On the properties of matrices defining some classes of BVMs (Q1434043)

Consider a band symmetric Toeplitz matrix \(T=[t_{ij}]\in \mathbb{R}^{n\times n}\), where \(t_{ij}=a_{m-| i-j| }\) for \(| i-j| \leq m\), else \(t_{ij}=0\) for \(i,j=0,1,\dots ,n-1\), \(m\) is a nonnegative integer. Denote the polynomial associated to \(T\) by \(p(z)=\sum_{i=0}^{2m}a_iz^i\), where \(a_{m+j}=a_{m-j}\), \(j=1,\dots,m\). The paper contributes by an alternative approach to the characterization of positive definitness of these matrices. It requires only the knowledge of a half of the negative real roots of \(p(z)\). It can be obtained for instance by a classical Descartes rule of signs. Such information is easier obtainable than the usage of other approaches. Further, the positive definitness of the Toeplitz matrices arising in the discretization of continuous initial value problems of ordinary differential equations is proved using this result for two classes of \(k\)-step boundary value method (BVM) known as generalized backward differentiation formulae and top order methods.