Spline2D Class

Spline2D Class

Represents and evaluates tensor-product splines.

Inheritance Hierarchy

System.Object
  Imsl.Math.Spline2D
    Imsl.Math.Spline2DInterpolate
    Imsl.Math.Spline2DLeastSquares

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

Syntax

C++

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[SerializableAttribute]
public abstract class Spline2D

<SerializableAttribute>
Public MustInherit Class Spline2D

[SerializableAttribute]
public ref class Spline2D abstract

[<AbstractClassAttribute>]
[<SerializableAttribute>]
type Spline2D =  class end

The Spline2D type exposes the following members.

Constructors

	Name	Description
	Spline2D	Initializes a new instance of the Spline2D class

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Methods

	Name	Description
	Derivative(Double, Double, Int32, Int32)	Returns the value of the partial derivative of the tensor-product spline at the point (x, y).
	Derivative(Double[],Double[], Int32, Int32)	Returns the values of the partial derivative of the tensor-product spline of an array of points.
	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.
	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.
	GetYKnots	Returns the knot sequences in the y-direction.
	Integral	Returns the value of an integral of a tensor-product spline on a rectangular domain.
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	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).
	Value(Double[],Double[])	Returns the values of the tensor-product spline of an array of points.

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Remarks

The simplest method of obtaining multivariate interpolation and approximation functions is to take univariate methods and form a multivariate method via tensor products. In the case of two-dimensional spline interpolation, the derivation proceeds as follows: Let be a knot sequence for splines of order , and be a knot sequence for splines of order . Let be the length of , and be the length of . Then, the tensor-product spline has the following form:

$\sum\limits_{m = 0}^{N_y - 1} {\sum\limits_{n = 0}^{N_x - 1} {c_{nm} B_{n,k_x ,t_x } \left( x \right)B_{m,k_y ,t_y } \left( y \right)} }$

Given two sets of points

$\left\{ {x_i } \right\}_{i = 1}^{N_x }$

and

$\left\{ {y_j } \right\}_{j = 1}^{N_y }$

for which the corresponding univariate interpolation problem can be solved, the tensor-product interpolation problem finds the coefficients $c_{nm}$ so that

$\sum\limits_{m = 0}^{N_y - 1} {\sum\limits_{n = 0}^{N_x - 1} {c_{nm} B_{n,k_x ,t_x } \left( {x_i } \right)B_{m,k_y ,t_y } \left( {y_j } \right)} } = f_{ij}$

This problem can be solved efficiently by repeatedly solving univariate interpolation problems as described in de Boor (1978, p. 347).

Reference

Imsl.Math Namespace

Other Resources

Spline2DInterpolate Example