;; Series acceleration. ;; Liam Healy, Wed Nov 21 2007 - 18:41 ;; Time-stamp: <2008-03-09 19:21:47EDT series-acceleration.lisp> ;; \$Id\$ (in-package :gsl) ;;;;**************************************************************************** ;;;; Creation and calculation of Levin series acceleration ;;;;**************************************************************************** (cffi:defcstruct levin "The definition of Levin series acceleration for GSL." (size size) (position-in-array size) (terms-used size) (sum-plain :double) (q-num :pointer) (q-den :pointer) (dq-num :pointer) (dq-den :pointer) (dsum :pointer)) (defgo-s (levin order) allocate-levin free-levin) (defmfun allocate-levin (order) "gsl_sum_levin_u_alloc" ((order size)) :c-return :pointer :export nil :index (letm levin) :documentation ; FDL "Allocate a workspace for a Levin u-transform of n terms. The size of the workspace is O(2n^2 + 3n).") (defmfun free-levin (levin) "gsl_sum_levin_u_free" ((levin :pointer)) :c-return :void ; Error in GSL manual, should be void? :export nil :index (letm levin) :documentation ; FDL "Free a previously allocated Levin acceleration.") (defmfun accelerate (array levin) "gsl_sum_levin_u_accel" (((gsl-array array) :pointer) ((dim0 array) size) (levin :pointer) (accelerated-sum :double) (absolute-error :double)) :documentation ; FDL "From the terms of a series in array, compute the extrapolated limit of the series using a Levin u-transform. Additional working space must be provided in levin. The extrapolated sum is returned with an estimate of the absolute error. 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.") ;;;;**************************************************************************** ;;;; Acceleration with error estimation from truncation ;;;;**************************************************************************** (defgo-s (levin-truncated order) allocate-levin-truncated free-levin-truncated) (defmfun allocate-levin-truncated (order) "gsl_sum_levin_utrunc_alloc" ((order size)) :c-return :pointer :export nil :index (letm levin-truncated) :documentation ; FDL "Allocate a workspace for a Levin u-transform of n terms, without error estimation. The size of the workspace is O(3n).") (defmfun free-levin-truncated (levin) "gsl_sum_levin_utrunc_free" ((levin :pointer)) :export nil :index (letm levin-truncated) :documentation ; FDL "Free a previously allocated Levin acceleration.") (defmfun accelerate-truncated (array levin) "gsl_sum_levin_utrunc_accel" (((gsl-array array) :pointer) ((dim0 array) size) (levin :pointer) (accelerated-sum :double) (absolute-error :double)) :documentation ; FDL "From the terms of a series in array, compute the extrapolated limit of the series using a Levin u-transform. Additional working space must be provided in levin. The extrapolated sum is returned with an estimate of the absolute error. The actual term-by-term sum is returned in 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 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.") ;;;;**************************************************************************** ;;;; Example ;;;;**************************************************************************** ;;; From Sec. 29.3 in the GSL manual. (defun acceleration-example () (let ((maxterms 20) (sum 0.0d0) (zeta2 (/ (expt pi 2) 6))) (letm ((levin (levin maxterms)) (array (vector-double-float maxterms))) (dotimes (n maxterms) (setf (maref array n) (coerce (/ (expt (1+ n) 2)) 'double-float)) (incf sum (maref array n))) (multiple-value-bind (accelerated-sum error) (accelerate array levin) (format t "~&term-by-term sum =~f using ~d terms" sum maxterms) (format t "~&term-by-term sum =~f using ~d terms" (cffi:foreign-slot-value levin 'levin 'sum-plain) (cffi:foreign-slot-value levin 'levin 'terms-used)) (format t "~&exact value = ~f" zeta2) (format t "~&accelerated sum = ~f using ~d terms" accelerated-sum (cffi:foreign-slot-value levin 'levin 'terms-used)) (format t "~&estimated error = ~f" error) (format t "~&actual error = ~f" (- accelerated-sum zeta2)))))) ;; term-by-term sum =1.5961632439130233 using 20 terms ;; term-by-term sum =1.5759958390005426 using 13 terms ;; exact value = 1.6449340668482264 ;; accelerated sum = 1.6449340669228176 using 13 terms ;; estimated error = 0.00000000008883604962761638 ;; actual error = 0.00000000007459122208786084