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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) throws DimensionMismatchException
UnivariateDifferentiableFunction
classes compute both the
value and the first derivative of the function.
value
in interface UnivariateDifferentiableFunction
t
- function input value
DimensionMismatchException
- if t is inconsistent with the
function's free parameters or order
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