The determination of derivative parameters for a monotonic rational quadratic interpolant

Explicit formulae are developed for determining the derivative parameters of a monotonic interpolation method of Gregory and Delbourgo (1982)

Rational quadratic spline interpolation to monotonic data.

In an earlier paper by Gregory & Delbourgo (1982), a piecewise rational quadratic function is developed which produces a monotonic interpolant to monotonic data. This interpolant gives visually pleasing curves and is of continuity class C1 . In the present paper, the data is restricted to be strictly monotonic and it is shown that it is possible to obtain a monotonic rational quadratic spline interpolant which is of continuity class .C2 An ○(h4) convergence analysis is included

An arbitrary mesh network scheme using rational splines

A C1 surface scheme is described which interpolates points defined on an arbitrary mesh network. The scheme involves the blending of strip ‘functions’ developed from a rational spline method. The rational spline provides interval and point tension weights which can be used to control the shape of the surface scheme

An introduction to regular splines and their application for initial value problems of ordinary differential equations

This report describes an application of the general method of integrating initial value problems by means of regular splines for equations with movable singularities. By defining the families of functions that make up the regular splines such that they closely resemble the behaviour of the solutions of the differential equation, it is possible to trace the location of the singularities very precisely. To demonstrate this we treat Riccati differential equations. These are known to possess solutions with poles, usually of the first order. This type of differential equation or system arises in describing chemical or biological processes or more general control processes. To make the report self contained it starts with an introduction to regular splines and develops the algebraic tools for the manipulation of rational splines. After the description of the integration procedure, the asymptotic behaviour of the systematic error is investigated. An example exhibits the results obtained from the program given in Appendix A. Then Riccati equations are introduced and methods for the determination of the singularities are developed. These methods are tested numerically with several examples. The results are given in Appendix B

Piecewise rational quadratic interpolation to monotonic data

An explicit representation of a piecewise rational quadratic function is developed which produces a monotonic interpolant to given monotonic data. The explicit representation means that the piecewise monotonic interpolant is easily constructed and numerical experiments indicate that the method produces visually pleasing curves. Furthermore, the use of the method is justified by an 0(h4) convergence result

Biorthogonal partners and applications

Two digital filters H(z) and F(z) are said to be biorthogonal partners of each other if their cascade H(z)F(z) satisfies the Nyquist or zero-crossing property. Biorthogonal partners arise in many different contexts such as filterbank theory, exact and least squares digital interpolation, and multiresolution theory. They also play a central role in the theory of equalization, especially, fractionally spaced equalizers in digital communications. We first develop several theoretical properties of biorthogonal partners. We also develop conditions for the existence of biorthogonal partners and FIR biorthogonal pairs and establish the connections to the Riesz basis property. We then explain how these results play a role in many of the above-mentioned applications