This paper reviews the Kalman (or H_2) filter and the recent development of H_∞ filter, with a preliminary remark on earlier mathematical works related to the least-squares estimation. For the linear Gaussian state-space model, it is shown that the Kalman filter is a dynamical system that produces the least-squares estimates of the states given observations and that the Kalman gain is determined by the associated Riccati equation. We then show that the H_∞ filtering problem is a minimax estimation problem that minimizes the energy of estimation errors maximized over all possible disturbances and initial states. A comparison of the Kalman filter and the H_∞ filter is made in frequency domain to illustrate the basic feature of two filters. After a brief introduction of Krein space, we also show that the H_∞ (central) filter is the Kalman filter for a Krein space state-space model with indefinite noise Gramian.

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Numerical parallel processing has been a matter of concern for a considerable time. Nevertheless, general-purpose parallel computing is still difficult to many end-users, because programming for parallel processors is not an easy task. This survey article indicates what mainly causes the impediments on one hand, and what patterns of numerical computation have been successful on the other, in order to have substantial parallel speedup.

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After touching upon methods of acquiring data to be fitted, an outline of splines and their important algorithms are explained. Then, the problem of data fitting with a spline is formulated. From the viewpoint of obtaining fair or aesthetic curves, some historical descriptions on interpolation and approximation using splines are given. It is shown that location of good knots is an important issue in the data fitting using splines, and representative approaches for determining good knots are shown. Finally, a brief description of reverse engineering is given, and an effective example of data fitting in reverse engineering is introduced.

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