com.imsl.math

Class Spline

java.lang.Object

com.imsl.math.Spline

All Implemented Interfaces:

Serializable, Cloneable

Direct Known Subclasses:

CsAkima, CsInterpolate, CsPeriodic, CsShape, CsSmooth, CsSmoothC2, CsTCB

public abstract class Spline
extends Object
implements Serializable, Cloneable

Spline represents and evaluates univariate piecewise polynomial splines.

A univariate piecewise polynomial (function) p(x) is specified by giving its breakpoint sequence \(\xi \in {\bf R}^n\), the order k (degree k-1) of its polynomial pieces, and the \(k \times (n-1)\) matrix c of its local polynomial coefficients. In terms of this information, the piecewise polynomial (ppoly) function is given by $$ p(x) = \sum_{j=1}^k c_{ji} \frac{(x-\xi_i)^{j-1}}{(j-1)!} \;\;{\rm for}\; \xi_i \le x \le \xi_{i+1} $$ The breakpoint sequence \(\xi\) is assumed to be strictly increasing, and we extend the ppoly function to the entire real axis by extrapolation from the first and last intervals.

See Also:

Akima Example, Interpolate Example, Periodic Example, Shape Preserving Example, Smoothing Example 1, Smoothing Example 2, Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
protected double[]	breakPoint The breakpoint array of length n, where n is the number of piecewise polynomials.
protected double[][]	coef Coefficients of the piecewise polynomials.
protected static double	EPSILON_LARGE The largest relative spacing for double.

Constructor Summary

Constructors

Constructor and Description
Spline()

Method Summary

Modifier and Type	Method and Description
protected void	copyAndSortData(double[] xData, double[] yData) Copy and sort xData into breakPoint and yData into the first column of coef.
protected void	copyAndSortData(double[] xData, double[] yData, double[] weight) Copy and sort xData into breakPoint and yData into the first column of coef.
double	derivative(double x) Returns the value of the first derivative of the spline at a point.
double[]	derivative(double[] x, int ideriv) Returns the value of the derivative of the spline at each point of an array.
double	derivative(double x, int ideriv) Returns the value of the derivative of the spline at a point.
double[]	getBreakpoints() Returns a copy of the breakpoints.
double	integral(double a, double b) Returns the value of an integral of the spline.
double	value(double x) Returns the value of the spline at a point.
double[]	value(double[] x) Returns the value of the 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

Coefficients of the piecewise polynomials. This is an n by k array, where n is the number of piecewise polynomials and k is the order (degree+1) of the piecewise polynomials.

coef[i] contains the coefficients for the piecewise polynomial valid in the interval [x[k],x[k+1]).

breakPoint

protected double[] breakPoint

The breakpoint array of length n, where n is the number of piecewise polynomials.

EPSILON_LARGE

protected static final double EPSILON_LARGE

The largest relative spacing for double.

See Also:

Constant Field Values

Constructor Detail

Spline

public Spline()

Method Detail

copyAndSortData

protected void copyAndSortData(double[] xData,
                               double[] yData)

Copy and sort xData into breakPoint and yData into the first column of coef.

copyAndSortData

protected void copyAndSortData(double[] xData,
                               double[] yData,
                               double[] weight)

Copy and sort xData into breakPoint and yData into the first column of coef.

value

public double value(double x)

Returns the value of the spline at a point.

Parameters:

x - a double, the point at which the spline is to be evaluated

Returns:

a double giving the value of the spline at the point x

derivative

public double derivative(double x)

Returns the value of the first derivative of the spline at a point.

Parameters:

x - a double, the point at which the derivative is to be evaluated

Returns:

a double containing the value of the first derivative of the spline at the point x

derivative

public double derivative(double x,
                         int ideriv)

Returns the value of the derivative of the spline at a point.

Parameters:

x - a double, the 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 spline at the point x

value

public double[] value(double[] x)

Returns the value of the spline at each point of an array.

Parameters:

x - a double array of points at which the spline is to be evaluated

Returns:

a double array containing the value of the spline at each point of the array x

derivative

public double[] derivative(double[] x,
                           int ideriv)

Returns the value of the derivative of the 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 of the spline at each point of the array x

getBreakpoints

public double[] getBreakpoints()

Returns a copy of the breakpoints.

Returns:

a double array containing a copy of the breakpoints

integral

public double integral(double a,
                       double b)

Returns the value of an integral of the spline.

Parameters:

a - a double specifying the lower limit of integration

b - a double specifying the upper limit of integration

Returns:

a double, the integral of the spline from a to b