Spline2DLeastSquares (JMSL Numerical Library)

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JMSL^TM Numerical Library 6.1

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com.imsl.math
Class Spline2DLeastSquares

java.lang.Object
  com.imsl.math.Spline2D
      com.imsl.math.Spline2DLeastSquares

All Implemented Interfaces:: Serializable, Cloneable

public class Spline2DLeastSquares
extends Spline2D
extends Spline2D

Computes a two-dimensional, tensor-product spline approximant using least squares.

The Spline2DLeastSquares class computes a tensor-product spline least-squares approximation to weighted tensor-product data. The input consists of data vectors to specify the tensor-product grid for the data, two vectors with the weights, the values of the surface on the grid, and the specification for the tensor-product spline. The grid is specified by the two vectors x = xData and y = yData of length
n = xData.length and m = yData.length, respectively. A two-dimensional array f = fData contains the data values which are to be fit. The two vectors = xWeights and = yWeights contain the weights for the weighted least-squares problem. The information for the approximating tensor-product spline can be provided using the setXOrder, setYOrder, setXKnots and setYKnots methods. This information is contained in = xOrder, = xKnots, and N = xSplineSpaceDim for the spline in the first variable, and in = yOrder, = yKnots and M = ySplineSpaceDim for the spline in the second variable. This class computes coefficients for the tensor-product spline by solving the normal equations in tensor-product form as discussed in de Boor (1978, Chapter 17). The interested reader might also want to study the paper by Grosse (1980).

As the computation proceeds, we obtain coefficients c minimizing

$sumlimits_{i = 0}^{n - 1} {sumlimits_{j = 0}^{m - 1} {w_x left( i right)w_y left( j right)} left[ {sumlimits_{k = 0}^{N - 1} {sumlimits_{l = 0}^{M - 1} {c_{kl} } B_{kl} left( {x_i ,y_i } right) - f_{ij} } } right]} ^2$

where the function $B_{kl}$ is the tensor-product of two B-splines of order

and

. Specifically, we have

$B_{kl} left( {x,y} right) = B_{k,k_x ,t_x } left( x right)B_{l,k_y ,t_y } left( y right)$

The spline

$sumlimits_{k = 0}^{N - 1} {sumlimits_{l = 0}^{M - 1} {c_{kl} } } B_{kl}$

and its partial derivatives can be evaluated using the value method.

See Also:: Example 1, Serialized Form

Constructor Summary
`Spline2DLeastSquares(double[] xData, double[] yData, double[][] fData, int xSplineSpaceDim, int ySplineSpaceDim)` Constructor for `Spline2DLeastSquares`.

Method Summary
`void`	`compute()` Computes a two-dimensional, tensor-product spline approximant using least squares.
`double`	`getErrorSumOfSquares()` Returns the weighted error sum of squares.
`int`	`getXOrder()` Returns the order of the spline in the x-direction.
`double[]`	`getXWeights()` Returns the weights for the least-squares fit in the x-direction.
`int`	`getYOrder()` Returns the order of the spline in the y-direction.
`double[]`	`getYWeights()` Returns the weights for the least-squares fit in the y-direction.
`void`	`setXKnots(double[] xKnots)` Sets the knot sequences of the spline in the x-direction.
`void`	`setXOrder(int xOrder)` Sets the order of the spline in the x-direction.
`void`	`setXWeights(double[] xWeights)` Sets the weights for the least-squares fit in the x-direction.
`void`	`setYKnots(double[] yKnots)` Sets the knot sequences of the spline in the y-direction.
`void`	`setYOrder(int yOrder)` Sets the order of the spline in the y-direction.
`void`	`setYWeights(double[] yWeights)` Sets the weights for the least-squares fit in the y-direction.

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