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Shanks' convergence acceleration transform, Padé approximants and partitions

George E Andrews, Ian P Goulden, David M Jackson
1986 Journal of combinatorial theory. Series A  
Shanks developed a method for accelerating the convergence of sequences. When applied to classical sequences in number theory, Shanks' transform yields some famous identities of Euler and Gauss.  ...  It is shown here that the Pad&. approximants for the little q-Jacobi polynomials can be used to explain and extend Shanks' observations.  ...  We rely on the close relationship of Shanks' transform to Pade approximants [lo; p. 211, and reveal that the little q-Jacobi polynomials and the PadC approximants related to their moment generating function  ... 
doi:10.1016/0097-3165(86)90024-5 fatcat:xhvkyfscgnccpclo3rgslzjnyu

Page 1526 of Mathematical Reviews Vol. , Issue 88c [page]

1988 Mathematical Reviews  
Molchanov (Kiev) 88c:65005 65B99 11N99 11Y35 Andrews, George E. (1-PAS); Goulden, Ian P. (3-WTRL); Jackson, David M. (3-WTRL) Shanksconvergence acceleration transform, Padé approximants and partitions  ...  The authors use the close relationship of the Shanks transform to Padé approximants and reveal that the lit- tle g-Jacobi polynomials [see W. C. Hahn, Math.  ... 

Extrapolation methods for Sommerfeld integral tails

K.A. Michalski
1998 IEEE Transactions on Antennas and Propagation  
Attention is limited to partition-extrapolation procedures in which the Sommerfeld integral is evaluated as a sum of a series of partial integrals over finite subintervals and is accelerated by an extrapolation  ...  A review is presented of the extrapolation methods for accelerating the convergence of Sommerfeld-type integrals (i.e., semi-infinite range integrals with Bessel function kernels), which arise in problems  ...  Weniger of the Universität Regensburg and S. Lucas of the University of Southern Australia for their helpful communications and A.  ... 
doi:10.1109/8.725271 fatcat:7xjfk3fr4zbrfdhzeixhffjhlq

The genesis and early developments of Aitken's process, Shanks' transformation, the ε–algorithm, and related fixed point methods

Claude Brezinski, Michela Redivo–Zaglia
2018 Numerical Algorithms  
In this paper, we trace back the genesis of Aitken's Δ 2 process and Shanks' sequence transformation.  ...  They had, and still have, many important applications in numerical analysis and in applied mathematics. They are related to continued fractions and Padé approximants.  ...  Anderson for sending us a long commentary about the discovery of his acceleration method, and, moreover, to carefully reading our paper and suggesting many important improvements.  ... 
doi:10.1007/s11075-018-0567-2 fatcat:ax3bhxndajbmbhhvpe7ogde7vm

Perturbation Theory in the Complex Plane: Exceptional Points and Where to Find Them

Antoine Marie, Hugh G A Burton, Pierre-Francois Loos
2021 Journal of Physics: Condensed Matter  
We also discuss several resummation techniques (such as Padé and quadratic approximants) that can improve the overall accuracy of the Møller-Plesset perturbative series in both convergent and divergent  ...  In particular, we highlight seminal work on the convergence behaviour of perturbative series obtained within Møller-Plesset perturbation theory, and its links with quantum phase transitions.  ...  Acknowledgments This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 863481).  ... 
doi:10.1088/1361-648x/abe795 fatcat:jssjxpkdsjfg3fo5525ronv75m

Perturbation Theory in the Complex Plane: Exceptional Points and Where to Find Them [article]

Antoine Marie and Hugh G. A. Burton and Pierre-François Loos
2021 arXiv   pre-print
We also discuss several resummation techniques (such as Padé and quadratic approximants) that can improve the overall accuracy of the Møller–Plesset perturbative series in both convergent and divergent  ...  In particular, we highlight seminal work on the convergence behaviour of perturbative series obtained within Møller–Plesset perturbation theory, and its links with quantum phase transitions.  ...  ACKNOWLEDGMENTS This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 863481).  ... 
arXiv:2012.03688v2 fatcat:lwp7pld7fjfuza4nwhi66ya5qe

Nonlinear Sequence Transformations: Computational Tools for the Acceleration of Convergence and the Summation of Divergent Series [article]

Ernst Joachim Weniger (Institute for Physical and Theoretical Chemistry, University of Regensburg, Regensburg, Germany)
2001 arXiv   pre-print
be on alternative sequence transformations which for some problems provide better results than Padé approximants.  ...  Padé approximants, which convert the partial sums of a power series to a doubly indexed sequence of rational functions, are the best known sequence transformations, but the emphasis of the review will  ...  Acknowledgments I would like to thank Professor Dzevad Belkić for his invitation to contribute to this topical issue of Journal of Computational Methods in Sciences and Engineering.  ... 
arXiv:math/0107080v1 fatcat:rsge4qizzvea7etakrbqq6kqje

Construction of New Generalizations of Wynn's Epsilon and Rho Algorithm by Solving Finite Difference Equations in the Transformation Order [article]

