On the properties of matrices defining some classes of BVMs

Abstract

It is known that the matrices defining the discrete problem generated by a k-step Boundary Value Method (BVM) have a quasi-Toeplitz band structure. In particular, when the boundary conditions are skipped, they become Toeplitz matrices. In this paper, by introducing a characterization of positive definiteness for such matrices, we shall prove that the Toeplitz matrices which arise when using the methods in the classes of BVMs known as Generalized BDF and Top Order Methods have such property

Simplification of a result on banded Toeplitz matrices and BVM methods

“…In order to have a fair comparison we will follow the notation in [1]. Let A n be a Toeplitz matrix of order n …”

“…In [1] the authors suppose that the only zeros of p of unitary modulus (if any) are real that is either 1 or −1. Under these assumptions they S. Serra-Capizzano (B) Dipartimento di Fisica e Matematica, Università dell'Insubria, Via Valleggio 11, 22100 Como, Italy e-mail: stefano.serrac@uninsubria.it; serra@mail.dm.unipi.it give a criterion on the positive definiteness of every A n which is based on a parameter γ appearing in the Wiener-Hopf factorization of the infinite matrix ( [1], pages 3-7, with explicit computations) and then on the application of the Descartes rule which implies further computations ( [1], pages 13-15).…”

“…or in other words the coefficients of the matrix are the Fourier coefficients of f (the unusual ordering is due to the notation adopted in [1], that is natural in a BVM setting but becomes unusual in the context of Toeplitz literature). Moreover, by a direct observation,…”

“…As an example of application, we consider the very same one-level (d = 1) BVM matrices considered in [1]. We recall that the BVM methods are essentially linear multistep formulae for Initial Value Problems, where instead of initial conditions, a blending of initial and final conditions (i.e., boundary conditions) are used.…”

“…In all the considered cases it is trivial to prove that the only unit roots of the associated polynomials are real (see [1], page 12, line 5 and page 13, lines 5-6, for the considered examples). In the case of the generalized BDF formulae (see Sect.…”