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

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.0.0

Syntax

C#
[SerializableAttribute] public class Spline2DInterpolate : Spline2D

Visual Basic (Declaration)
<SerializableAttribute> _ Public Class Spline2DInterpolate _ Inherits Spline2D

Visual C++
[SerializableAttribute] public ref class Spline2DInterpolate : public Spline2D

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).

Inheritance Hierarchy

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

Syntax

Remarks

Inheritance Hierarchy

See Also