Spline functions

The analytic PES function is usually a summation of two- and three-body terms. Spline functions have also been used. Three-body terms are often polynomials. Some of the two-body terms used are Morse functions, Rydberg... 

The best fit, as measured by statistics, was achieved by one participant in the International Workshop on Kinetic Model Development (1989), who completely ignored all kinetic formalities and fitted the data by a third order spline function. While the data fit well, his model didn t predict temperature runaway at all. Many other formal models made qualitatively correct runaway predictions, some even very close when compared to the simulation using the true kinetics. 

These functions are truncated and shifted to zero at a cutoff-distance between the third and fourth nearest neighbor shell. N ai is the number of valence electrons and U4s is a parameter. Following Daw and Baskes further on we use cubic spline functions to represent the functions and Z(r). The splii s have been fitted to... 

In a high-order polynomial, the highly inflected character of the function can more accurately be reproduced by the cubic spline function. Given a series... 

L. L. Schumaker 1981, Spline Functions Basic Theory, John Wiley and Sons, New York. 

Wold S (1974) Spline functions in data analysis. Technometrics 16 1... 

To circumvent this, the function A can be approximated using spline functions... 

E. B. Smith, The Virial Coefficients of Pure Gases and Mixtures, Clarendon Press, Oxford, 1980. By interpolation (e.g., with cubic spline functions), virial coefficients can be determined for any temperature. 

The utility of spline functions to molecular dynamic studies has been tested by Sathyamurthy and Raff by carrying out quassiclassical trajectory and quantum mechanical calculations for various surfaces. However, the accuracy of spline interpolation deteriorated with an increase in dimension from 1 to 2 to 3. Various other numerical interpolation methods, such as Akima s interpolation in filling ab initio PES for reactive systems, have been used. 

Wold, S. Kemisk Tidskrift 3, 1972, 34—37. Spline functions, a new tool in data analysis. 

Craven, P. and Wahba, G., Smoothing noisy data with spline functions, Num. Math., 31, 377 03, 1979. 

There are a number of ways to model calibration data by regression. Host researchers have attempted to describe data with a linear function. Others ( 4,5 ) have chosen a higher order or a polynomial method. One report ( 6 ) compared the error in the interpolation using linear segments over a curved region verses using a curvilinear regression. Still others ( 7,8 ) chose empirical or spline functions. Mixed model descriptions have also been used ( 4,7 ). 

Use of Cubic Spline Functions in Solving Calibration Problems... 

The analytical exploitation of the full dynamic range of a detection principle invariably encompasses nonlinear portion of the concentration response function. The use of cubic spline functions for the description of this relationship is discussed after a short introduction to the theoretical principles of spline approximations. 

It is the aim of this contribution to review the principles of the very flexible curve fitting approach by cubic spline functions for construction of concentration/response curves, to demonstrate the applicability of this methodology for several data sets relevant in pesticide residue analysis... 

Functions expressing physical relationship are frequently of a disjointed or disassociated nature- their behavior in one region may be totally unrelated to their behavior in another region. Polynomials and most other mathematical functions have just the opposite property. The same is not true for spline functions. 

We are trying to determine the curve S, called spline function, with... 

Calibration graphs defined by data with non-negligible error have to be constructed by some kind of smoothing operation. In cases, in which the form of the underlying curve is known a priori, the latter can be approximated by minimizing the squares of deviations. Otherwise a spline function can be used (JJ[, 1 ). The spline function S(x) is constructed to minimize a measure of smoothness defined by... 

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