com.imsl.math

Class CsInterpolate

java.lang.Object

com.imsl.math.Spline

com.imsl.math.CsInterpolate

All Implemented Interfaces:

Serializable, Cloneable

public class CsInterpolate
extends Spline

Extension of the Spline class to interpolate data points.

CsInterpolate 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. Endpoint conditions can be automatically determined by the program, or explicitly specified by using the appropriate constructor. Constructors are provided that allow setting specific values for first or second derivative values at the endpoints, or for specifying conditions that correspond to the "not-a-knot" condition (see de Boor 1978).

The "not-a-knot" conditions require that the third derivative of the spline be continuous at the second and next-to-last breakpoint. If n is 2 or 3, then the linear or quadratic interpolating polynomial is computed, respectively.

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

$$ |f-s|_{[\xi_0,\xi_n]} \le C \left\|f^{(4)}\right\|_{[{\xi_0 ,\xi_n }]} |\xi|^4 $$

where

$$|\xi|\;: = \max_{i=0,\ldots,n-1} \left|\xi_{i+1} - \xi_i \right|$$

For more details, see de Boor (1978, pages 55-56).

See Also:

Example, Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
static int	FIRST_DERIVATIVE
static int	NOT_A_KNOT
static int	SECOND_DERIVATIVE

Fields inherited from class com.imsl.math.Spline

breakPoint, coef, EPSILON_LARGE

Constructor Summary

Constructors

Constructor and Description
CsInterpolate(double[] xData, double[] yData) Constructs a cubic spline that interpolates the given data points.
CsInterpolate(double[] xData, double[] yData, int typeLeft, double valueLeft, int typeRight, double valueRight) Constructs a cubic spline that interpolates the given data points with specified derivative endpoint 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

Field Detail

NOT_A_KNOT

public static final int NOT_A_KNOT

See Also:

Constant Field Values

FIRST_DERIVATIVE

public static final int FIRST_DERIVATIVE

See Also:

Constant Field Values

SECOND_DERIVATIVE

public static final int SECOND_DERIVATIVE

See Also:

Constant Field Values

Constructor Detail

CsInterpolate

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

Constructs a cubic spline that interpolates the given data points. The interpolant satisfies the "not-a-knot" condition.

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. The arrays xData and yData must have the same length.

CsInterpolate

public CsInterpolate(double[] xData,
                     double[] yData,
                     int typeLeft,
                     double valueLeft,
                     int typeRight,
                     double valueRight)

Constructs a cubic spline that interpolates the given data points with specified derivative endpoint conditions.

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. The arrays xData and yData must have the same length.

typeLeft - An int denoting the type of condition at the left endpoint. This can be NOT_A_KNOT, FIRST_DERIVATIVE or SECOND_DERIVATIVE.

valueLeft - A double value at the left endpoint. If typeLeft is NOT_A_KNOT this is ignored, Otherwise, it is the value of the specified derivative.

typeRight - An int denoting the type of condition at the right endpoint. This can be NOT_A_KNOT, FIRST_DERIVATIVE or SECOND_DERIVATIVE.

valueRight - A double value at the right endpoint.