Interpolation cubic spline

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 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.

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

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. 

Figure l. Observed CFSTR flow rates during the course of the desorption experiments for selected CFSTRs. For the 30 mg/L Tween 20 CFSTR. the cubic spline interpolation of the flow rate variations is shown. 

Fig. 4. (A) Parametrization of a shaped pulse envelope with the help of a cubic spline interpolation between a small number of anchor points (circles). (B) Approximation of the smooth pulse envelope by rectangular pulses with piecewise constant rf amplitude. (Adapted from Ewing et al., 1990, p. 123, with kind permission from Elsevier Science—NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)...

Fig. 7.4 Phonon dispersion of graphite from inelastic X-ray scattering along the in-plane high symmetry directions (full circles). The dashed line is a cubic-spline interpolation to the TO-derived branch. The open circles show an ab-initio calculation for graphene, downscaled by 1% (Reprinted with permission from Maultzsch J, Reich S, Thomsen C, Requardt H, Ordejon P (2004) Phonon dispersion in graphite. Physical Review Letters 92 (7) 075501-1-4. Copyright (2004) by the American Physical Society)...

The unpredictable shifts of the pure component peaks respect to the mixtures, mainly due to the dififerent surface charging of the samples, made it necessary to find the optimal alignment of the peak doublets, before determining the percentage composition of the components in the mixtures through the Kalman filter. The smoothed Pb 4f peaks of each pure component were used in the filter model after background ranoval from the peak doublets by using the cubic splines interpolation. Also the unresolved Pb 4f peaks relative to the mixtures... 

Figure 16 The result of applying a cubic spline interpolation model to the stream magnesium data...

Fig. 6.24. Gas-phase reaction coordinates for addition of hydroxide ion to formaldehyde (lower curve) and of hydrosulfide ion to formaldehyde (upper curve). The curves were constructed by using a cubic spline interpolation of the calculated ab initio points. Both reaction coordinates were calculated at the RHF/4-3IG//RHF/4-31G ab initio level...

Consider a countercurrent mass-transfer device for which the equilibrium-distribution relation consists of a set of discrete values XiD,YiD instead of a continuous model such as Henry s law. An analysis to determine minimum flow rates similar to that presented in Problem 3.14 is still possible using the cubic spline interpolation capabilities of Mathcad. Cubic spline interpolation passes a smooth curve through a set of points in such a way that the first and second derivatives are continuous across each point. Once the cubic spline describing a data set is assembled, it can be used as if it were a continuous model relating the two variables to predict accurately interpolated... 

Calculate the number of transfer units using cubic spline interpolation and numerical integration... 

A cubic spline interpolation formula is generated with these data, which is then integrated according to equation (6-13) to determine the residue concentration, x when... 

Write a Mathcad program to calculate the minimum reflux ratio under these conditions and test it with the data of Example 6.4. In this case, the equilibrium-distribution relationship is a table of VLE values therefore, an algebraic relationship of the form y =J x) must be developed by cubic spline interpolation. 

Using the data from Table 7.1, generate a cubic spline interpolation formula for the saturated raffinate curve, one for the saturated extract curve, and one for the tie-line data (see the Mathcad program in Appendix G-l). Equations in the form... 

Because of the time curves of gas emission is not very complex (He et al. 2008), so in strike the upper and lower envelope, the study used cubic spline interpolation. In addition, the data mirror extension method is applied to eliminate end effect (Lee et al. 2009). 

The overpotential functions on anode and cathode were natural cubic spline interpolations of measured values taken from J. 0 M Bockris et al. [ 17]. They are repeated in table 3.6. 

The linear algebraic system consists of 3n - 3 equations and 3n - 3 unknowns that can be solved to produce the optimal piecewise cubic spline. Press, Teukolsky, Vetterling, and Flannery describe a routine for cubic spline interpolation. " ... 

The outlined model carefully preserves variations in market observations, thereby maintaining important pricing information. However, linear interpolation can introduce inaccuracies when there is significant curvature in the term structure, or sparse or noisy data. Cubic spline interpolation, on the other hand, may produce inconsistent or implausible forward term structures. 

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