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:

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

Field Summary

Fields

Modifier and Type	Field and 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 and Description
BSpline()

Method Summary

Modifier and Type	Method and 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 Detail

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 Detail

BSpline

public BSpline()

Method Detail

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