Spline function natural condition

With the end condition flag EC = 0 on the input, the module determines the natural cubic spline function interpolating the function values stored in vector F. Otherwise, D1 and DN are additional input parameters specifying the first derivatives at the first and last points, respectively. Results are returned in the array S such that S(J,1), J = 0, 1, 2, 3 contain the 4 coefficients of the cubic defined on the I-th segment between Z(I) and ZII+l). Note that the i-th cubic is given in a coordinate system centered at Z(I). The module also calculates the area under the curve from the first point Z(l) to each grid point Z(I), and returns it in S(4,I). The entries in the array S can be directly used in applications, but we provide a further module to facilitate this step. 

A cubic spline function is mechanically simulated by a flexible plastic strip. Mathematically, a spline function is a cubic in each interval between two experimental points. Thus, for n points, a spline includes n — 1 pieces of cubic each cubic having 4 unknown parameters, there are 4(n — 1) parameters to determine. The following conditions are imposed, (i) Continuity of the spline function and of its first and second derivatives at each of the n — 2 nodes (3n — 6 conditions), (ii) The spline function is an interpolating function (n conditions), (iii) The second derivatives at each extremity are null (2 conditions) this condition corresponds to the natural spline. It may be shown that the natural spline obtained is the smoothest interpolation function. Details concerning the construction of a spline and corresponding programs can be found in Forsythe et al. [127]. Of course, after a spline has been built up, it can be used to calculate derivatives. 

Sphnes are functions that match given values at the points X, . . . , x t and have continuous derivatives up to some order at the knots, or the points X9,. . . , x vr-i-Cubic sphnes are most common see Ref. 38. The function is represented by a cubic polynomial within each interval Xj, X, +1) and has continuous first and second derivatives at the knots. Two more conditions can be specified arbitrarily. These are usually the second derivatives at the two end points, which are commonly taken as zero this gives the natural cubic splines. 

The input is similar to that of the module 1463. No end condition flag is used since only natural splines can be fitted. On the other hand, you should specify the maximum number IM of iterations. The module returns the array S defined in the description of the module M63, and hence the function value, the derivatives and the integral at a specified X can be computed by calling the module 1464. The important additional inputs needed by the module 1465 are the standard errors given in the vector D. With all D(I) = 0, the module... 

---