
 @end itemize

 @noindent
 A review paper on the Levin Transform is available online,
 @itemize @asis
 @item
 Herbert H. H. Homeier, Scalar Levin-Type Sequence Transformations,
 @uref{http://arxiv.org/abs/math/0005209}.
 @end itemize
	@cindex acceleration of series
	@cindex summation, acceleration
	@cindex series, acceleration
	@cindex u-transform for series
	@cindex Levin u-transform
	@cindex convergence, accelerating a series

	The functions described in this chapter accelerate the convergence of a
	series using the Levin @math{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
	@math{u}-transform works for both convergent and divergent series, including
	asymptotic series.

	These functions are declared in the header file @file{gsl_sum.h}.

	@menu
	* Acceleration functions::
	* Acceleration functions without error estimation::
	* Example of accelerating a series::
	* Series Acceleration References::
	@end menu

	@node Acceleration functions
	@section Acceleration functions

	The following functions compute the full Levin @math{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 @code{GSL_DBL_EPSILON} for each term.

	The calculation of the error in the extrapolated value is an
	@math{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 @math{O(N)} must be computed and stored, but this
	does give a reliable error estimate.

	@deftypefun {gsl_sum_levin_u_workspace *} gsl_sum_levin_u_alloc (size_t @var{n})
	This function allocates a workspace for a Levin @math{u}-transform of @var{n}
	terms. The size of the workspace is @math{O(2n^2 + 3n)}.
	@end deftypefun

	@deftypefun void gsl_sum_levin_u_free (gsl_sum_levin_u_workspace * @var{w})
	This function frees the memory associated with the workspace @var{w}.
	@end deftypefun

	@deftypefun int gsl_sum_levin_u_accel (const double * @var{array}, size_t @var{array_size}, gsl_sum_levin_u_workspace * @var{w}, double * @var{sum_accel}, double * @var{abserr})
	This function takes the terms of a series in @var{array} of size
	@var{array_size} and computes the extrapolated limit of the series using
	a Levin @math{u}-transform. Additional working space must be provided in
	@var{w}. The extrapolated sum is stored in @var{sum_accel}, with an
	estimate of the absolute error stored in @var{abserr}. The actual
	term-by-term sum is returned in @code{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 @var{array} should be non-zero.
	@end deftypefun


	@node Acceleration functions without error estimation
	@section Acceleration functions without error estimation

	The functions described in this section compute the Levin @math{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 @math{O(N)}
	process and only requires @math{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.

	@deftypefun {gsl_sum_levin_utrunc_workspace *} gsl_sum_levin_utrunc_alloc (size_t @var{n})
	This function allocates a workspace for a Levin @math{u}-transform of @var{n}
	terms, without error estimation. The size of the workspace is
	@math{O(3n)}.
	@end deftypefun

	@deftypefun void gsl_sum_levin_utrunc_free (gsl_sum_levin_utrunc_workspace * @var{w})
	This function frees the memory associated with the workspace @var{w}.
	@end deftypefun

	@deftypefun int gsl_sum_levin_utrunc_accel (const double * @var{array}, size_t @var{array_size}, gsl_sum_levin_utrunc_workspace * @var{w}, double * @var{sum_accel}, double * @var{abserr_trunc})
	This function takes the terms of a series in @var{array} of size
	@var{array_size} and computes the extrapolated limit of the series using
	a Levin @math{u}-transform. Additional working space must be provided in
	@var{w}. The extrapolated sum is stored in @var{sum_accel}. The actual
	term-by-term sum is returned in @code{w->sum_plain}. The algorithm
	terminates when the difference between two successive extrapolations
	reaches a minimum or is sufficiently small. The difference between these
	two values is used as estimate of the error and is stored in
	@var{abserr_trunc}. To improve the reliability of the algorithm the
	extrapolated values are replaced by moving averages when calculating the
	truncation error, smoothing out any fluctuations.
	@end deftypefun


	@node Example of accelerating a series
	@section Examples

	The following code calculates an estimate of @math{\zeta(2) = \pi^2 / 6}
	using the series,
	@tex
	\beforedisplay
	$$
	\zeta(2) = 1 + 1/2^2 + 1/3^2 + 1/4^2 + \dots
	$$
	\afterdisplay
	@end tex
	@ifinfo

	@example
	\zeta(2) = 1 + 1/2^2 + 1/3^2 + 1/4^2 + ...
	@end example

	@end ifinfo
	@noindent
	After @var{N} terms the error in the sum is @math{O(1/N)}, making direct
	summation of the series converge slowly.

	@example
	@verbatiminclude examples/sum.c
	@end example

	@noindent
	The output below shows that the Levin @math{u}-transform is able to obtain an
	estimate of the sum to 1 part in
	@c{$10^{10}$}
	@math{10^10} using the first eleven terms of the series. The
	error estimate returned by the function is also accurate, giving
	the correct number of significant digits.

	@example
	$ ./a.out
	@verbatiminclude examples/sum.out
	@end example

	@noindent
	Note that a direct summation of this series would require
	@c{$10^{10}$}
	@math{10^10} terms to achieve the same precision as the accelerated
	sum does in 13 terms.

	@node Series Acceleration References
	@section References and Further Reading

	The algorithms used by these functions are described in the following papers,

	@itemize @asis
	@item
	T. Fessler, W.F. Ford, D.A. Smith,
	@sc{hurry}: An acceleration algorithm for scalar sequences and series
	@cite{ACM Transactions on Mathematical Software}, 9(3):346--354, 1983.
	and Algorithm 602 9(3):355--357, 1983.
	@end itemize

	@noindent
	The theory of the @math{u}-transform was presented by Levin,

	@itemize @asis
	@item
	D. Levin,
	Development of Non-Linear Transformations for Improving Convergence of
	Sequences, @cite{Intern.@: J.@: Computer Math.} B3:371--388, 1973.
	@end itemize

	@noindent
	A review paper on the Levin Transform is available online,
	@itemize @asis
	@item
	Herbert H. H. Homeier, Scalar Levin-Type Sequence Transformations,
	@uref{http://arxiv.org/abs/math/0005209}.
	@end itemize