Package 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:

  • Constructor Summary

    Constructors
    Constructor
    Description
    Spline2D()
     

  • Method Summary

    Modifier and Type
    Method
    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 Details

    • Spline2D

      public Spline2D()

  • Method Details

    • 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].