Parabolic interpolation

Smoothed data presented at rounded temperatures, such as are available in Tables 6.2 and 6.4, plus the C° values at 298 K listed in Table 6.1 and 6.3, are especially suitable for substitution in the foregoing parabolic equations. The use of such a parabolic fit is appropriate for interpolation, but data extrapolated outside the original temperature range should not be sought. 

Parabolic interpolation is more effective than golden section search for this problem, because the function is of parabolic character in the vicinity of the minimum. To show a counterexample we slightly change the approximate objective function (2.19) and define by... 

The interpolated curves are given as horizontal lines in Figure 3, and the Kc/Re values at constant time are given by A. The vertical parabolic curves result at their lowest end in the lime o Kc/Re values, given by . 

Fitting to Scatter Outside the Object In this method, the scatter is estimated by fitting an analytic function (Gaussian or parabolic) to the activity outside the source and interpolating the function to the source. The interpolated scatter contributions are then subtracted from the measured source counts to obtain scatter-corrected data for reconstruction of PET images. This method is based on the assumption that (1) the events outside the source are only scatter events, (2) the scatter distribution is a low-frequency function across the FOV, and (3) they are independent of activity distribution in the source. 

Reconsider Ex. 43. Use a parabolic interpolation for the initial temperatures. Comment on the number of iteration steps. 

Owing to the size of the molecules considered in this paper, it would have been impractical to optimize fully the geometry of each molecule. Instead we adopted a set of standard geometries (Table I) which were chosen to approximate the ST0-3G equilibrium values. Thus, all bond angles and C-H bond lengths were fixed at standard values, but the C-C bond lengths were optimized fully in many cases by parabolic interpolation (14), Total energies obtained in the calculations are reported in Appendix 1, whereas the experimental heats of formation... 

To calculate y(x) at any point x, Eq. 11.9 uses three points Xo,x,X2 with X() < X < X2, and is called the Lagrange three-point interpolation formula. The three-point formula amounts to a parabolic representation of the function y(x) between any three points. 

The discretization of the ordinary differential equation, Eq. (36), and of the two mentioned boundary conditions leads finally to a complete linear equation system whose inhomogeneity results from the discretized normalization condition, Eq. (35). An efficient resolution of this system becomes possible if those terms obtained by the parabolic interpolation are iteratively treated in the resolution procedure. 

The parabolic interpolation method is never used in the BzzMaA hbrary classes dedicated to root-finding. 

Parabolic interpolation usually converges faster than comparison methods in a ( ) sequential search. 

If it has two real solutions and (2, it is necessary to check the second derivative 6o3t y 2u2 in correspondence with them. If this is positive and if the solution is within the interval of uncertainty, this point can be used as a prediction of the minimum. When none of these points is satisfactory, parabolic interpolation must be used. 

Cubic interpolation has the same pros and cons as parabolic interpolation and is often used instead of it. 

In the BzzMath library, if the efficient method (parabolic or cubic interpolation) does not reduce the interval of uncertainty any better than the golden section, a new point is inserted in the middle of the largest subintervals. 

When several processors are available and parallel computing can be exploited, it is possible to merge an efficient (cubic or parabolic interpolations) with a robust (comparison) method in a very simple, effective way. 

The first processor is used to calculate the function in the point selected by the efficient method (cubic or parabolic interpolation), whereas the remaining np — 1 processors evaluate the function in the points opportunely positioned in the three subintervals into which the interval is split. 

The search within the range of uncertainty adopts cubic interpolation as its basic method. If such a method is inefficient, the parabolic method is adopted. If this method is also not as high performance as the golden section method, the new point is inserted into the middle of the largest subintervals. 

A second modification is also needed since the function or its derivatives may present either some discontinuities or nonevaluable regions in correspondence with ti. Parabolic and cubic interpolation methods may be unreliable and, therefore, it is opportune to give a smaller weight to the prediction coming from them. 

---