The Computable Differential Equation Lecture ... - Bruce E. Shapiro

CHAPTER 5. RUNGE-KUTTA METHODS 115
Theorem 5.4. For every Runge-Kutta method there exists a rational function P ν/ν
such that
y n = [ P ν/ν (z) ] n
(5.233)
Proof. Start with
R(z) = 1 + zb T (I − zA) −1 1 (5.234)
which is a result we have already demonstrated. Then
(I − zA) −1 =
adj(I − zA)
det(I − zA)
(5.235)
where adj(M) is the adjunct matrix of M,
[adj(M)] ij = (−1) i+j minor(M ji ) (5.236)
We observe that each entry in the adjunct matrix is linear in z, hence each of the
principal minor determinants is a polynomial in z of order ν − 1. Therefore
zb T adj(I − zA) −1 1 (5.237)
is a polynomial of degree ν. Hence zb T (I − zA) −1 1 is the quotient of two polynomials,
each of order ν, which immediately yields the desired result.
Corollary 5.1. For an explicit Runge-Kutta method, y n = [P ν ] n , a polynomial of
order ν.
Proof. If the method is explicit then A is strictly lower triangular; hence det(I −
zA) = 1.
Corollary 5.2. Explicit Runge-Kutta methods may not be A-stable.
Proof. A non-constant polynomial cannot be uniformly bounded.
Theorem 5.5. Let R(z) be any rational functions. Then |R(z)| < 1 for all z ∈ C −
if and only if both of the following conditions hold:
1. All the poles of R(z) have positive real parts; and
2. |R(it)| ≤ 1 for all t ∈ R.
Theorem 5.6. R(z) = e z + O(z ν 1
) for any Runge-Kutta method for order ν.
Proof. By definition, y n = R(z)y n−1 ; for the test equation, y(t n ) = y n = e z y n−1 .
Theorem 5.7. For any two integers m, n there is a Padé Approximant to the exponential
for the stability function such that
P m/n (t) = P m,n(t)
Q m,n (t)
(5.238)
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge