![]() ![]() and Redivo Zaglia, M., Extrapolation Methods (North-Holland, Amsterdam, 1991). C., “ A derivation of extrapolation algorithms based on error estimates,” J. Reprinted in Numerical Analysis 2000, edited by Brezinski, C. Brezinski, C., “ Convergence acceleration during the 20th century,” J. Brezinski, C., “ Difference and differential equations, and convergence acceleration algorithms,” CRM Proc. Brezinski, C., “Error estimates and convergence acceleration,” in Error Control and Adaptivity in Scientific Computing, edited by Bulgak, H. Brezinski, C., Projection Methods for Systems of Equations (Elsevier, Amsterdam, 1997). Brezinski, C., “ Extrapolation algorithms and Padé approximations: A historical survey,” Appl. Brezinski, C., History of Continued Fractions and Padé Approximants (Springer-Verlag, Berlin, 1991). Brezinski, C., “ Prediction properties of some extrapolation methods,” Appl. Brezinski, C., “ A general extrapolation algorithm,” Numer. Brezinski, C., Padé-Type Approximation and General Orthogonal Polynomials (Birkhäuser, Basel, 1980). Brezinski, C., “ Rational approximation to formal power series,” J. Brezinski, C., Algorithmes d’Accélération de la Convergence–Étude Numérique (Éditions Technip, Paris, 1978). Brezinski, C., Accélération de la Convergence en Analyze Numérique (Springer-Verlag, Berlin, 1977). Bouferguene, A.and Fares, M., “ Convergence accelerators in the computation of molecular integrals over Slater-type basis functions in the two-range one-center expansion method,” Phys. Borghi, R.and Santarsiero, M., “ Summing Lax series for nonparaxial beam propagation,” Opt. Vakar, Lectures on Divergent Series, Translation LA-6140-TR (Los Alamos Scientific Laboratory, Los Alamos, 1975). Reprinted by Éditions Jacques Gabay (Paris, 1988). Borel, E., Leçons sur les Séries Divergentes, 2nd ed. Borel, E., “ Mémoires sur les séries divergentes,” Ann. Bhattacharya, R., Roy, D., and Bhowmick, S., “ Rational interpolation using Levin–Weniger transforms,” Comput. Bhattacharya, R., Roy, D., and Bhowmick, S., “ On the regularity of the Levin u -transform,” Comput. Bhagat, V., Bhattacharya, R., and Roy, D., “ On the evaluation of generalized Bose–Einstein and Fermi–Dirac integrals,” Comput. A study in perturbation theory in large order,” Phys. T., “ Large-order behavior of perturbation theory,” Phys. J., “ Numerical evidence that the perturbation expansion for a non-Hermitian PT-symmetric Hamiltonian is Stieltjes,” J. A., Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978). V., “ Large-order perturbation theory for a non-Hermitian PT-symmetric Hamiltonian,” J. Belkić, D., “ New hybrid nonlinear transformations of divergent perturbation series for quadratic Zeeman effects,” J. Bar-Shalom, A., Klapisch, M., and Oreg, J., “ HULLAC, an integrated computer package for atomic processes in plasmas,” J. Bar-Shalom, A., Klapisch, M., and Oreg, J., “ Phase-amplitude algorithms for atomic continuum orbitals and radial integrals,” Comput. ![]() (Cambridge University Press, Cambridge, 1996). and Graves-Morris, P., Padé Approximants, 2nd ed. J., “ Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics,” Comput. C., “ On Bernoulli’s numerical solution of algebraic equations,” Proc. This leads to a considerable formal simplification and unification. The algebraic theory of these transformations-explicit expressions, recurrence formulas, explicit expressions in the case of special remainder estimates, and asymptotic order estimates satisfied by rational approximants to power series-is formulated in terms of hitherto unknown mathematical properties of the new transformation introduced by Čı́žek, Zamastil, and Skála. Special cases of the new transformation are sequence transformations introduced by Levin and Weniger and also a variant of Richardson extrapolation. The explicit incorporation of the information contained in the truncation error estimates makes this and related transformations potentially much more powerful than, for instance, Padé approximants. Čı́žek, Zamastil, and Skála introduced in connection with the summation of the divergent perturbation expansion of the hydrogen atom in an external magnetic field a new sequence transformation which uses as input data not only the elements of a sequence n=0 ∞ for the truncation errors. ![]()
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