Natural cubic spline

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. 

Since the continuity conditions apply only for i = 2,. . . , NT — 1, we have only NT — 2 conditions for the NT values of y. Two additional conditions are needed, and these are usually taken as the value of the second derivative at each end of the domain, y, y f. If these values are zero, we get the natural cubic splines they can also be set to achieve some other purpose, such as making the first derivative match some desired condition at the two ends. With these values taken as zero in the natural cubic spline, we have a NT — 2 system of tridiagonal equations, which is easily solved. Once the second derivatives are known at each of the knots, the first derivatives are given by... 

In order to study this question in a more systematic way, we have recently optimized 144 different structures of ALA at the HF/4-21G level, covering the entire 4>/v )-space by a 30° grid (Schafer et al. 1995aG, 1995bG). From the resulting coordinates of ALA analytical functions were derived for the most important main chain structural parameters, such as N-C(a), C(a)-C, and N-C(a)-C, expanding them in terms of natural cubic spline parameters. In fact, Fig. 7.18 is an example of the type of conformational geometry map that can be derived from this procedure. 

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. 

In addition, we are interested in functions that are at least twice continuously differentiable. One can draw several such curves satisfying (4.27), and the "smoothest" of them is the one minimizing the integral (4.19). It can be shown that the solution of this constrained minimization problem is a natural cubic spline (ref. 12). We call it smoothing spline. 

Let S (t) denote the ny-vector of natural cubic splines interpolating the... 

The overpotential functions on anode and cathode were natural cubic spline interpolations of measured values taken from J. 0 M Bockris et al. [ 17]. They are repeated in table 3.6. 

The load capacity was obtained using library routines which integrated a natural cubic spline representation of the pressure distribution (IMSL- ICSICU and DCSQDU)... 

The pressure values from the previous time step were represented by a natural cubic spline (IMSL-ICSICU), then integrated (IMSL-DCSQDU) and the values of the interal at various X-locations were themselves represented as a natural cubic spline (ISML-ICSICU). Thus, the values of the definite integral involving the pressure values from the previous time step were available to the function subroutine of DGEAR at any X-location using a cubic spline interpolation (IMSL-ICSEVU),... 

These equations now provide the formulation of the equations for a cubic spline interpolation. The set of relationships of Eq. (6.19) provide n-2 equations while for n data points there are n unknown second derivative terms. Thus two additional equations are needed in order to obtain a set of valid solutions. These are the two second derivatives at the two end points of the data set. In most applications of cubic splines the second derivatives are set equal to zero at the boundaries and this results in what are known as a natural cubic spline . A second choice is to set the derivatives at the two neighbor end points equal. These two choices are expressed mathematically as ... 

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