Spline interpolating splines

Halang, W. A., Langlais, R., and Kugler, E., Cubic Spline Interpolation for the Calculation of Retention Indices in Temperature-Programmed Gas-Liquid Chromatography, Ana/. Chem. 50, 1978, 1829-1832. 

Figure 3.7 Spline interpolation of the chondrite normalized Ce/Yb ratio in recent lava flows of the Piton de la Fournaise volcano (Albarede and Tamagnan, 1988). The end derivatives are supposed to be zero.

The calculation can be made for an arbitrary number of points provided their abscissa lie inside the range of x values. Figure 3.7 shows the characteristic features of spline interpolation, a very smooth aspect although with some overshooting problems, i.e., extrema located between the data points. Alternative interpolation schemes are discussed by Wiggins (1976). o... 

Sandwell, D. T. (1987). Biharmonic spline interpolation of GEOS-3 and SEASAT altimeter data. Geophys. Res. Letters, 2, 139-42. 

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. 

Figure 13.11. Dissolved oxygen profile during a typical Escherichia coli fermentation. Using a cubic spline interpolation of the data shown in Figure 13.3, the responses of the two sensors have been converted into percent oxygen and plotted as shown. The optical sensor closely tracks the response of the Clark-type electrode throughout the fermentation.

If every calibration point (with the exception of replicates that can be treated by averaging their response values) is treated as a separate knot, two different situations can be distinguished. In case of very precisely defined response values, y., obtained in practice by a high number of replicates in presence of small random errors, it is possible to use interpolating splines. Presumbly, the more frequent case envisaged will be the one, where relatively few data points whose random errors are not negligible and/or that are not highly replicated span the concentration (or mass) domain. 

Obviously, S = 0 for a straight line. If S is small for a given function, it indicates that f does not wildly oscillate over the interval [xi,xn] of interest. It can be shown that among all functions that are twice continuously differentiable and interpolate the given points, S takes its minimum value on the natural cubic interpolating spline (ref. 12). 

It remains to calculate the coefficients that define the interpolating spline. One can obviously solve the 4(n-l) constraint equations directly, but there exists a much more efficient algorithm. Let and mi+1 denote the... 

Example 4.2.1 Enthalpy and heat capacity by spline interpolation... 

The additional problem we face is determining the optimal value for p. It is important to note that the squared distance F2(p) increases with the value of p. Therefore, the algorithm can be viewed as starting with an interpolating spline obtained at p = 0, and then "streching" this function by gradually increasing the value of p until (4.27) holds. To find this particular p we solve the nonlinear equation /2... 

Using inverse linear interpolation the two titration equivalence points are obtained as the zero-crossing points of the second derivative at V = 3.78 ml and V = 7.14 ml. On Fig. 4.4 the second derivative curve of the interpolating spline (SD = ) and that of the smoothing spline (SD = 8.25) are shown. The false zero-crossing of the second derivative present at interpolation is eliminated by smoothing. 

Spline interpolation is a global method, and this property is not necessarily advantageous for large samples. Several authors proposed interpolating formulas that are "stiffer" than the local polynomial interpolation, thereby reminding spline interpolation, but are local in nature. The cubic polynomial of the form... 

Interpolate the titration curve implementing Akima s method. Compare the interpolating curve with the results of local cubic interpolation and spline interpolation. 

Let S (t) denote the ny-vector of natural cubic splines interpolating the... 

Since spline interpolation and integration is mucht faster than solving the sensitivity equations and the original differential equations, the direct method is superior to the indirect one in terms of numerical efficiency, whenever it is feasible. 

In spite of its simplicity the direct integral method has relatively good statistical properties and it may be even superior to the traditional indirect approach in ill-conditioned estimation problems (ref. 18). Good performance, however, can be expected only if the sampling is sufficiently dense and the measurement errors are moderate, since otherwise spline interpolation may lead to severely biased estimates. 

The following program module is a modification of the nonlinear least squares module M45. Because of spline interpolation and differential equation solution involved it is rather lengthy. 

The values for 12 laboratories are missing from the original tables These values are obtained from the other tabulated data via cubic spline interpolation. 

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. 

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