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The epsilon algorithm and related topics. (English) Zbl 0970.65004

The authors study the application of the epsilon algorithm with an emphasis on the case of complex-valued sequences. The importance of studying the vector epsilon algorithm lies partly in its potential for application to the acceleration of convergence of iterative solution of discretised partial differential equations.

MSC:

65B05 Extrapolation to the limit, deferred corrections
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
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[1] Baker, G. A., Essentials of Padé Approximants (1975), Academic Press: Academic Press New York · Zbl 0315.41014
[2] Baker, G. A.; Graves-Morris, P. R., Padé approximants. Padé approximants, Encyclopedia of Mathematics and its Applications, 2nd Edition, Vol. 59 (1996), Cambridge University Press: Cambridge University Press New York
[3] Berlekamp, E. R., Algebraic Coding Theory (1968), McGraw-Hill: McGraw-Hill New York · Zbl 0199.54101
[4] Brezinski, C., Etude sur les \(ε\)-et \(ρ\)-algorithmes, Numer. Math., 17, 153-162 (1971) · Zbl 0204.48305
[5] Brezinski, C., Généralisations de la transformation de Shanks, de la table de Padé et de l’\(ε\)-algorithme, Calcolo, 12, 317-360 (1975) · Zbl 0329.65006
[6] Brezinski, C., Accélération de la Convergence en Analyse Numérique. Accélération de la Convergence en Analyse Numérique, Lecture Notes in Mathematics, Vol. 584 (1977), Springer: Springer Berlin · Zbl 0352.65003
[7] Brezinski, C., Convergence acceleration of some sequences by the \(ε\)-algorithm, Numer. Math., 29, 173-177 (1978) · Zbl 0352.65004
[8] Brezinski, C., Padé-Type Approximation and General Orthogonal Polynomials (1980), Birkhäuser: Birkhäuser Basel · Zbl 0418.41012
[9] Brezinski, C.; Redivo-Zaglia, M., Extrapolation Methods, Theory and Practice (1991), North-Holland: North-Holland Amsterdam · Zbl 0744.65004
[10] Chandler-Wilde, S. N.; Hothersall, D., Efficient calculation of the Green function for acoustic propagation above a homogeneous impedance plane, J. Sound Vibr., 180, 705-724 (1995) · Zbl 1237.76166
[11] Chandler-Wilde, S. N.; Rahman, M.; Ross, C. R., A fast, two-grid method for the impedance problem in a half-plane. A fast, two-grid method for the impedance problem in a half-plane, Proceedings of the Fourth International Conference on Mathematical Aspects of Wave Propagation, SIAM (1998), Philadelphia: Philadelphia PA · Zbl 0946.65504
[12] Colton, D.; Kress, R., Integral Equations Methods in Scattering Theory (1983), Wiley: Wiley New York
[13] F. Cordellier, L’\(ε\); F. Cordellier, L’\(ε\)
[14] F. Cordellier, Démonstration algébrique de l’extension de l’identité de Wynn aux tables de Padé non-normales, in: L. Wuytack (Ed.), Padé Approximation and its Applications, Springer, Berlin, Lecture Notes in Mathematics, Vol. 765, 1979, pp. 36-60.; F. Cordellier, Démonstration algébrique de l’extension de l’identité de Wynn aux tables de Padé non-normales, in: L. Wuytack (Ed.), Padé Approximation and its Applications, Springer, Berlin, Lecture Notes in Mathematics, Vol. 765, 1979, pp. 36-60.
[15] Cordellier, F., Utilisation de l’invariance homographique dans les algorithmes de losange, (Werner, H.; Bünger, H. J., Padé Approximation and its Applications, Bad Honnef 1983. Padé Approximation and its Applications, Bad Honnef 1983, Lecture Notes in Mathematics, Vol. 1071 (1984), Springer: Springer Berlin), 62-94 · Zbl 0575.65003
[16] F. Cordellier, Thesis, University of Lille, 1989.; F. Cordellier, Thesis, University of Lille, 1989.
