Class Sinc

    • 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.