org.apache.commons.math3.analysis.function

Class Sinc

java.lang.Object

org.apache.commons.math3.analysis.function.Sinc

All Implemented Interfaces:

DifferentiableUnivariateFunction, UnivariateDifferentiableFunction, UnivariateFunction

public class Sinc
extends java.lang.Object
implements UnivariateDifferentiableFunction, DifferentiableUnivariateFunction

Sinc function, defined by

   sinc(x) = 1            if x = 0,
             sin(x) / x   otherwise.
 

Since:

3.0

Field Summary

Fields

Modifier and Type	Field and Description
private boolean	normalized For normalized sinc function.
private static double	SHORTCUT Value below which the computations are done using Taylor series.

Constructor Summary

Constructors

Constructor and Description
Sinc() The sinc function, sin(x) / x.
Sinc(boolean normalized) Instantiates the sinc function.

Method Summary

Methods

Modifier and Type	Method and Description
UnivariateFunction	derivative() Deprecated. as of 3.1, replaced by value(DerivativeStructure)
DerivativeStructure	value(DerivativeStructure t) Simple mathematical function.
double	value(double x) Compute the value of the function.

Methods inherited from class java.lang.Object

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

Field Detail

SHORTCUT

private static final double SHORTCUT

Value below which the computations are done using Taylor series.

The Taylor series for sinc even order derivatives are:

 d^(2n)sinc/dx^(2n)     = Sum_(k>=0) (-1)^(n+k) / ((2k)!(2n+2k+1)) x^(2k)
                        = (-1)^n     [ 1/(2n+1) - x^2/(4n+6) + x^4/(48n+120) - x^6/(1440n+5040) + O(x^8) ]
 

The Taylor series for sinc odd order derivatives are:

 d^(2n+1)sinc/dx^(2n+1) = Sum_(k>=0) (-1)^(n+k+1) / ((2k+1)!(2n+2k+3)) x^(2k+1)
                        = (-1)^(n+1) [ x/(2n+3) - x^3/(12n+30) + x^5/(240n+840) - x^7/(10080n+45360) + O(x^9) ]
 

So the ratio of the fourth term with respect to the first term is always smaller than x^6/720, for all derivative orders. This implies that neglecting this term and using only the first three terms induces a relative error bounded by x^6/720. The SHORTCUT value is chosen such that this relative error is below double precision accuracy when |x| <= SHORTCUT.

See Also:

Constant Field Values

normalized

private final boolean normalized

For normalized sinc function.

Constructor Detail

Sinc

public Sinc()

The sinc function, sin(x) / x.

Sinc

public Sinc(boolean normalized)

Instantiates the sinc function.

Parameters:

normalized - If true, the function is sin(πx) / πx, otherwise sin(x) / x.

Method Detail

value

public double value(double x)

Compute the value of the function.

Specified by:

value in interface UnivariateFunction

Parameters:

x - Point at which the function value should be computed.

Returns:

the value of the function.

derivative

@Deprecated
public UnivariateFunction derivative()

Deprecated. as of 3.1, replaced by value(DerivativeStructure)

Returns the derivative of the function

Specified by:

derivative in interface DifferentiableUnivariateFunction

Returns:

the derivative function

value

public DerivativeStructure value(DerivativeStructure t)
                          throws DimensionMismatchException

Simple mathematical function.

UnivariateDifferentiableFunction classes compute both the value and the first derivative of the function.

Specified by:

value in interface UnivariateDifferentiableFunction

Parameters:

t - function input value

Returns:

function result

Throws:

DimensionMismatchException - if t is inconsistent with the function's free parameters or order

Since:

3.1