On approximating multiple roots

Dimitri Breda, Davide Liessi

To approximate characteristic roots of linear delay differential equations one can use collocation to discretize either the solution operator semigroup or its infinitesimal generator, [1]. Both methods approximate an exact root \(\lambda\) with algebraic multiplicity \(m\) by \(m\) eigenvalues \(\lambda_{M,1},\ldots,\lambda_{M,m}\), counted with multiplicities, \(M\) being the index of discretization (say the degree of collocation). The results are of the form

\[ \max_{i=1,\ldots,m}|\lambda_{M,i}-\lambda|=\epsilon(M)\]

where \(\epsilon(M)\rightarrow0\) with spectral accuracy as \(M\rightarrow\infty\). The main difference is that

\[ \epsilon(M)=O(M^{-M/m})\]

for the infinitesimal generator approach, while

\[ \epsilon(M)=O(M^{-M/\nu})\]

for the solution operator approach, with \(\nu\leq m\) the ascent of \(\lambda\) (the length of its longest Jordan chain).

Why is that and what is the effect at the computational level?

Is it true that roots of linear scalar DDEs have always geometric multiplicity \(1\), and at most \(m=\nu=2\)? Or is it just a generic property?