com.imsl.math

Class CsPeriodic

java.lang.Object

com.imsl.math.Spline

com.imsl.math.CsPeriodic

All Implemented Interfaces:

Serializable, Cloneable

public class CsPeriodic
extends Spline

Extension of the Spline class to interpolate data points with periodic boundary conditions.

Class CsPeriodic computes a \(C^2\) cubic spline interpolant to a set of data points \((x_i, f_i)\) for \(i =0,\ldots n-1\). The breakpoints of the spline are the abscissas. The program enforces periodic endpoint conditions. This means that the spline s satisfies s(a) = s(b), \({\it{s'}}\left( {\it{a}} \right) = {\it{s'}}\left( {\it{b}} \right)\), and \(s''\left( a \right) = s''\left( b \right)\), where a is the leftmost abscissa and b is the rightmost abscissa. If the ordinate values corresponding to a and b are not equal, then a warning message is issued. The ordinate value at b is set equal to the ordinate value at a and the interpolant is computed.

If the data points arise from the values of a smooth (say \(C^4\)) periodic function f, i.e. \(f_i = f(x_i)\), then the error will behave in a predictable fashion. Let \(\xi\) be the breakpoint vector for the above spline interpolant. Then, the maximum absolute error satisfies

$$ |f-s|_{[\xi_0,\xi_{n-1}]} \le C |f^{(4)}|_{[{\xi_0 ,\xi_{n-1} }]} |\xi|^4 $$

where

$$|\xi|\;: = \max_{i=1,\ldots,n-1} |\xi_i - \xi_{i - 1} |$$

For more details, see de Boor (1978, pages 320-322).

See Also:

Example, Serialized Form

Field Summary

Fields inherited from class com.imsl.math.Spline

breakPoint, coef, EPSILON_LARGE

Constructor Summary

Constructors

Constructor and Description
CsPeriodic(double[] xData, double[] yData) Constructs a cubic spline that interpolates the given data points with periodic boundary conditions.

Method Summary

Methods inherited from class com.imsl.math.Spline

copyAndSortData, copyAndSortData, derivative, derivative, derivative, getBreakpoints, integral, value, value

Methods inherited from class java.lang.Object

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

Constructor Detail

CsPeriodic

public CsPeriodic(double[] xData,
                  double[] yData)

Constructs a cubic spline that interpolates the given data points with periodic boundary conditions.

Parameters:

xData - A double array containing the x-coordinates of the data. There must be at least 4 data points and values must be distinct.

yData - A double array containing the y-coordinates of the data. The arrays xData and yData must have the same length.