[17] Cuyt, A.; Wuytack, L., Nonlinear Methods in Numerical Analysis (1987), North-Holland: North-Holland Amsterdam · Zbl 0609.65001
[18] Delahaye, J.-P.; Germain-Bonne, B., The set of logarithmically convergent sequences cannot be accelerated, SIAM J. Numer. Anal., 19, 840-844 (1982) · Zbl 0495.65001
[19] Delahaye, J.-P., Sequence Transformations (1988), Springer: Springer Berlin · Zbl 0468.65001
[20] W. Gander, E.H. Golub, D. Gruntz, Solving linear systems by extrapolation in Supercomputing, Trondheim, Computer Systems Science, Vol. 62, Springer, Berlin, 1989, pp. 279-293.; W. Gander, E.H. Golub, D. Gruntz, Solving linear systems by extrapolation in Supercomputing, Trondheim, Computer Systems Science, Vol. 62, Springer, Berlin, 1989, pp. 279-293.
[21] Graffi, S.; Grecchi, V., Borel summability and indeterminancy of the Stieltjes moment problem: Application to anharmonic oscillators, J. Math. Phys., 19, 1002-1006 (1978) · Zbl 0432.40007
[22] Graves-Morris, P. R., Vector valued rational interpolants I, Numer. Math., 42, 331-348 (1983) · Zbl 0525.41014
[23] Graves-Morris, P. R.; Beckermann, B., The compass (star) identity for vector-valued rational interpolants, Adv. Comput. Math., 7, 279-294 (1997) · Zbl 0888.41007
[24] Graves-Morris, P. R.; Jenkins, C. D., Generalised inverse vector-valued rational interpolation, (Werner, H.; Bünger, H. J., Padé Approximation and its Applications, Vol. 1071 (1984), Springer: Springer Berlin), 144-156 · Zbl 0538.41006
[25] Graves-Morris, P. R.; Jenkins, C. D., Vector-valued rational interpolants III, Constr. Approx., 2, 263-289 (1986) · Zbl 0614.41016
[26] Graves-Morris, P. R.; Roberts, D. E., From matrix to vector Padé approximants, J. Comput. Appl. Math., 51, 205-236 (1994) · Zbl 0819.41015
[27] Graves-Morris, P. R.; Roberts, D. E., Problems and progress in vector Padé approximation, J. Comput. Appl. Math., 77, 173-200 (1997) · Zbl 0951.65011
[28] Graves-Morris, P. R.; Saff, E. B., Row convergence theorems for generalised inverse vector-valued Padé approximants, J. Comput. Appl. Math., 23, 63-85 (1988) · Zbl 0651.41009
[29] Graves-Morris, P. R.; Saff, E. B., An extension of a row convergence theorem for vector Padé approximants, J. Comput. Appl. Math., 34, 315-324 (1991) · Zbl 0747.41013
[30] Graves-Morris, P. R.; Van Iseghem, J., Row convergence theorems for vector-valued Padé approximants, J. Approx. Theory, 90, 153-173 (1997) · Zbl 0878.41012
[31] Gutknecht, M. H., Lanczos type solvers for non-symmetric systems of linear equations, Acta Numer., 6, 271-397 (1997) · Zbl 0888.65030
[32] Heyting, A., Die Theorie der linear Gleichungen in einer Zahlenspezies mit nichtkommutatives Multiplikation, Math. Ann., 98, 465-490 (1927) · JFM 53.0120.01
[33] H.H.H. Homeier, Scalar Levin-type sequence transformations, this volume, J. Comput. Appl. Math. 122 (2000) 81-147.; H.H.H. Homeier, Scalar Levin-type sequence transformations, this volume, J. Comput. Appl. Math. 122 (2000) 81-147. · Zbl 0976.65004
[34] Jentschura, U. C.; Mohr, P. J.; Soff, G.; Weniger, E. J., Convergence acceleration via combined nonlinear-condensation transformations, Comput. Phys. Comm., 116, 28-54 (1999) · Zbl 0995.81526
