LCB interpolation

Although it is certainly not an obvious conclusion, the use of these derivative evaluations in Eq. (6.10) through (6.12) will give exactly the same cubic interpolation equation as previously given by the LCB approach of Eq. (6.7). How can one know that this is the case Well they are both cubic polynomials passing through the same data points. Also they have the same identical first derivatives at the end data points. This gives four identical conditions for the cubic polynomials and a cubic polynomial has only 4 parameters. Thus they must be the same cubic polynomial equations. Thus the previous LCB interpolation technique should be more accurately called a cubic Hermite interpolation with end point derivative values evaluated by a three point numerical derivative . For the sake of simplicity the term local cubic or LCB will be used to describe the approach since the cubic equation is determined by the data points at the ends of the interval and the two adjacent data points. 

Examples of the use of the LCB interpolation function will be given following the development of the cubie spline approach in the next section. This is a second general approach to a cubic data interpolation polynomial. 

Finally, Figures 6.21 and 6.22 show interpolation functions applied to two sets of experimental data, one for the vapor pressure of water vs. temperature (lines 20 and 21 of listing) and the other for thermoeouple voltage vs. temperatures (lines 23 and 24 of listing). Both sets of data show very smooth variations and the CSPl and LCB interpolation functions fall essentially on top of each other. Either teeh-nique could be used equally successfully with these sets of experimental data. Interpolating experimental data such as these sets is one of the most important applications of interpolation. 

Figure 7.23. Illustration of least squares fitting with Local Cubic Function (LCB) interpolation.

The local cubic function can apply to interior data points, but how about the first and last intervals where there are not four surrounding data points For these intervals, the best that can be done is to use a single fitting quadratic function over the first three or last three data points. Thus an LCB algorithm must check for the first and last interval and use a single quadratic there and use the weighted quadratic or local cubic function for interior points. If the data has only two data points, the best that can be done is to fall back on linear interpolation and if there are only three data points, then only a single quadratic function can be used. 

An important feature of the LCB algorithm is continuity of the first derivative of the interpolation function. A serious limitation of the linear interpolation algorithm is the abrupt change in slope of the approximating function at each tabular data point. It is certainly highly desirable to have an interpolation function with a continuous derivative (or perhaps derivatives). From Equation (6.7) the derivative can be expressed as ... 

This means that the first derivative of the LCB polynomial is continuous across the data points, i.e. has the same value on each side of a data point and it is continuous within an interval. Thus the LCB algorithm provides a cubic interpolating... 

Listing 6.3. Code for Local Cubic Function interpolation (LCB). 

Figure 6.4. Comparison of the yj (x) funetion with LCB and CSPl interpolated values.

Larger differences are expeeted to be seen in the seeond derivative of the interpolating functions and this is shown in Figures 6.8 and 6.9. It is seen that the second derivative of the LCB and LCB4 funetions shows abrupt ehanges at the table data points while the CSPl and CSP2 funetions shows a continuous second derivative. All of the interpolation funetions provide a reasonably good fit to the... 

Figure 6.6. First derivative of/j(x) compared with LCB and CSPl interpolated derivative values.

Figure 6.8. Second derivative of/J(x) compared with values from the LCB and CSPl interpolated functions.

For most applications, both techniques give very good interpolation results. When fitting the interpolations to known functions, sometimes the LCB technique produces smaller RMS errors and sometimes the CSP technique produces smaller RMS errors. In terms of computer resources, for a single point interpolation, the LCB technique is faster since it does not require the solution of a set of coupled equations. For calculations at many data points, both techniques are somewhat comparable, since the coupled equations need only be solved once for a given table of data and the equation coefficients can be stored for subsequent calculations. 

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