Note: The main interest of implicit Runge-Kutta methods lies in their absolute stability characteristics, which are much superior to those of explicit methods.

Example: For example, if we analyze the method given by (6.14) for absolute stability, we obtain

, where

(6.15)

This is a fourth order rational approximation (or (2, 2) Pade approximation) to whereas a fourth order explicit Runge-Kutta method produces a fourth order polynomial approximation to the exponential. It follows from (6.15) that the interval of absolute stability for the method (6.14) is

Remark: One can conclude that implicit Runge-Kutta method can offer substantially improved regions of absolute stability but at such a high computational cost. The demand for such methods arises when weak stability considerations are paramount; such as solving stiff systems of initial value problems to be discussed later.