CsSmoothC2 Class

CsSmoothC2 Class

Extension of the Spline class used to construct a spline for noisy data points using an alternate method.

Inheritance Hierarchy

SystemObject
Imsl.MathSpline
Imsl.MathCsSmoothC2

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

Syntax

C++

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[SerializableAttribute]
public class CsSmoothC2 : Spline

<SerializableAttribute>
Public Class CsSmoothC2
	Inherits Spline

[SerializableAttribute]
public ref class CsSmoothC2 : public Spline

[<SerializableAttribute>]
type CsSmoothC2 =  
    class
        inherit Spline
    end

The CsSmoothC2 type exposes the following members.

Constructors

	Name	Description
	CsSmoothC2(Double, Double, Double)	Constructs a smooth cubic spline from noisy data using an algorithm based on Reinsch (1967). All of the points have equal weights.
	CsSmoothC2(Double, Double, Double, Double)	Constructs a smooth cubic spline from noisy data using an algorithm based on Reinsch (1967) with weights supplied by the user.

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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 CsSmoothC2 is designed to produce a cubic spline approximation to a data set in which the function values are noisy. This spline is called a smoothing spline. It is a natural cubic spline with knots at all the data abscissas x, but it does not interpolate the data . The smoothing spline $S_\sigma$ is the unique function that minimizes

$\int\limits_a^b {s''_\sigma \left( x \right)^2 dx}$

subject to the constraint

$\sum\limits_{i=0}^{n-1} {\left| {s_\sigma \left( {x_i } \right) - f_i } \right|} ^2 \le \sigma$

Recommended values for $\sigma$ depend on the weights, w. If an estimate for the standard deviation of the error in the y-values is availiable, then should be set to this value and the smoothing parameter should be choosen in the confidence interval corresponding to the left side of the above inequality. That is,

$n-\sqrt{2n} \le \sigma \le n+\sqrt{2n}$

CsSmoothC2 is based on an algorithm of Reinsch (1967). This algorithm is also discussed in de Boor (1978, pages 235-243).

Reference

Imsl.Math Namespace

Other Resources

Example