Extension of the Spline class to handle the Akima cubic spline.

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

Syntax

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

Remarks

Class CsAkima computes a C^1 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; see Akima (1970) or de Boor (1978).

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

\left\| {f - s} \right\|_{\left[ {\xi_0 , \xi_{n-1} } \right]} \le C\left\| f^{(2)}\right\|_{[\xi_0,\xi_{n-1}} \left| \xi \right|^2

where

|\xi| \;: = \max\limits_{i = 1,\ldots,n-1} |\xi_i -\xi_{i-1}|

CsAkima is based on a method by Akima (1970) to combat wiggles in the interpolant. The method is nonlinear; and although the interpolant is a piecewise cubic, cubic polynomials are not reproduced. (However, linear polynomials are reproduced.)

Inheritance Hierarchy

See Also