Univariate Interpolation on a Regular Finite Grid by a Multiquadric Plus a Linear Polynomial

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Abstract
Univariate multiquadric interpolation to a twice continuously differentiable function on a regular infinite grid enjoys second order convergence and some excellent localization properties, but numerical calculations suggest that, if the grid is finite, then usually the convergence rate deteriorates to first order near the grid boundaries, ibis conjecture is proved. It is also shown that one can recover superlinear convergence by adding a linear polynomial term to the multiquadric approximation. Making such additions is a standard technique, but we find that the usual way of choosing the polynomial fails to provide superlinear convergence m general. Therefore some new procedures are given that pick a suitable polynomial automatically. Thus it is not unusual to reduce the maximum error of the interpolation by a factor of 10 3 . Further, it is straightforward to include one of the new procedures in multiquadric interpolation to functions of several variables when the data points are in general position.

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