CsAkima Class

CsAkima Class

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

Inheritance Hierarchy

System.Object
Imsl.Math.Spline
Imsl.Math.CsAkima

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

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[SerializableAttribute]
public class CsAkima : Spline

<SerializableAttribute>
Public Class CsAkima
	Inherits Spline

[SerializableAttribute]
public ref class CsAkima : public Spline

[<SerializableAttribute>]
type CsAkima =  
    class
        inherit Spline
    end

The CsAkima type exposes the following members.

Constructors

	Name	Description
	CsAkima	Constructs the Akima cubic spline interpolant to the given data points.

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Methods

	Name	Description
	Derivative(Double)	Returns the value of the first derivative of the spline at a point. (Inherited from Spline.)
	Derivative(Double, Int32)	Returns the value of the derivative of the spline at a point. (Inherited from Spline.)
	Derivative(Double[], Int32)	Returns the value of the derivative of the spline at each point of an array. (Inherited from Spline.)
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Eval(Double)	Returns the value of the spline at a point. (Inherited from Spline.)
	Eval(Double[])	Returns the value of the spline at each point of an array. (Inherited from Spline.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetBreakpoints	Returns a copy of the breakpoints. (Inherited from Spline.)
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	Integral	Returns the value of an integral of the spline. (Inherited from Spline.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Remarks

Class CsAkima computes a cubic spline interpolant to a set of data points 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 , function f, i.e. , 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.)

Reference

Imsl.Math Namespace

Other Resources

Example