BIT Numerical Mathematics

, Volume 21, Issue 2, pp 175–189 | Cite as

A generalization of singly-implicit methods

  • J. C. Butcher
Part II Numerical Mathematics


Singly-implicit Runge-Kutta methods are characterized by a one-point spectrum property of the coefficient matrix. If a method of this type is also a collocation method, then its abscissae are proportional to the zeros of a Laguerre polynomial. The generalization introduced here is a multistep method in the style of Nordsieck and also a multistage method under the one-point spectrum constraint. It is found that much of the theory of singly-implicit methods carries over but with Laguerre polynomials replaced by their usual generalizations. Amongst the formal properties of the new method which are studied is a derivation of the similarity transformations which allow their efficient implementation. A preliminary investigation is made of the stability of the new methods.

CR classification


AMS classification



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Copyright information

© BIT Foundations 1981

Authors and Affiliations

  • J. C. Butcher
    • 1
  1. 1.Department of Computer ScienceUniversity of AucklandAucklandNew Zealand

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