Spline2DInterpolate (JMSL Numerical Library)

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com.imsl.math
Class Spline2DInterpolate

java.lang.Object
  com.imsl.math.Spline2D
      com.imsl.math.Spline2DInterpolate

All Implemented Interfaces:: Serializable, Cloneable

public class Spline2DInterpolate
extends Spline2D
extends Spline2D

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

The class Spline2DInterpolate computes a tensor-product spline interpolant. The tensor-product spline interpolant to data ${(x_i, y_j, f_{ij})}$ , where and 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

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:

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

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

Setting

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

note that for each fixed i from 0 to

, we have

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

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

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).

See Also:: Example 1, Example 2, Example 3, Example 4, Serialized Form

Constructor Summary
`Spline2DInterpolate(double[] xData, double[] yData, double[][] fData)` Constructor for `Spline2DInterpolate`.
`Spline2DInterpolate(double[] xData, double[] yData, double[][] fData, int xOrder, int yOrder)` Constructor for `Spline2DInterpolate`.
`Spline2DInterpolate(double[] xData, double[] yData, double[][] fData, int xOrder, int yOrder, double[] xKnots, double[] yKnots)` Constructor for `Spline2DInterpolate`.

Method Summary

Methods inherited from class com.imsl.math.Spline2D
`derivative, derivative, getCoefficients, getXKnots, getYKnots, integral, value, value`

Methods inherited from class java.lang.Object
`clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`

Constructor Detail