31 research outputs found

    Mathematical Properties of a New Levin-Type Sequence Transformation Introduced by \v{C}\'{\i}\v{z}ek, Zamastil, and Sk\'{a}la. I. Algebraic Theory

    Full text link
    \v{C}\'{\i}\v{z}ek, Zamastil, and Sk\'{a}la [J. Math. Phys. \textbf{44}, 962 - 968 (2003)] 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 {sn}n=0∞\{s_n \}_{n=0}^{\infty} of partial sums, but also explicit estimates {ωn}n=0∞\{\omega_n \}_{n=0}^{\infty} for the truncation errors. The explicit incorporation of the information contained in the truncation error estimates makes this and related transformations potentially much more powerful than for instance Pad\'{e} approximants. Special cases of the new transformation are sequence transformations introduced by Levin [Int. J. Comput. Math. B \textbf{3}, 371 - 388 (1973)] and Weniger [Comput. Phys. Rep. \textbf{10}, 189 - 371 (1989), Sections 7 -9; Numer. Algor. \textbf{3}, 477 - 486 (1992)] and also a variant of Richardson extrapolation [Phil. Trans. Roy. Soc. London A \textbf{226}, 299 - 349 (1927)]. 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 \v{C}\'{\i}\v{z}ek, Zamastil, and Sk\'{a}la. This leads to a considerable formal simplification and unification.Comment: 41 + ii pages, LaTeX2e, 0 figures. Submitted to Journal of Mathematical Physic

    Some expansions for integrals with weight functions

    No full text

    Derivation of explicit expressions for the error terms in the ordinary and the modified Romberg algorithms

    No full text

    Remarks on an expansion for integrals of rapidly oscillating functions

    No full text

    Some algorithms for numerical quadrature using the derivatives of the integrand in the integration interval

    No full text
    Some quadrature formulae using the derivatives of the integrand are discussed. As special cases are obtained generalizations of both the ordinary and the modified Romberg algorithms. In all cases the error terms are expressed in terms of Bernoulli polynomials and functions

    Expansions for integrals with weight functions

    No full text

    Error deviation in Romberg intergration

    No full text

    On a modification of the Clenshaw-Curtis quadrature formula

    No full text
    corecore