We derive a closed form for the Jordan decomposition of companion matrices including properties of generalized eigenvectors. As a consequence, we provide a formula for the inverse of confluent Vandermonde matrices and results on sensitivity of multiple roots of polynomials.

Vandermonde, Cauchy, and Cauchy–Vandermonde totally positive linear systems can be solved extremely accurately in O(n2) time using Björck–Pereyra-type methods. We prove that Björck–Pereyra-type methods exist not only for the above linear systems but also for any totally positive linear system as long as the initial minors (i.e., contiguous minors that include the first row or column) can be com...

Lower bounds on the condition number κp(Vc ) of a real confluent Vandermonde matrix Vc are established in terms of the dimension n or n and the largest absolute value among all nodes that define the confluent Vandermonde matrix or n and the interval that contains the nodes. In particular, it is proved that for any modest kmax (the largest number of equal nodes), κp(Vc ) behaves no smaller than ...

We present parallel algorithms for fast polynomial interpolation. These algorithms can be used for constructing and evaluating polynomials interpolating the function values and its derivatives of arbitrary order (Hermite interpolation). For interpolation, the parallel arithmetic complexity is 0(log2 M + log N) for large M and N, where M 1 is the order of the highest derivative information and N...

In this paper we study Stein equations where the coefficient matrices are in companion form. Solutions to such equations are relatively easy to compute as soon as one knows how to invert a Vandermonde matrix (in the generic case where all eigenvalues have multiplicity one) or a confluent Vandermonde matrix (in the general case). As an application we present a way to compute the Fisher informati...