Interpolation spline

The most common spline used in engineering problems is the cubic spline. In this method, a cubic polynomial is used to approximate the curve between each two adjacent base points. Because there would be an infinite number of third-degree polynomials passing through each pair of points, additional constraints are necessary to make the spline unique. Therefore, it is set that all the polynomials should have equal first and second derivatives at the base points. These conditions imply that the slope and the curvature of the spline polynomials are continuous across the base points. 

The cubic spline of the interval [. ( jcj has the following general form 

There are four unknown coefficients in Eq. (3.138) and n such polynomials for the whole range of data points [jr, x ]. Therefore, there are 4n unknown coefficients and we need 4n equations to evaluate these coefficients. The required equations come from the following conditions  

Each spline passes fi om the base points of the edge of its interval (2n equations). The first derivative of the splines are continuous across the interior base points (n - 1 equations). 

The second derivative of the splines are continuous across the interior base points 

The command v = polyval(p, w) evaluates the polynomial p at the w points (w can be a vector). Plot the data points (y) and the polynomial (i ) 

Plotting Results from Integration of Partial Differential Equations 

The result is the value of the solution at several fixed positions and all times  

This gives a plot of each variable y(t, i) vs time. Here, y t, i) is the solution at point x i) and time t. To plot the solution vs x(j), at various times, specify that you want the solution at a select number of times  

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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.

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

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. 

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. 

We consider a square area as shown in Figure 11. This area is divided into 40,000 square elements. When summing the areas of the square elements in the circle in Figure 11a, the value obtained differs from the circular area by 0.1%. The locations of transducers 1, 2, and 3 and the locations of the gas-liquid interface heights are X, x 2, and x3 and Si( i,i/i), s2(x2,y2), and s3( 3,1/3), respectively. In the range x, free surface is calculated by a cubic polynomial function (spline interpolation), as shown by Kreyszig (1999). In the ranges 0 < x < Xj and x3 liquid interface is calculated by a linear extrapolation. The slope of... 

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