### Abstract

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 e^{μ}. 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 e^{Ah}. However, if any particularly useful integration forms R can be found, they can be performed by the IRK method.

Original language | English (US) |
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Pages (from-to) | 20-22 |

Number of pages | 3 |

Journal | BIT |

Volume | 10 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 1970 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Software
- Computer Graphics and Computer-Aided Design
- Computational Mathematics
- Applied Mathematics

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## Cite this

*BIT*,

*10*(1), 20-22. https://doi.org/10.1007/BF01940888