Convergence acceleration of alternating series

We propose a new simple convergence acceleration method for a wide range class of convergent alternating series. It has some common features with Smith’s and Ford’s modification of Levin’s and Weniger’s sequence transformations, but its computational and memory cost is lower. We compare all three methods and give some common theoretical results. Numerical examples confirm a similar performance of all of them.

[1]  Herbert H. H. Homeier,et al.  On remainder estimates for Levin-type sequence transformations , 1995 .

[2]  William H. Press,et al.  Numerical Recipes 3rd Edition: The Art of Scientific Computing , 2007 .

[3]  David Levin,et al.  Development of non-linear transformations for improving convergence of sequences , 1972 .

[4]  Avram Sidi,et al.  A new method for deriving Padé approximants for some hypergeometric functions , 1981 .

[5]  David Levin,et al.  Two New Classes of Nonlinear Transformations for Accelerating the Convergence of Infinite Integrals and Series , 1981 .

[6]  Avram Sidi,et al.  Analysis of convergence of the $T$-transformation for power series , 1980 .

[7]  Avram Sidi,et al.  Acceleration of convergence of some infinite sequences {An} whose asymptotic expansions involve fractional powers of n via the ${\tilde {d}}^{(m)}$ transformation , 2020, Numer. Algorithms.

[8]  David Levin,et al.  On the choice of β in the u - transformation for convergence acceleration , 2014, Numerical Algorithms.

[9]  Avram Sidi,et al.  Practical Extrapolation Methods - Theory and Applications , 2003, Cambridge monographs on applied and computational mathematics.

[10]  Avram Sidi,et al.  Convergence properties of some nonlinear sequence transformations , 1979 .

[11]  Ernst Joachim Weniger,et al.  Interpolation between sequence transformations , 1992, Numerical Algorithms.

[12]  David A. Smith,et al.  Acceleration of linear and logarithmic convergence , 1979 .

[13]  Ernst Joachim Weniger,et al.  Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series , 1989 .

[14]  Claude Brezinski,et al.  Extrapolation methods - theory and practice , 1993, Studies in computational mathematics.

[15]  A. C. Aitken.  XXV.—On Bernoulli's Numerical Solution of Algebraic Equations , 1927 .