Hermite matrix in Lagrange basis for scaling static output feedback polynomial matrix inequalities

Using Hermite's formulation of polynomial stability conditions, static output feedback (SOF) controller design can be formulated as a polynomial matrix inequality (PMI), a (generally nonconvex) nonlinear semidefinite programming problem that can be solved (locally) with PENNON, an implementation of a penalty method. Typically, Hermite SOF PMI problems are badly scaled and experiments reveal that this has a negative impact on the overall performance of the solver. In this note we recall the algebraic interpretation of Hermite's quadratic form as a particular Bezoutian and we use results on polynomial interpolation to express the Hermite PMI in a Lagrange polynomial basis, as an alternative to the conventional power basis. Numerical experiments on benchmark problem instances show the substantial improvement brought by the approach, in terms of problem scaling, number of iterations and convergence behavior of PENNON

Newton's method in practice II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees

We present a practical implementation based on Newton's method to find all roots of several families of complex polynomials of degrees exceeding one billion ($10^9$) so that the observed complexity to find all roots is between $O(d\ln d)$ and $O(d\ln^3 d)$ (measuring complexity in terms of number of Newton iterations or computing time). All computations were performed successfully on standard desktop computers built between 2007 and 2012.Comment: 24 pages, 19 figures. Update in v2 incorporates progress on polynomials of even higher degrees (greater than 1 billion

A note on generalized companion pencils in the monomial basis

In this paper, we introduce a new notion of generalized companion pencils for scalar polynomials over an arbitrary field expressed in the monomial basis. Our definition is quite general and extends the notions of companion pencil in De Terán et al. (Linear Algebra Appl 459:264&-333, 2014), generalized companion matrix in Garnett et al. (Linear Algebra Appl 498:360&-365, 2016), and Ma&-Zhan companion matrices in Ma and Zhan (Linear Algebra Appl 438: 621&-625, 2013), as well as the class of quasi-sparse companion pencils introduced in De Terán and Hernando (INdAM Series, Springer, Berlin, pp 157&-179, 2019). We analyze some algebraic properties of generalized companion pencils. We determine their Smith canonical form and we prove that they are all nonderogatory. In the last part of the work we will pay attention to the sparsity of these constructions. In particular, by imposing some natural conditions on its entries, we determine the smallest number of nonzero entries of a generalized companion pencilThis work has been partially supported by the Ministerio de Economía y Competitividad of Spain through Grants MTM2017-90682-REDT and MTM2015-65798-P