No abstract

Abstract-A polynomial eigenvalue decomposition of parahermitian matrices can be calculated approximately using iterative approaches such as the sequential matrix diagonalisation (SMD) algorithm. In this paper, we present an improved SMD algorithm which, compared to existing SMD approaches, eliminates more off-diagonal energy per step. This leads to faster convergence while incurring only a marginal increase in complexity. We motivate the approach, prove its convergence, and demonstrate some results that underline the algorithm's performance.

In this paper, we show that the paraunitary (PU) matrices that arise from the polynomial eigenvalue decomposition (PEVD) of a parahermitian matrix are not unique. In particular, arbitrary shifts (delays) of polynomials in one row of a PU matrix yield another PU matrix that admits the same PEVD. To keep the order of such a PU matrix as low as possible, we pro- pose a row-shift correction. Using the example of an iterative PEVD algorithm with previously proposed truncation of the PU matrix, we demonstrate that a considerable shortening of the PU order can be accomplished when using row-corrected truncation

We address an open question of Francis and Steel about phylogenetic networks and trees. They give a polynomial time algorithm to decide if a phylogenetic network, N , is tree-based and pose the problem: given a fixed tree T and network N , is N based on T ? We show that it is N P -hard to decide, by reduction from 3-Dimensional Matching (3DM), and further, that the problem is fixed parameter tractable.