Extension of the Spline class to interpolate data points.

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.0.0

Syntax

C#
[SerializableAttribute]
public class CsInterpolate : Spline
Visual Basic (Declaration)
<SerializableAttribute> _
Public Class CsInterpolate _
	Inherits Spline
Visual C++
[SerializableAttribute]
public ref class CsInterpolate : public Spline

Remarks

CsInterpolate computes a C^2 cubic spline interpolant to a set of data points (x_i, f_i) for i = 0, \ldots, n-1. The breakpoints of the spline are the abscissas. Endpoint conditions are automatically determined by the program. These conditions correspond to the "not-a-knot" condition (see de Boor 1978), which requires that the third derivative of the spline be continuous at the second and next-to-last breakpoint. If n is 2 or 3, then the linear or quadratic interpolating polynomial is computed, respectively.

If the data points arise from the values of a smooth, say, C^4 function f, i.e. f_i = f(x_i), then the error will behave in a predictable fashion. Let \xi be the breakpoint vector for the above spline interpolant. Then, the maximum absolute error satisfies

|f-s|_{[\xi_0,\xi_n]} \le C \left\|f^{(4)}\right\|_{[{\xi_0 ,\xi_n }]} |\xi|^4
where
|\xi|\;: = \max_{i=0,\ldots,n-1} \left|\xi_{i+1} - \xi_i \right|

For more details, see de Boor (1978, pages 55-56).

Inheritance Hierarchy

System..::.Object
Imsl.Math..::.Spline
Imsl.Math..::.CsInterpolate

See Also