Spline2DLeastSquares Class

Computes a two-dimensional, tensor-product spline approximant using least squares.

Inheritance Hierarchy

System.Object
Imsl.Math.Spline2D
Imsl.Math.Spline2DLeastSquares

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

Syntax

C++

Copy

[SerializableAttribute]
public class Spline2DLeastSquares : Spline2D

<SerializableAttribute>
Public Class Spline2DLeastSquares
	Inherits Spline2D

[SerializableAttribute]
public ref class Spline2DLeastSquares : public Spline2D

[<SerializableAttribute>]
type Spline2DLeastSquares =  
    class
        inherit Spline2D
    end

The Spline2DLeastSquares type exposes the following members.

Constructors

	Name	Description
	Spline2DLeastSquares	Constructor for Spline2DLeastSquares.

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Methods

	Name	Description
	Compute	Computes a two-dimensional, tensor-product spline approximant using least squares.
	Derivative(Double, Double, Int32, Int32)	Returns the value of the partial derivative of the tensor-product spline at the point (x, y). (Inherited from Spline2D.)
	Derivative(Double[],Double[], Int32, Int32)	Returns the values of the partial derivative of the tensor-product spline of an array of points. (Inherited from Spline2D.)
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetCoefficients	Returns the coefficients for the tensor-product spline. (Inherited from Spline2D.)
	GetErrorSumOfSquares	Returns the weighted error sum of squares.
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	GetXKnots	Returns the knot sequences in the x-direction. (Inherited from Spline2D.)
	GetXOrder	Returns the order of the spline in the x-direction.
	GetXWeights	Returns the weights for the least-squares fit in the x-direction.
	GetYKnots	Returns the knot sequences in the y-direction. (Inherited from Spline2D.)
	GetYOrder	Returns the order of the spline in the y-direction.
	GetYWeights	Returns the weights for the least-squares fit in the y-direction.
	Integral	Returns the value of an integral of a tensor-product spline on a rectangular domain. (Inherited from Spline2D.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	SetXKnots	Sets the knot sequences of the spline in the x-direction.
	SetXOrder	Sets the order of the spline in the x-direction.
	SetXWeights	Sets the weights for the least-squares fit in the x-direction.
	SetYKnots	Sets the knot sequences of the spline in the y-direction.
	SetYOrder	Sets the order of the spline in the y-direction.
	SetYWeights	Sets the weights for the least-squares fit in the y-direction.
	ToString	Returns a string that represents the current object. (Inherited from Object.)
	Value(Double, Double)	Returns the value of the tensor-product spline at the point (x, y). (Inherited from Spline2D.)
	Value(Double[],Double[])	Returns the values of the tensor-product spline of an array of points. (Inherited from Spline2D.)

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Remarks

The Spline2DLeastSquares class computes a tensor-product spline least-squares approximation to weighted tensor-product data. The input consists of data vectors to specify the tensor-product grid for the data, two vectors with the weights, the values of the surface on the grid, and the specification for the tensor-product spline. The grid is specified by the two vectors x = xData and y = yData of length
n = xData.Length and m = yData.Length, respectively. A two-dimensional array f = fData contains the data values which are to be fit. The two vectors = xWeights and = yWeights contain the weights for the weighted least-squares problem. The information for the approximating tensor-product spline can be provided using the SetXOrder, SetYOrder, SetXKnots and SetYKnots methods. This information is contained in = xOrder, = xKnots, and N = xSplineSpaceDim for the spline in the first variable, and in = yOrder, = yKnots and M = ySplineSpaceDim for the spline in the second variable. This class computes coefficients for the tensor-product spline by solving the normal equations in tensor-product form as discussed in de Boor (1978, Chapter 17). The interested reader might also want to study the paper by Grosse (1980).

As the computation proceeds, we obtain coefficients c minimizing

$\sum\limits_{i = 0}^{n - 1} {\sum\limits_{j = 0}^{m - 1} {w_x \left( i \right)w_y \left( j \right)} \left[ {\sum\limits_{k = 0}^{N - 1} {\sum\limits_{l = 0}^{M - 1} {c_{kl} } B_{kl} \left( {x_i ,y_i } \right) - f_{ij} } } \right]} ^2$

where the function $B_{kl}$ is the tensor-product of two B-splines of order

and

. Specifically, we have

$B_{kl} \left( {x,y} \right) = B_{k,k_x ,t_x } \left( x \right)B_{l,k_y ,t_y } \left( y \right)$

The spline

$\sum\limits_{k = 0}^{N - 1} {\sum\limits_{l = 0}^{M - 1} {c_{kl} } } B_{kl}$

and its partial derivatives can be evaluated using the Value method.

Reference

Imsl.Math Namespace

Other Resources

Example 1