Computing

, Volume 9, Issue 4, pp 267–273

Convergence acceleration by a method of intercalation

Article

Summary

Many convergence acceleration techniques function successfully when applied to series of real terms which alternate in sign, but are unsuccessful when applied to series of terms having the same sign. Given a series Σuv of terms having the same sign and whose sumU is required, it is often possible to construct a series of the form u0−v0+u1−v1+..., where Σvv is a series whose terms have the same sign as the {uv} and whose sumV is known, which is amenable to transformation. The required estimate ofU is then recovered from the transformed estimate ofU−V and the known value ofV. Use of this artifice is illustrated with reference to the ⃛-algorithm, two numerical examples being given.

Konvergenzbeschleunigung durch Einschiebung

Zusammenfassung

Gewisse Konvergenzbeschleunigungsmethoden erweisen sich als besonders wirksam im Falle von Reihen mit reellen Gliedern, deren Vorzeichen alternieren. Es wird gezeigt, daß unter günstigen Umständen mit Hilfe eines sehr einfachen Kunstgriffes eine gegebene Reihe mit reellen Gliedern und gleichen Vorzeichen mit einer alternierenden Reihe in Beziehung gebracht werden kann und daß daher die gegebene Reihe mit Erfolg transformiert werden kann. Zwei numerische Beispiele werden angegeben.

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References

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    Wynn, P.: On a device for Computing thee m (S n) Transformation. Maths. of Comp.10, 91–96 (1956).Google Scholar
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    Wynn, P.: On the Convergence and Stability of the Epsilon Algorithm. SIAM Jour. for Num. Anal.3, 91–122 (1966).Google Scholar
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    Wynn, P.: Acceleration Techniques in Numerical Analysis, with Particular Reference to Problems in One Independent Variable, Proc. IFIP Congress 1962, pp. 149–156. Amsterdam; North Holland. 1962.Google Scholar
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    Barkley-Rosser, J.: Theory and Application\(\int\limits_0^z {e^{ - x^2 } dx} \) and\(\int\limits_0^z {e^{ - p^2 y^2 } dy} \int\limits_0^y {e^{ - x^2 } dz} \). Brooklyn, N. Y.: Mapleton Press. 1948.Google Scholar

Copyright information

© Springer-Verlag 1972

Authors and Affiliations

  • P. Wynn
    • 1
  1. 1.Mathematics DepartmentLouisiana State UniversityNew OrleansU.S.A.

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