Spline2D (JMSL Numerical Library)

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
```
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

Constructors
Constructor and Description

Spline2D()

Constructors
Constructor and Description
`Spline2D()`

Method Summary

Methods
Modifier and Type	Method and Description
`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.

Class Spline2D

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

Spline2D

Method Detail

derivative

derivative

getCoefficients

getXKnots

getYKnots

integral

value

value