Editing Runge–Kutta methods (section)

===Stability===
The advantage of implicit Runge–Kutta methods over explicit ones is their greater stability, especially when applied to [[stiff equation]]s. Consider the linear test equation <math> y' = \lambda y </math>. A Runge–Kutta method applied to this equation reduces to the iteration <math> y_{n+1} = r(h\lambda) \, y_n </math>, with ''r'' given by

:<math> r(z) = 1 + z b^T (I-zA)^{-1} e = \frac{\det(I-zA+zeb^T)}{\det(I-zA)}, </math> <ref>{{harvnb|Hairer|Wanner|1996|pp=40–41}}</ref>

where ''e'' stands for the vector of ones. The function ''r'' is called the ''stability function''.<ref>{{harvnb|Hairer|Wanner|1996|p=40}}</ref> It follows from the formula that ''r'' is the quotient of two polynomials of degree ''s'' if the method has ''s'' stages. Explicit methods have a strictly lower triangular matrix ''A'', which implies that det(''I'' − ''zA'') = 1 and that the stability function is a polynomial.<ref name="Iserles 1996 60">{{harvnb|Iserles|1996|p=60}}</ref>

The numerical solution to the linear test equation decays to zero if | ''r''(''z'') | < 1 with ''z'' = ''h''λ. The set of such ''z'' is called the ''domain of absolute stability''. In particular, the method is said to be [[Stiff equation#A-stability|absolute stable]] if all ''z'' with Re(''z'') < 0 are in the domain of absolute stability. The stability function of an explicit Runge–Kutta method is a polynomial, so explicit Runge–Kutta methods can never be A-stable.<ref name="Iserles 1996 60"/>

If the method has order ''p'', then the stability function satisfies <math> r(z) = \textrm{e}^z + O(z^{p+1}) </math> as <math> z \to 0 </math>. Thus, it is of interest to study quotients of polynomials of given degrees that approximate the exponential function the best. These are known as [[Padé approximant]]s. A Padé approximant with numerator of degree ''m'' and denominator of degree ''n'' is A-stable if and only if ''m'' ≤ ''n'' ≤ ''m'' + 2.<ref>{{harvnb|Iserles|1996|pp=62–63}}</ref>

The Gauss–Legendre method with ''s'' stages has order 2''s'', so its stability function is the Padé approximant with ''m'' = ''n'' = ''s''. It follows that the method is A-stable.<ref>{{harvnb|Iserles|1996|p=63}}</ref> This shows that A-stable Runge–Kutta can have arbitrarily high order. In contrast, the order of A-stable [[linear multistep method]]s cannot exceed two.<ref>This result is due to {{harvtxt|Dahlquist|1963}}.</ref>