Showing papers on "Monotone cubic interpolation published in 1971"

Lattice Green's Functions for the Cubic Lattices in Terms of the Complete Elliptic Integral

Abstract: The real and imaginary parts of the lattice Green's functions for the simple cubic (actually the tetragonal), body‐centered cubic, and face‐centered cubic lattices, at the variable from −∞ to +∞, are expressed as a sum of simple integrals of the complete elliptic integral of the first kind. The results of the numerical calculations obtained with the aid of the formulas are shown by graphs.

Objective analysis of a two dimensional data field by the cubic spline technique

Abstract: : A procedure to use the Spline interpolation technique on an arbitrarily prescribed two-dimensional data field is described. In order to use this technique it is necessary to obtain an initial approximation to the data at the grid points. This is achieved by fitting spherical surfaces to the data. Bi-directional Spline interpolation was then applied repeatedly on the grid point estimates of the data to produce convergence to the true surface. The Spline interpolation technique and another objective analysis technique developed by Gilchrist and Cressman are tested against an exact solution and the resulting analyses are compared. Real temperature, geopotential height, and wind data for various pressure surfaces are analysed by the Spline method and the results are compared to the subjective analyses of the same data.

On equidistant cubic spline interpolation

Abstract: and is seen to depend linearly o n w + 3 parameters. Here we have used the function # + = max(0, x). The interpolatory properties of the elements of the class 5[0, n] have recently attracted considerable attention. The two main kinds of this so-called cubic spline interpolation are as follows. 1. Natural cubic spline interpolation. We are required to find •SOxOGSfO, n] such as to satisfy the conditions

Interpolation of germanium resistor measurements at low temperatures with spline functions

Abstract: A germanium resistor used for temperature measurements is calibrated at a number of temperatures. In order to interpolate between these points and make a convenient conversion table the use of spline functions is proposed. This method of calculation has some advantages compared with other methods often used. The results of test calculations are given and discussed. Spline functions seem to be superior to other interpolation methods if the calibration is of sufficient accuracy.

Comments on "Objective Analysis of a Two- Dimensional Data Field by the Cubic Spline Technique "

Abstract: Fritsch (1971) concludes, “Objective analysis by the spline technique appears to be a satisfactory method for two-dimensional data analysis.” It may be that the spline technique can be engineered to produce satisfactory analyqes, but Fritsch has certainly not demonstrated this, and in fact, his applications suggest the contrary. We agree with Fritsch that it is, “. . . imperative to begin any numerical weather prediction with the ‘best possible’ representation of the real data.” However, the definition of “best possible” should consider the numerical model t o be used. That is, “initialization” for the model and the objective analysis of all pertinent data available should be an integrated procedure. This is a very difficult problem and one that deserves much study. Certainly, the objective analysis methods used operationally now should be replaced when ‘(better” techniques become available (computer time is, of course, a factor). However, the spline technique, as described by Fritsch, does not address the initialization problem and can, therefore, be satisfactory only to the extent that some measure of the difference between the data and the analysis is satisfactory. The examples shown by Fritsch do not appear to meet this criterion. The evidence presented by Fritsch to support his conclusion consists of four spline analyses and their comparison with analyses produced by other methods. Each of these comparisons will be discussed below. The first example is the analysis of data a t the points shown in figure 5 interpolated from the field of values shown in figure 6 and known at 5’ latitude-longitude intersections.2 (Fritsch does not state the longitudinal grid length but it is evidentlj5 O since the array size is 18 X 72.) Actually, there are two interpolation problems here, that of interpolating from grid points to stations and that of interpolating from stations to grid points. The spline analysis shown in figure 7 is considerably different in detail from the “true” analysis in figure 6.