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java.lang.Objectorg.apache.commons.math3.analysis.function.Sinc
public class Sinc
Sinc function, defined by
sinc(x) = 1 if x = 0,
sin(x) / x otherwise.
Field Summary | |
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private boolean |
normalized
For normalized sinc function. |
private static double |
SHORTCUT
Value below which the computations are done using Taylor series. |
Constructor Summary | |
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Sinc()
The sinc function, sin(x) / x . |
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Sinc(boolean normalized)
Instantiates the sinc function. |
Method Summary | |
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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 |
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clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait |
Field Detail |
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private static final double SHORTCUT
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.
private final boolean normalized
Constructor Detail |
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public Sinc()
sin(x) / x
.
public Sinc(boolean normalized)
normalized
- If true
, the function is
sin(πx) / πx
, otherwise sin(x) / x
.Method Detail |
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public double value(double x)
value
in interface UnivariateFunction
x
- Point at which the function value should be computed.
@Deprecated public UnivariateFunction derivative()
value(DerivativeStructure)
derivative
in interface DifferentiableUnivariateFunction
public DerivativeStructure value(DerivativeStructure t)
UnivariateDifferentiableFunction
classes compute both the
value and the first derivative of the function.
value
in interface UnivariateDifferentiableFunction
t
- function input value
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