cubSplineValue

Computes the value of a cubic spline or the value of one of its derivatives.

Synopsis

cubSplineValue (x, ppoly)

Required Arguments

float x (Input)

Evaluation point for the cubic spline.

Imsl_d_ppoly ppoly (Input)

Pointer to the piecewise polynomial structure that represents the cubic spline.

Return Value

The value of a cubic spline or one of its derivatives at the point x. If no value can be computed, then NaN is returned.

Optional Arguments

deriv, int (Input)

Let d = deriv and let s be the cubic spline that is represented by the structure Imsl_d_ppoly, then this option produces the d-th derivative of s at x, \(s^{(d)}(x)\).

grid, float xvec, float value (Input/Output)

The array xvec contains the points at which the cubic spline is to be evaluated. The d-th derivative of the spline at the points in xvec is returned in value. The values in array xvec must appear sorted and non-decreasing. Arranging for this requirement may benefit by use of the function sort, Chapter 12.

Description

The function cubSplineValue computes the value of a cubic spline or one of its derivatives. The first and last pieces of the cubic spline are extrapolated. As a result, the cubic spline structures returned by the cubic spline routines are defined and can be evaluated on the entire real line. This routine is based on the routine PPVALU by de Boor (1978, p. 89).

Examples

Example 1

In this example, a cubic spline interpolant to a function f is computed. The values of this spline are then compared with the exact function values. Since the default settings are used, the interpolant is determined by the “not-a-knot” condition (see de Boor 1978).

Output

     x        F(x)     Interpolant     Error
 0.250      -0.572       -0.549       0.0228
 0.300      -0.978       -0.978       0.0000
 0.350      -0.859       -0.843       0.0162
 0.400      -0.279       -0.279       0.0000
 0.450       0.450        0.441       0.0093
 0.500       0.938        0.938       0.0000
 0.550       0.923        0.903       0.0199
 0.600       0.412        0.412       0.0000
 0.650      -0.320       -0.315       0.0049
 0.700      -0.880       -0.880       0.0000
 0.750      -0.968       -0.938       0.0295

Example 2

Recall that in the first example, a cubic spline interpolant to a function f is computed. The values of this spline are then compared with the exact function values. This example compares the values of the first derivatives.

Output

     x        F(x)     Interpolant     Deriv Error
 0.250     -12.308      -12.559       0.2510
 0.300      -3.162       -3.218       0.0560
 0.350       7.681        7.796       0.1151
 0.400      14.403       13.919       0.4833
 0.450      13.395       13.530       0.1346
 0.500       5.200        5.007       0.1926
 0.550      -5.786       -5.840       0.0535
 0.600     -13.667      -13.201       0.4660
 0.650     -14.214      -14.393       0.1798
 0.700      -7.133       -6.734       0.3990
 0.750       3.775        3.911       0.1359