## Jens Keiner (University of Lübeck)

We develop an efficient algorithm for computing the eigendecomposition of diagonal plus upper or
lower triangular semiseparable matrices for which the eigenvalues are always known a-priori.
By using fast summation techniques, we obtain an *O(n*^{2}) algorithm for
computing the eigenvectors explicitly and an *O(n log n)* algorithm for applying
the eigenvector matrix, or optionally its inverse, to an arbitrary vector.
Both algorithms are approximate, i.e. accurate up to a prefixed accuracy *ε*.
This extends an existing divide-and-conquer algorithm for symmetric diagonal plus semiseparable
matrices. As an application, we develop a fast algorithm for converting expansion coefficients
between different representations in terms of Gegenbauer polynomials.