Spline2DInterpolate Class

Computes a two-dimensional, tensor-product spline interpolant from two-dimensional, tensor-product data.

Inheritance Hierarchy

SystemObject
Imsl.MathSpline2D
Imsl.MathSpline2DInterpolate

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

Syntax

C++

Copy

[SerializableAttribute]
public class Spline2DInterpolate : Spline2D

<SerializableAttribute>
Public Class Spline2DInterpolate
	Inherits Spline2D

[SerializableAttribute]
public ref class Spline2DInterpolate : public Spline2D

[<SerializableAttribute>]
type Spline2DInterpolate =  
    class
        inherit Spline2D
    end

The Spline2DInterpolate type exposes the following members.

Constructors

	Name	Description
	Spline2DInterpolate(Double, Double, Double)	Constructor for Spline2DInterpolate.
	Spline2DInterpolate(Double, Double, Double, Int32, Int32)	Constructor for Spline2DInterpolate.
	Spline2DInterpolate(Double, Double, Double, Int32, Int32, Double, Double)	Constructor for Spline2DInterpolate.

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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). (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.)
	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.)
	GetYKnots	Returns the knot sequences in the y-direction. (Inherited from Spline2D.)
	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.)
	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 class Spline2DInterpolate computes a tensor-product spline interpolant. The tensor-product spline interpolant to data $\{(x_i, y_j, f_{ij})\}$ , where $0 \le i \le (n_x - 1)$ and $0 \le j \le (n_y - 1)$ has the form

$\sum_{m=0}^{n_y - 1} \sum_{n=0}^{n_x-1} c_{nm}B_{n,k_x,t_x}(x) B_{m,k_y,t_y}(y)$

where

and

are the orders of the splines. These numbers are defaulted to be 4, but can be set to any positive integer using xOrder and yOrder in the constructor. Likewise,

and

are the corresponding knot sequences (xKnots and yKnots). These default values are selected by Spline2DInterpolate. The algorithm requires that

$t_x(k_x - 1) \le x_i \le t_x(n_x)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \le i \le n_x -1$

$t_y(k_y - 1) \le y_j \le t_y(n_y - 1)\,\,\,\,\,\,\,\,\,\,\,0 \le j \le n_y -1$

Tensor-product spline interpolants in two dimensions can be computed quite efficiently by solving (repeatedly) two univariate interpolation problems.

The computation is motivated by the following observations. It is necessary to solve the system of equations

$\sum_{m=0}^{n_y - 1} \sum_{n=0}^{n_x-1} c_{nm}B_{n,k_x,t_x}(x_i) B_{m,k_y,t_y}(y_j) = f_{ij}$

Setting

$h_{mi} = \sum_{n=0}^{n_x-1} c_{nm}B_{n,k_x,t_x}(x_i)$

note that for each fixed i from 0 to

, we have

linear equations in the same number of unknowns as can be seen below:

$\sum_{m=0}^{n_y - 1} h_{mi}B_{n,k_y,t_y}(y_j) = f_{ij}$

$\sum_{m=0}^{n_y - 1} \sum_{n=0}^{n_x-1} c_{nm}B_{n,k_x,t_x}(x_i) B_{m,k_y,t_y}(y_j) = f_{ij}$

Setting

$h_{mi} = \sum_{n=0}^{n_x-1} c_{nm}B_{n,k_x,t_x}(x_i)$

note that for each fixed i from 0 to

, we have

linear equations in the same number of unknowns as can be seen below:

$\sum_{m=0}^{n_y - 1} h_{mi}B_{m,k_y,t_y}(y_j) = f_{ij}$

The same matrix appears in all of the equations above:

$\bigl[ B_{m,k_y,t_y}(y_j)\bigr]\,\,\,\,\,\,\,\,\,\,\, 0 \le m,j \le n_y - 1$

Thus, only factor this matrix once and then apply this factorization to the

right-hand sides. Once this is done and $h_{mi}$ is computed, then solve for the coefficients $c_{nm}$ using the relation

$\sum_{n=0}^{n_x-1} c_{nm}B_{n,k_x,t_x}(x_i) = h_{mi}$

for n from 0 to

, which again involves one factorization and

solutions to the different right-hand sides. The class Spline2DInterpolate is based on the routine SPLI2D by de Boor (1978, p. 347).

Reference

Imsl.Math Namespace

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