The approximation of a hnction by a sum of complex exponentials is a problem that is at least two centuries old. Fundamentally, all techniques discussed in this article proceed from using the same sequence of data samples and vary only, but importantly, in how those samples are used in achieving the parameter estimation. All of these techniques, in other words, seek the same quantitative parameters to represent the sampled data, but use different routes to get there. The techniques for estimating the parameters are either linear or nonlinear. The linear techniques are emphasized in this presentation. In particular, the Matrix Pencil Method is described, which is more robust to noise in the sampled data. The Matrix Pencil approach has a lower variance of the estimates of the parameters of interest than a polynomial-type method (Prony's method belongs to this category), and is also computationally more efficient. A bandpass version of the Matrix Pencil can be implemented in hardware, utilizing an AT&T DSP32C chip operating in real time. A copy of the computer program implementing the Matrix Pencil technique is given in Appendix.

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In this paper an approach to the formulation of equilibrium elements for the analysis of three-dimensional elasticity problems is presented.This formulation is an extension of the approach previously proposed l S z for the analysis of two-dimensional elasticity problems. The general aspects of the formulation remain unchanged when applied to the new problem, but new points are considered, namely the way to perform volume integrations for general elements and the techniques used to obtain the self-equilibrated three-dimensional stress approximation functions.The numerical behaviour of such elements is presented and discussed. KEY WORDS finite elements; three-dimensional elasticity; equilibrium formulations; hybrid finite elements(1) by direct approximation of the stress field in the elements;(2) by approximation of a stress potential (the Airy function in the 2-D case).The first approach was initially used by de Veubeke for 2-D elasticity. Because it is possible to obtain linearly dependent systems of equations in these formulations, most of the research in this field has been devoted to the search of alternative ways of controlling or eradicating the corresponding spurious This approach, to the authors' knowledge, has never been used for three-dimensional elasticity problems.The second approach, used by de Veubeke and Zienckiewicz* for 2-D elasticity, is suitable for Neumann problems, when the kinematic boundary does not exist, but is difficult to use whenever constraints on displacements are imposed and has been the subject of limited attention.

SUMMARYEquilibrated solutions, locally satisfying all the equilibrium conditions, may be obtained by using a special case of the hybrid finite element formulation. Unlike simplicial super-elements in 2D and 3D, which are free from spurious kinematic modes, and the quadrilateral super-element with diagonal subdivision, which has exactly one internal spurious kinematic mode, no hexahedral super-element free of external spurious kinematic modes is yet known as far as the author is aware.