Peter Lancaster

Learn More
We study iterative methods for finding the maximal Hermitian positive definite solutions of the matrix equations X + A∗X−1A = Q and X − A∗X−1A = Q, where Q is Hermitian positive definite. General convergence results are given for the basic fixed point iteration for both equations. Newton’s method and inversion free variants of the basic fixed point(More)
When Newton’s method is applied to find the maximal symmetric solution of an algebraic Riccati equation, convergence can be guaranteed under moderate conditions. In particular, the initial guess need not be close to the solution. The convergence is quadratic if the Fréchet derivative is invertible at the solution. In this paper we examine the behaviour of(More)
We consider the quadratic eigenvalue problem (QEP) (λ2A+λB+ C)x = 0, where A,B, and C are Hermitian with A positive definite. The QEP is called hyperbolic if (x∗Bx)2 > 4(x∗Ax)(x∗Cx) for all nonzero x ∈ Cn. We show that a relatively efficient test for hyperbolicity can be obtained by computing the eigenvalues of the QEP. A hyperbolic QEP is overdamped if B(More)
A paper of Tan and Pugh [TP] raises the question of ambiguity in a frequently used form of linearization when applied to regular matrix polynomials. Here, further insight into this question is provided, as well as a reminder of a stronger form of linearization (for which ambiguities are removed) introduced by Gohberg et al. [GKL]. Let A0, A1, . . . , An ∈ C(More)
Earlier work of the authors concerning the generation of isospectral families of second order (vibrating) systems is generalized to higher-order systems (with no spectrum at infinity). Results and techniques are developed first for systems without symmetries, then with Hermitian symmetry and, finally, with palindromic symmetry. The construction of(More)