com.imsl.math

Class CsAkima

java.lang.Object

com.imsl.math.Spline

com.imsl.math.CsAkima

All Implemented Interfaces:

Serializable, Cloneable

public class CsAkima
extends Spline

Extension of the Spline class to handle the Akima cubic spline.

Class CsAkima computes a \(C^1\) 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. Endpoint conditions are automatically determined by the program; see Akima (1970) or de Boor (1978).

If the data points arise from the values of a smooth, say \(C^4\), 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

$$ \left\| {f - s} \right\|_{\left[ {\xi_0 ,\xi_{n-1} } \right]} \le C\left\| f^{(2)}\right\|_{[\xi_0,\xi_{n-1}} \left| \xi \right|^2 $$

where

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

CsAkima is based on a method by Akima (1970) to combat wiggles in the interpolant. The method is nonlinear; and although the interpolant is a piecewise cubic, cubic polynomials are not reproduced. (However, linear polynomials are reproduced.)

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
CsAkima(double[] xData, double[] yData) Constructs the Akima cubic spline interpolant to the given data points.

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

CsAkima

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

Constructs the Akima cubic spline interpolant to the given data points.

Parameters:

xData - a double array containing the x-coordinates of the data. Values must be distinct.

yData - a double array containing the y-coordinates of the data.

Throws:

IllegalArgumentException - This exception is thrown if the arrays xData and yData do not have the same length.