Peter Lancaster

Surfaces generated by moving least squares methods

An analysis of moving least squares (m.l.s.) methods for smoothing and interpolating scattered data is presented. In particular, theorems are proved concerning the smoothness of interpolants and the description of m.l.s. processes as projection methods. Some properties of compositions of the m.l.s. projector, with projectors associated with finiteelement schemes, are also considered. The analysis is accompanied by examples of univariate and bivariate problems.

Iterative solution of two matrix equations

We study iterative methods for finding the maximal Hermitian positive definite solutions of the matrix equations X + A * X -1 A = Q and X - A * X -1 A = Q, where Q is Hermitian positive definite. General convergence results are given for the basic fixed point iteration for both equations. Newtons method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation. Numerical results are reported to illustrate the convergence behaviour of various algorithms.

Existence and uniqueness theorems for the algebraic Riccati equation

Abstract Necessary and sufficient conditions for existence and uniqueness of hermitian solutions of the algebraic n×n matrix Riccati equation (D≥0,C∗=C,(A, D) controllable) are obtained. The conditions are formulated in terms of the spectral structure of a certain 2n × 2n matrix. A description is also given of the set of solutions in a geometrical language of invariant subspaces which are neutral with respect to a certain indefinite scalar product. This technique is then applied to provide some results on existence and uniqueness of solutions which are not necessarily hermitian. The problem is also approached (when Dz;> 0)via a related unilateral equation. for Z where K1 ∗ equals K1, K0 ∗ equals K0

Derivatives of eigenvalues and eigenvectors of matrix functions

For an

Linear matrix equations from an inverse problem of vibration theory

n \times n

Canonical Forms for Hermitian Matrix Pairs under Strict Equivalence and Congruence

matrix-valued function

Spectral analysis of matrix polynomials— I. canonical forms and divisors

L( {\boldsymbol \rho} ,\lambda )

Complex interval arithmetic

, where

Factored forms for solutions of AX − XB = C and X − AXB = C in companion matrices

{\boldsymbol \rho}

Linear complexity parallel algorithms for linear systems of equations with recursive structure

is a vector of independent parameters and