Cubic spline method

Several basis schemes are used for very-high-accuracy calculations. The highest-accuracy HF calculations use numerical basis sets, usually a cubic spline method. For high-accuracy correlated calculations with an optimal amount of computing effort, correlation-consistent basis sets have mostly replaced ANO... 

M65 Determination of smoothing cubic spline method of C.H. Reinsch 6500 6662... 

Figure 1. Titration curves for monovalent cation-hydrogen exchange. (A) Na at 0.11 (B) Na at 0.3 I (C)Na at 0.5 I (D) K" at 0.11. Every third data point is shown. Interpolation is by a cubic spline method.

Thermodynamic properties for As( ,cr) were calculated from a least-squares representation using a cubic-spline method described previously (Archer, 1992 Archer et al., 1996). Briefly, a function f(T) was used, where ... 

THE CUBIC SPLINE METHOD FOR ESTIMATING AND FITTING THE YIELD CURVE... 

FIGURE 5.3 Yield curves fitted using cubic spline method and Svensson parametric method, hypothetical bond yields. Reproduced with permission from the Bank of England Quarterly Bulletin, November 1999.)... 

Fitting forms obtained from other motivations such as standard cubic-spline methods. Morse-spline and rotated Morse-spline interpolation methods, reproducing kernel Hilbert space interpolation methods, distributed approximating functionals, and hybrid methods combining spline fits with simple empirical functions. ... 

In addition, the program used for data smoothing with cubic splines for the shortcut methods are given for the example ... 

In the present study we have extracted the EXAFS from the experimentally recorded X-ray absorption spectra following the method described in detail in Ref. (l , 20). In this procedure, a value for the energy threshold of the absorption edge is chosen to convert the energy scale into k-space. Then a smooth background described by a set of cubic splines is subtracted from the EXAFS in order to separate the non-osciHatory part in ln(l /i) and, finally, the EXAFS is multiplied by a factor k and divided by a function characteristic of the atomic absorption cross section (20). 

Differences in calibration graph results were found in amount and amount interval estimations in the use of three common data sets of the chemical pesticide fenvalerate by the individual methods of three researchers. Differences in the methods included constant variance treatments by weighting or transforming response values. Linear single and multiple curve functions and cubic spline functions were used to fit the data. Amount differences were found between three hand plotted methods and between the hand plotted and three different statistical regression line methods. Significant differences in the calculated amount interval estimates were found with the cubic spline function due to its limited scope of inference. Smaller differences were produced by the use of local versus global variance estimators and a simple Bonferroni adjustment. 

Wegscheider fitted a cubic spline function to the logarithmically transformed sample means of each level. This method obviates any lack of fit, and so it is not possible to calculate a confidence band about the fitted curve. Instead, the variance in response was estimated from the deviations of the calibration standards from their means at an Ot of 0.05. The intersection of this response interval with the fitted calibration line determined the estimated amount interval. 

REN DETERMINATION OF SH00THIN6 CUBIC SPLINE I 6504 REN METHOD OF . H. REINSCH t... 

A method for interpolation of calculated vapor compositions obtained from U-T-x data is described. Barkers method and the Wilson equation, which requires a fit of raw T-x data, are used. This fit is achieved by dividing the T-x data into three groups by means of the miscibility gap. After the mean of the middle group has been determined, the other two groups are subjected to a modified cubic spline procedure. Input is the estimated errors in temperature and a smoothing parameter. The procedure is tested on two ethanol- and five 1-propanol-water systems saturated with salt and found to be satisfactory for six systems. A comparison of the use of raw and smoothed data revealed no significant difference in calculated vapor composition. 

The time-resolved measurements were made using standard time-correlated single photon counting techniques [9]. The instrument response function had a typical full width at half-maximum of 50 ps. Time-resolved spectra were reconstructed by standard methods and corrected to susceptibilities on a frequency scale. Stokes shifts were calculated as first moments of cubic-spline interpolations of these spectra. 

By solving these equations using the method proposed by Mauri[10], a and (3 can be easily calculated, then the discrete components compositions and continuous fraction distribution in the outlet can be obtained. In the calculation, the distribution function was calculated by a cubic spline fit method by Ying [9],... 

The answer to this difficulty lies in the use of piecewise approximants, such as cubic splines, which are in general use in the mathematics literature (11). Carey and Finlayson (12) have introduced a finite-element collocation method along these lines, which uses polynomial approximants on sub-intervals of the domain, and apply continuity conditions at the break-points to smooth the solution. It would seem more straight-forward, however, to use piecewise polynomials which do not require explicit continuity... 

A common method of extracting f K) from Eq. 3.82 is to assume a form of the distribution function by differentiation of a smooth fimction describing the data. The function obtained by this method is called the affinity spectrum (AS) and the method, the AS method [71]. The most general approach uses a cubic spline to approximate the data. However, a simpler procedure uses a Langmuir-Freundlich (LF) isotherm model and the AS distribution is derived from the best parameters of a fit of the experimental isotherm data to the LF model [71]. This approach yields a unimodal distribution of binding affinity with a central peak, if the range... 

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