The speed of convergence of stationary iterative techniques for solving simultaneous linear equations may be increased by using a method similar to conjugate gradients but which does not require the stationary iterative technique to be symmetrisable. The method of refinement is to find linear combinations of iterates from a stationary technique which minimise a quadratic form. This basic method may be used in several ways to construct refined versions of the simple technique. In particular, quadratic forms of much less than full rank may be used. It is suggested that the method is likely to be competitive with other techniques when the number of linear equations is very large and little is known about the properties of the system of equations. A refined version of the Gauss-Seidel technique was found to converge satisfactorily for two large systems of equations arising in the estimation of genetic merit of dairy cattle.

Consider the prototype ill-posed problem of a first kind integral equation ℛ with discrete noisy data di, = f(xi) + εi, i = 1, …, n. Let u0 be the true solution and unα a regularised solution with regularisation parameter α. Under certain assumptions, it is known that if α → 0 but not too quickly as n → ∞, then unα converges to u0. We examine the dependence of the optimal sequence of α and resulting optimal convergence rate on the smoothness of f or u0, the kernel K, the order of regularisation m and the error norm used. Some important implications are made, including the fact that m must be sufficiently high relative to the smoothness of u0 in order to ensure optimal convergence. An optimal filtering criterion is used to determine the order where is the maximum smoothness of u0. Two practical methods for estimating the optimal α, the unbiased risk estimate and generalised cross validation, are also discussed.

We consider a generalisation of the stochastic formulation of smoothing splines, and discuss the smoothness properties of the resulting conditional expectation (generalised smoothing spline), and the sensitivity of the numerical algorithms. One application is to the calculation of smoothing splines with less than the usual order of continuity at the data points.

The square-root fixed-interval discrete-time smoother has been used extensively in discrete recursive estimation since it was first developed by Rauch, Tung and Streibel [10]. Various people, for example Bierman [2], [3], have recognized the inherent instability in employing this kind of smoother in its original form; they have investigated implementing the recursion more stably. Bierman's paper [3] is one such contribution. In this paper we plan to present a more comprehensive development of Bierman's approach, and to show that this algorithm can be implemented more stably as a square-root smoother. Throughout this paper the fixed-interval discrete-time smoother will be referred to as the RTS smoother. Numerical results are given for the usual form of the RTS smoother, Bierman's algorithm and our square-root formulation of his algorithm. These confirm that the square-root formulation is more desirable than Bierman's algorithm, which performs better than the usual implementation of the RTS smoother.

The state controllability for generalised dynamical systems with constrained control is discussed in this paper. The main results of the paper are the following:

(1) a necessary and sufficient condition of the state controllability in the sense of control energy or amplitude constrained for generalised dynamical systems is obtained;

(2) a control function u(t) is constructed such that

a) u(t) satisfies constrained energy or amplitude condition,

b) the state driven by u(t) moves from an arbitrary x(0−) = x0 to x(T(x0)) = 0,

c) the trajectory driven by u(t) has no impulsive behaviour within (0, T(x0)].

The behaviour of duopolists is considered within a framework that allows for flexibility of the adopted strategy against the rival. In a difficult external climate, a firm may concentrate on its own profit, whereas in a more favourable external climate, it may adopt a more aggressive attitude towards the rival. The strategy considered in this paper permits this flexible approach. The market functions are kept general to allow the widest interpretation of the results.

The modelling of the combustion of dust suspensions leads to a nonlinear eigenvalue problem for a system of ordinary differential equations defined over an infinite interval. The equations contain a number of parameters. In this study, the shooting method is used to prove the existence of a solution. Linearisation is then used to provide an approximate solution, from which an estimate of the eigenvalue and its dependence on the given parameters can be obtained.

A method based on the method of images is described for the solution of the linear equation modelling diffusion and elimination of substrate in a fluid flowing through a chemical reactor of finite length, when the influx of substrate is prescribed at the point of entry and Danckwerts' zero-gradient condition is imposed at the point of exit. The problem is shown to be transformable to an equivalent problem in heat conduction. Associated with the images appearing in the method of solution is a sequence of functions which form a vector space carrying a representation of the Lie group SO(2, 1) generated by three third-order differential operators. The functions are eigenfunctions of one of these operators, with integer-spaced eigenvalues, and they satisfy a third-order recurrence relation which simplifies their successive determination, and hence the determination of the Green's function for the problem, to any desired degree of approximation. Consequently, the method has considerable computational advantages over commonly used methods based on the use of Laplace and related transforms. Associated with these functions is a sequence of polynomials satisfying the same third-order differential equation and recurrence relation. The polynomials are shown to bear a simple relationship to Laguerre polynomials and to satisfy the ordinary diffusion equation, for which SO(2, 1) is therefore revealed as an invariance group. These diffusion polynomials are distinct from the well-known heat polynomials, but a relationship between them is derived. A generalised set of diffusion polynomials, based on the associated Laguerre polynomials, is also described, having similar properties.

A Demianski-type metric is investigated in connection with the Einstein-Maxwell fields. Using complex vectorial formalism, some exact solutions of Einstein-Maxwell field equations for source-free electromagnetic fields plus pure radiation fields are obtained. The radiating Demianski solution, the Debney-Kerr-Schild solution and the Brill solution are derived as particular cases.