Rational approximations by implicit Runge-Kutta schemes

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.