Xiang-Ke Chang, Yi He, Xing-Biao Hu, Jian-Qing Sun, Ernst Joachim Weniger
2019 arXiv   pre-print
We construct new sequence transformations based on Wynn's epsilon and rho algorithms.  ...  We demonstrate the performance of our algorithms numerically by applying them to some linearly and logarithmically convergent sequences as well as some divergent series.  ...  and thus also of Padé approximants.  ... 
arXiv:1903.10198v1 fatcat:7ya3jg2rvrgazgez2gifc5tite

Page 799 of Mathematical Reviews Vol. , Issue 87b [page]

1987 Mathematical Reviews  
More complete results and the application of the Shanks transformation (¢-algorithm) have been studied by the reviewer and A. Lembarki [Appl. Numer. Math. 2 (1986), no. 1, 1-8).  ...  The approximating function r is no longer restricted to be rational as is the case with Padé approximants.  ... 

Extrapolation Methods for Improving the Convergence of Oligomer Calculations to the Infinite Chain Limit of Quasi-Onedimensional Stereoregular Polymers [article]

Ernst Joachim Weniger (Institute for Physical and Theoretical Chemistry, University of Regensburg, Regensburg, Germany), Bernard Kirtman (Department of Chemistry and Biochemistry, University of California, Santa Barbara, U.S.A.)
2000 arXiv   pre-print
The other method is essentially a quantum chemical method since it approximates the polymer by oligomers consisting of a finite number of monomer units.  ...  Several different extrapolation methods are discussed which are able to accomplish an extrapolation of energies and properties of oligomers to the infinite chain limit.  ...  Shanks introduced a sequence transformation which in the case of a power series produces Padé approximants [62] , and Wynn showed that this transformation can be computed conveniently by a nonlinear recursive  ... 
arXiv:math/0004115v2 fatcat:6a76q46s5ba6dhl6shflknxh2m

Indices

1984 Mathematics of Computation  
Gilewicz, J., Editor 2 45-XX Integral Equations 45D05 Vblterra integral Equations First French-Polish Meeting on Padé Approximation and Conver- gence Acceleration Techniques Lakshmikantham  ...  and Systems With Randomness Lakshmikantham, V., Editor 12 Trends in Theory and Practice of Nonlinear Differential Equations 41-XX Approximations and Expansions 41A21 Padé Approximation  ... 
doi:10.1090/s0025-5718-84-99775-8 fatcat:tqlxvxvze5ez7d663xxgl2zewq

The Degenerate Form of the Adomian Polynomials in the Power Series Method for Nonlinear Ordinary Differential Equations

Jun-Sheng Duan, Randolph Rach
2015 Journal of Mathematics and System Science  
Convergence acceleration techniques including the diagonal Padé approximants are considered, and new numeric algorithms for the multistage decomposition are deduced using the degenerate Adomian polynomials  ...  Several expository examples are investigated to demonstrate its reliability and efficiency.  ...  Acknowledgements This work was supported by the Natural Science Foundation of Shanghai (No.14ZR1440800) and the Innovation Program of the Shanghai Municipal  ... 
doi:10.17265/2159-5291/2015.10.003 fatcat:4bfb2qnklbhnrpbwxn5hoatojm

Indices

1974 Mathematics of Computation  
Padé Approximants and Their Applications 859 Orthogonal Polynomials for Engineers and Physicists Padé Approximants and Their Applications GRAVES-MORRIS, P.  ...  Z. & WAHBA, Convergence Rates of Approximate Least Squares Solu- GRACE tions of Linear Integral and Operator Equations of the First Kind . 69 NEILD, CAROL & SHANKS, On the 3-Rank of Quadratic  ... 
doi:10.1090/s0025-5718-74-99681-1 fatcat:rvypdtbgd5d2xjjvelgwva7a5y

Non-perturbative Quantum Propagators in Bounded Spaces [article]

James P. Edwards and Víctor A. González-Domínguez and Idrish Huet and María Anabel Trejo
2021 arXiv   pre-print
of the propagator studied in previous work (the so-called "hit function"), and a convergent sequence of Padé approximants.  ...  We outline a new approach to calculating the quantum mechanical propagator in the presence of geometrically non-trivial Dirichlet boundary conditions based upon a generalisation of an integral transform  ...  The authors thank Christian Schubert and Axel Weber for helpful discussions and Fiorenzo Bastianelli and Petr Jizba for comments on the manuscript and various suggestions.  ... 
arXiv:2110.04969v1 fatcat:wwhsrdsynjbp5j4y7jlapamy7u

Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series [article]

Ernst Joachim Weniger
2003 arXiv   pre-print
The theoretical properties of the sequence transformations in convergence acceleration and summation processes are analyzed.  ...  In this report, a large number of mainly nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series are discussed.  ...  Special thanks to Professor J.Čížek who aroused my interest in asymptotic techniques, the summation of divergent series, and the construction of rational approximants. Many thanks also to H.  ... 
arXiv:math/0306302v1 fatcat:5k6k6yfafbcd7o4fmukevdjg4q
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