CsPeriodic Class

CsPeriodic Class

Extension of the Spline class to interpolate data points with periodic boundary conditions.

Inheritance Hierarchy

System.Object
Imsl.Math.Spline
Imsl.Math.CsPeriodic

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

Syntax

C++

Copy

[SerializableAttribute]
public class CsPeriodic : Spline

<SerializableAttribute>
Public Class CsPeriodic
	Inherits Spline

[SerializableAttribute]
public ref class CsPeriodic : public Spline

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

The CsPeriodic type exposes the following members.

Constructors

	Name	Description
	CsPeriodic	Constructs a cubic spline that interpolates the given data points with periodic boundary conditions.

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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 CsPeriodic 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. The program enforces periodic endpoint conditions. This means that the spline s satisfies s(a) = s(b), ${\it{s'}}\left( {\it{a}} \right) = {\it{s'}}\left( {\it{b}} \right)$ , and $s''\left( a \right) = s''\left( b \right)$ , where a is the leftmost abscissa and b is the rightmost abscissa. If the ordinate values corresponding to a and b are not equal, then a warning message is issued. The ordinate value at b is set equal to the ordinate value at a and the interpolant is computed.

If the data points arise from the values of a smooth (say ) periodic 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

$|f-s|_{[\xi_0,\xi_{n-1}]} \le C |f^{(4)}|_{[{\xi_0 ,\xi_{n-1} }]} |\xi|^4$

where

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

For more details, see de Boor (1978, pages 320-322).

Reference

Imsl.Math Namespace

Other Resources

Example