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

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 $t_x$ be a knot sequence for splines of order $k_x$, and $t_y$ be a knot sequence for splines of order $k_y$. Let $N_x + k_x$ be the length of $t_x$, and $N_y + k_x$ be the length of $t_y$. Then, the tensor-product spline has the following form: $$\sum\limits_{m = 0}^{N_y - 1} {\sum\limits_{n = 0}^{N_x - 1} {c_{nm} B_{n,k_x ,t_x } \left( x \right)B_{m,k_y ,t_y } \left( y \right)} }$$ 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 $$\sum\limits_{m = 0}^{N_y - 1} {\sum\limits_{n = 0}^{N_x - 1} {c_{nm} B_{n,k_x ,t_x } \left( {x_i } \right)B_{m,k_y ,t_y } \left( {y_j } \right)} } = f_{ij}$$ 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()

Method Summary

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

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.

getCoefficients

public double[][] getCoefficients()

Returns the coefficients for the tensor-product spline.

Returns:

a double matrix containing the coefficients.

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.

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).

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.

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).

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 $$\int_a^b {\int_c^d {s\left( {x,y} \right)} } dydx$$ This method uses the (univariate integration) identity (22) in de Boor (1978, p. 151) $$\int_{t_0 }^x {\sum\limits_{i = 0}^{n - 1} {\alpha _i } } B_{i,k} \left( \tau \right)d\tau = \sum\limits_{i = 0}^{r - 1} {\left[ {\sum\limits_{j = 0}^i {\alpha _j \frac{{t_{j + k} - t_j }}{k}} } \right]} B_{i,k + 1} \left( x \right)$$ where $t_0 \le x \le t_r$.

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].