[35] K. Jbilou, H. Sadok, Vector extrapolation methods, Applications and numerical comparison, this volume, J. Comput. Appl. Math. 122 (2000) 149-165.; K. Jbilou, H. Sadok, Vector extrapolation methods, Applications and numerical comparison, this volume, J. Comput. Appl. Math. 122 (2000) 149-165. · Zbl 0974.65034
[36] Jones, W. B.; Thron, W., (Rota, G.-C., Continued Fractions, Encyclopedia of Mathematics and its Applications, Vol. 11 (1980), Addison-Wesley, Reading: Addison-Wesley, Reading MA, USA) · Zbl 0445.30003
[37] McLeod, J. B., A note on the \(ε\)-algorithm, Computing, 7, 17-24 (1972) · Zbl 0219.65094
[38] Roberts, D. E., Clifford algebras and vector-valued rational forms I, Proc. Roy. Soc. London A, 431, 285-300 (1990) · Zbl 0745.41017
[39] Roberts, D. E., On the convergence of rows of vector Padé approximants, J. Comput. Appl. Math., 70, 95-109 (1996) · Zbl 0851.41015
[40] Roberts, D. E., On a vector q-d algorithm, Adv. Comput. Math., 8, 193-219 (1998) · Zbl 0922.65005
[41] Roberts, D. E., A vector Chebyshev algorithm, Numer. Algorithms, 17, 33-50 (1998) · Zbl 0907.65013
[42] Roberts, D. E., On a representation of vector continued fractions, J. Comput. Appl. Math., 105, 453-466 (1999) · Zbl 0948.41011
[43] Salam, A., An algebraic approach to the vector \(ε\)-algorithm, Numer. Algorithms, 11, 327-337 (1996) · Zbl 0852.65004
[44] Salam, A., Formal vector orthogonal polynomials, Adv. Comput. Math., 8, 267-289 (1998) · Zbl 0906.42013
[45] Salam, A., What is a vector Hankel determinant? Linear Algebra Appl., 278, 147-161 (1998) · Zbl 0940.15006
[46] Salam, A., Padé-type approximants and vector Padé approximants, J. Approx. Theory, 97, 92-112 (1999) · Zbl 0921.41007
[47] Shanks, D., Non-linear transformations of divergent and slowly convergent sequences, J. Math. Phys., 34, 1-42 (1955) · Zbl 0067.28602
[48] Sidi, A.; Ford, W. F.; Smith, D. A., SIAM J. Numer. Anal., 23, 178-196 (1986)
[49] Tan, R. C.E., Implementation of the topological epsilon algorithm, SIAM J. Sci. Statist. Comput., 9, 839-848 (1988) · Zbl 0654.65005
[50] Weniger, E. J., Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Comput. Phys. Rep., 10, 371-1809 (1989)
[51] Weniger, E. J., A convergent, renormalised strong coupling perturbation expansion for the ground state energy of the quartic, sextic and octic anharmonic oscillator, Ann. Phys., 246, 133-165 (1996) · Zbl 0877.47041
[52] E.J. Weniger, Prediction properties of Aitken’s iterated \(Δ^2\); E.J. Weniger, Prediction properties of Aitken’s iterated \(Δ^2\) · Zbl 0974.65002
[53] Wimp, J., Sequence Transformations and Their Applications (1981), Academic Press: Academic Press New York · Zbl 0566.47018
[54] Wynn, P., On a device for calculating the \(e_m(S_{n\) · Zbl 0074.04601
[55] Wynn, P., The epsilon algorithm and operational formulas of numerical analysis, Math. Comp., 15, 151-158 (1961) · Zbl 0102.33205
[56] Wynn, P., L’\(ε\)-algoritmo e la tavola di Padé, Rendi. Mat. Roma, 20, 403-408 (1961) · Zbl 0104.34205
[57] Wynn, P., Acceleration techniques for iterative vector problems, Math. Comp., 16, 301-322 (1962) · Zbl 0105.10302
[58] Wynn, P., Continued fractions whose coefficients obey a non-commutative law of multiplication, Arch. Rational Mech. Anal., 12, 273-312 (1963) · Zbl 0122.30604
[59] Wynn, P., On the convergence and stability of the epsilon algorithm, SIAM J. Numer. Anal., 3, 91-122 (1966) · Zbl 0299.65003
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