Package com.imsl.math

Class BSpline

java.lang.Object
com.imsl.math.BSpline
All Implemented Interfaces:
Serializable, Cloneable
Direct Known Subclasses:
BsInterpolate, BsLeastSquares
public abstract class BSpline extends Object implements Serializable, Cloneable
BSpline represents and evaluates univariate B-splines.

B-splines provide a particularly convenient and suitable basis for a given class of smooth ppoly functions. Such a class is specified by giving its breakpoint sequence, its order k, and the required smoothness across each of the interior breakpoints. The corresponding B-spline basis is specified by giving its knot sequence \({\bf t} \in {\bf R}^M \). The specification rule is as follows: If the class is to have all derivatives up to and including the j-th derivative continuous across the interior breakpoint \(\xi_i\), then the number \(\xi_i\) should occur k - j - 1 times in the knot sequence. Assuming that \(\xi_1\) and \(\xi_n \) are the endpoints of the interval of interest, choose the first k knots equal to \(\xi_1\) and the last k knots equal to \(\xi_n\). This can be done because the B-splines are defined to be right continuous near \(\xi_1\) and left continuous near \(\xi_n\).

When the above construction is completed, a knot sequence \({\bf t}\) of length M is generated, and there are m: = M-k B-splines of order k, for example \(B_0, ..., B_{m-1}\), spanning the ppoly functions on the interval with the indicated smoothness. That is, each ppoly function in this class has a unique representation \( p = a_0B_0 + a_1B_1 + ... + a_{m-1}B_{m-1} \) as a linear combination of B-splines. A B-spline is a particularly compact piecewise polynomial function. \(B_i\) is a nonnegative function that is nonzero only on the interval \([{\bf t}_i,{\bf t}_{i+k}]\). More precisely, the support of the i-th B-spline is \(\left[ t_i,t_{i+k}\right]\). No piecewise polynomial function in the same class (other than the zero function) has smaller support (i.e., vanishes on more intervals) than a B-spline. This makes B-splines particularly attractive basis functions since the influence of any particular B-spline coefficient extends only over a few intervals.

See Also:

  • Field Summary

    Fields
    Modifier and Type
    Field
    Description
    protected double[]
    coef
    The B-spline coefficient array.
    protected double[]
    knot
    The knot array of length n + order, where n is the number of coefficients in the B-spline.
    protected int
    order
    Order of the spline.

  • Constructor Summary

    Constructors
    Constructor
    Description
    BSpline()
     

  • Method Summary

    Modifier and Type
    Method
    Description
    double
    derivative(double x)
    Returns the value of the first derivative of the B-spline at a point.
    double[]
    derivative(double[] x, int ideriv)
    Returns the value of the derivative of the B-spline at each point of an array.
    double
    derivative(double x, int ideriv)
    Returns the value of the derivative of the B-spline at a point.
    double[]
    getKnots()
    Returns a copy of the knot sequence.
    Spline
    getSpline()
    Returns a Spline representation of the B-spline.
    double
    integral(double a, double b)
    Returns the value of an integral of the B-spline.
    double
    value(double x)
    Returns the value of the B-spline at a point.
    double[]
    value(double[] x)
    Returns the value of the B-spline at each point of an array.

    Methods inherited from class java.lang.Object

    clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

  • Field Details

    • order

      protected int order
      Order of the spline.

    • knot

      protected double[] knot
      The knot array of length n + order, where n is the number of coefficients in the B-spline.

    • coef

      protected double[] coef
      The B-spline coefficient array.

  • Constructor Details

    • BSpline

      public BSpline()

  • Method Details

    • value

      public double value(double x)
      Returns the value of the B-spline at a point.
      Parameters:
      x - a double specifying the point at which the B-spline is to be evaluated
      Returns:
      a double giving the value of the B-spline at the point x

    • derivative

      public double derivative(double x)
      Returns the value of the first derivative of the B-spline at a point.
      Parameters:
      x - a double specifying a point at which the derivative is to be evaluated
      Returns:
      a double containing the value of the first derivative of the B-spline at the point x

    • derivative

      public double derivative(double x, int ideriv)
      Returns the value of the derivative of the B-spline at a point.
      Parameters:
      x - a double specifying a point at which the derivative is to be evaluated
      ideriv - an int specifying the derivative to be computed. If zero, the function value is returned. If one, the first derivative is returned, etc.
      Returns:
      a double containing the value of the derivative of the B-spline at the point x

    • value

      public double[] value(double[] x)
      Returns the value of the B-spline at each point of an array.
      Parameters:
      x - a double array of points at which the B-spline is to be evaluated
      Returns:
      a double array containing the value of the B-spline at each point of the array x

    • derivative

      public double[] derivative(double[] x, int ideriv)
      Returns the value of the derivative of the B-spline at each point of an array.
      Parameters:
      x - a double array of points at which the derivative is to be evaluated
      ideriv - an int specifying the derivative to be computed. If zero, the function value is returned. If one, the first derivative is returned, etc.
      Returns:
      a double array containing the value of the derivative the B-spline at each point of the array x

    • getKnots

      public double[] getKnots()
      Returns a copy of the knot sequence.
      Returns:
      a double array containing a copy of the knot sequence.

    • integral

      public double integral(double a, double b)
      Returns the value of an integral of the B-spline.
      Parameters:
      a - a double specifying the lower limit of integration
      b - a double specifying the upper limit of integration
      Returns:
      a double which specifies the B-spline integral value from a to b

    • getSpline

      public Spline getSpline()
      Returns a Spline representation of the B-spline.
      Returns:
      a Spline representation of the BSpline