Syntax

C#
[SerializableAttribute] public abstract class Spline2D

Visual Basic (Declaration)
<SerializableAttribute> _ Public MustInherit Class Spline2D

Visual C++
[SerializableAttribute] public ref class Spline2D abstract

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).

Inheritance Hierarchy

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

Syntax

Remarks

Inheritance Hierarchy

See Also