Rational approximations by implicit Runge-Kutta schemes

C. W. Gear

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


Ehle [3] has pointed out that the n-stage implicit Runge-Kutta (IRK) methods due to Butcher [1] are A-stable in the definition of Dahlquist [2] because they effect the operation R(Ah) where R(μ) is the diagonal Padé approximation to eμ. The purpose of this note is to point out that if R(μ)=P(μ)/Q(μ) is a rational polynomial whose n poles are distinct and nonzero, and if degree P(μ)≦degree Q(μ)=n, then an n-stage IRK method applied to y=Ay can be used for the operation {Mathematical expression} This will no longer be of order 2 n, nor necessarily the same order as the approximation of R(Ah) to eAh. However, if any particularly useful integration forms R can be found, they can be performed by the IRK method.

Original languageEnglish (US)
Pages (from-to)20-22
Number of pages3
Issue number1
StatePublished - Mar 1970

All Science Journal Classification (ASJC) codes

  • Software
  • Computer Networks and Communications
  • Computational Mathematics
  • Applied Mathematics


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