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Experimental Curves and Rates of Change from Piecewise Parabolic Fits¹

The determination of experimental curves and rates of change for nonlinear data is approached and solved without assuming an artificially restrictive mathematical form for the complete range of the data. This relatively form-free result is achieved by least-squares computer fitting of parabolic segments to short subranges of the experimental data. Two ways of doing this, referred to as the sliding-parabola and the parabolic-splines methods, are developed. These are tested on both smooth and scattered data generated basically from the function y = X^1/2, without and with random error, respectively. for smooth data the sliding-parabola method is slightly better than the parabolic splines, but in general both are subject to only very small errors, and are also in good agreement with a previously presented graphical prism method. For scattered data wherein the inherent errors of function and slope evaluation are much increased, the parabolic-splines method is distinctly superior to the sliding parabola method. Both methods require only relatively short computing times, on the order of 1 sec for 40 data points, and are of utility for determining nonconstant experimental rates of change and for least-squares curve fitting without specification of a complete-range mathematical curve.

Key Words: Form-free curve fitting • Spline • Least squares

² Student Assistant (now Assistant Professor of Mathematics, University of Kentucky, Lexington); Graduate Research Assistant; Post-doctoral Research Associate; and Professor of Soils, respectively.

Received for publication September 2, 1971.