Spline2D (JMSL Numerical Library)

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JMSL^TM Numerical Library 6.1

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com.imsl.math
Class Spline2D

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

All Implemented Interfaces:: Serializable, Cloneable

Direct Known Subclasses:: Spline2DInterpolate, Spline2DLeastSquares

public abstract class Spline2D
extends Object
implements Serializable, Cloneable
extends Object
implements Serializable, Cloneable

Represents and evaluates tensor-product splines.

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:

$sumlimits_{m = 0}^{N_y - 1} {sumlimits_{n = 0}^{N_x - 1} {c_{nm} B_{n,k_x ,t_x } left( x right)B_{m,k_y ,t_y } left( y right)} }$

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

$sumlimits_{m = 0}^{N_y - 1} {sumlimits_{n = 0}^{N_x - 1} {c_{nm} B_{n,k_x ,t_x } left( {x_i } right)B_{m,k_y ,t_y } left( {y_j } right)} } = f_{ij}$

This problem can be solved efficiently by repeatedly solving univariate interpolation problems as described in de Boor (1978, p. 347).

See Also:: Spline2DInterpolate Example, Serialized Form

Constructor Summary
`Spline2D()`

Method Summary
`double[][]`	`derivative(double[] xVec, double[] yVec, int xPartial, int yPartial)` Returns the values of the partial derivative of the tensor-product spline of an array of points.
`double`	`derivative(double x, double y, int xPartial, int yPartial)` Returns the value of the partial derivative of the tensor-product spline at the point (x, y).
`double[][]`	`getCoefficients()` Returns the coefficients for the tensor-product spline.
`double[]`	`getXKnots()` Returns the knot sequences in the x-direction.
`double[]`	`getYKnots()` Returns the knot sequences in the y-direction.
`double`	`integral(double a, double b, double c, double d)` Returns the value of an integral of a tensor-product spline on a rectangular domain.
`double[][]`	`value(double[] xVec, double[] yVec)` Returns the values of the tensor-product spline of an array of points.
`double`	`value(double x, double y)` Returns the value of the tensor-product spline at the point (x, y).

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

Constructor Detail

Spline2D

public Spline2D()

Method Detail

derivative

public double[][] derivative(double[] xVec,
                             double[] yVec,
                             int xPartial,
                             int yPartial)

Returns the values of the partial derivative of the tensor-product spline of an array of points.

Parameters:: xVec - a double array specifying the x-coordinates at which the spline is to be evaluated.; yVec - a double array specifying the y-coordinates at which the spline is to be evaluated.; xPartial - an int scalar specifying the x-partial derivative.; yPartial - an int scalar specifying the y-partial derivative.
Returns:: a double matrix containing the values of the partial derivatives
$frac{{partial ^{i + j} s}}{{partial ^i x,,partial ^j y}}$
where i = xPartial and j = yPartial, at each (x, y).

derivative

public double derivative(double x,
                         double y,
                         int xPartial,
                         int yPartial)

Returns the value of the partial derivative of the tensor-product spline at the point (x, y).

Parameters:: x - a double scalar specifying the x-coordinate of the evaluation point for the tensor-product spline.; y - a double scalar specifying the y-coordinate of the evaluation point for the tensor-product spline.; xPartial - an int scalar specifying the x-partial derivative.; yPartial - an int scalar specifying the y-partial derivative.
Returns:: a double scalar containing the value of the partial derivative
$frac{{partial ^{i + j} s}}{{partial ^i x,,partial ^j y}}$
where i = xPartial and j = yPartial, at (x, y).

getCoefficients

public double[][] getCoefficients()

Returns the coefficients for the tensor-product spline.

Returns:: a double matrix containing the coefficients.

getXKnots

public double[] getXKnots()

Returns the knot sequences in the x-direction.

Returns:: a double array containing the knot sequences of the spline in the x-direction.

getYKnots

public double[] getYKnots()

Returns the knot sequences in the y-direction.

Returns:: a double array containing the knot sequences of the spline in the y-direction.

integral

public double integral(double a,
                       double b,
                       double c,
                       double d)

Returns the value of an integral of a tensor-product spline on a rectangular domain.

If s is the spline, then the integral method returns

$int_a^b {int_c^d {sleft( {x,y} right)} } dydx$

This method uses the (univariate integration) identity (22) in de Boor (1978, p. 151)

$int_{t_0 }^x {sumlimits_{i = 0}^{n - 1} {alpha _i } } B_{i,k} left( tau right)dtau = sumlimits_{i = 0}^{r - 1} {left[ {sumlimits_{j = 0}^i {alpha _j frac{{t_{j + k} - t_j }}{k}} } right]} B_{i,k + 1} left( x right)$

where

It assumes (for all knot sequences) that the first and last k knots are stacked, that is, $t_0 = ldots = t_{k-1}$ and $t_n = ldots = t_{n+k-1}$ , where k is the order of the spline in the x or y direction.

Parameters:: a - a double specifying the lower limit for the first variable of the tensor-product spline.; b - a double specifying the upper limit for the first variable of the tensor-product spline.; c - a double specifying the lower limit for the second variable of the tensor-product spline.; d - a double specifying the upper limit for the second variable of the tensor-product spline.
Returns:: a double, the integral of the tensor-product spline over the rectangle [a, b] by [c, d].

value

public double[][] value(double[] xVec,
                        double[] yVec)

Returns the values of the tensor-product spline of an array of points.

Parameters:: xVec - a double array specifying the x-coordinates at which the spline is to be evaluated.; yVec - a double array specifying the y-coordinates at which the spline is to be evaluated.
Returns:: a double matrix containing the values evaluated.

value

public double value(double x,
                    double y)

Returns the value of the tensor-product spline at the point (x, y).

Parameters:: x - a double scalar specifying the x-coordinate of the evaluation point for the tensor-product spline.; y - a double scalar specifying the y-coordinate of the evaluation point for the tensor-product spline.
Returns:: a double scalar containing the value of the tensor-product spline.

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JMSL^TM Numerical Library 6.1

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com.imsl.math Class Spline2D

Spline2D

derivative

derivative

getCoefficients

getXKnots

getYKnots

integral

value

value

com.imsl.math
Class Spline2D