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 . 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 , then the number should occur k - j - 1 times in the knot sequence. Assuming that and are the endpoints of the interval of interest, choose the first k knots equal to and the last k knots equal to . This can be done because the B-splines are defined to be right continuous near and left continuous near .

When the above construction is completed, a knot sequence of length M is generated, and there are m: = M-k B-splines of order k, for example , spanning the ppoly functions on the interval with the indicated smoothness. That is, each ppoly function in this class has a unique representation as a linear combination of B-splines. A B-spline is a particularly compact piecewise polynomial function. is a nonnegative function that is nonzero only on the interval . More precisely, the support of the i-th B-spline is . 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:

BsInterpolate.html Example, BsLeastSquares.html Example, Serialized Form

Field Summary

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

BSpline()

Method Summary

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 Detail

coef

protected double[] coef

The B-spline coefficient array.

knot

protected double[] knot

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

order

protected int order

Order of the spline.

Constructor Detail

BSpline

public BSpline()

Method Detail

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 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

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

getKnots

public double[] getKnots()

Returns a copy of the knot sequence.

Returns:

a double array containing a copy of the knot sequence.

getSpline

public Spline getSpline()

Returns a Spline representation of the B-spline.

Returns:

a Spline representation of the BSpline

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

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

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