The spectrum of delay-differential equations: numerical methods - KTH

34 Chapter 2. Computing the spectrum
SOD(LMS) is a Padé approximation of the logarithm
The solution operator is a characterization of the solution of the DDE. We are interested
in the eigenvalues and not the solution of the DDE. Hence, the derivation
using the solution operator appears somewhat indirect. We now wish to search
alternative shorter derivations leading to (2.28) without using the solution operator.
The value of this discussion that follows is the simplicity of the derivation and
that we point out that the resulting approximation is a polynomial eigenvalue
problem. Polynomial eigenvalue problems can be solved using other methods
than the companion linearization which was implied by the discretization motivation
above. There are several methods for polynomial eigenvalue problems,
e.g. the different types of linearizations [MMMM06b] and Jacobi-Davidson for
polynomial eigenvalue problems [SBFvdV96] (see also Section 2.3.2).
It turns out that the linear multistep discretization of the solution operator
(SOD(LMS)) is equivalent to a Padé-approximation of the logarithm. The relation
between ODE-methods and polynomial (and rational) interpolation is not
new (see e.g. [Lam91, Section 3.3]). It is however worthwhile to clarify the interpretation
in the current context, i.e., how LMS applied to the solution operator
can be interpreted as a rational approximation.
The substitution s = 1
h ln(µ) for h = 1/N into the characteristic equation
yields
1
h µN ln µ ∈ σ(A0µ N + A1). (2.29)
We are interested in µ ≈ 1, as this corresponds to s close to the origin. We
will now approximate the logarithm close to this point. For instance, consider
the first-order Taylor expansion ln µ ≈ µ − 1. This yields
1
h ˜µN (˜µ − 1) ∈ σ(A0 ˜µ N + A1),
which reduces to Example 2.12 for A0 = 0, A1 = −3/2.
Similarly, we may consider a rational approximation of ln µ, i.e.,
ln µ ≈ α(µ)
β(µ) ,
where α, β are polynomials (where we denote the polynomials α and β as they will
coincide with the polynomials in Definition 2.14). If we insert this approximation
in (2.29) we have derived (2.28).