Eigen::internal::digamma_impl< Scalar > Struct Template Reference

#include <SpecialFunctionsImpl.h>

Static Public Member Functions
static EIGEN_DEVICE_FUNC Scalar	run (Scalar x)

Member Function Documentation

run()

template<typename Scalar >

static EIGEN_DEVICE_FUNC Scalar Eigen::internal::digamma_impl< Scalar >::run ( Scalar  x )

inlinestatic

                                                {
    /*
     *
     *     Psi (digamma) function (modified for Eigen)
     *
     *
     * SYNOPSIS:
     *
     * double x, y, psi();
     *
     * y = psi( x );
     *
     *
     * DESCRIPTION:
     *
     *              d      -
     *   psi(x)  =  -- ln | (x)
     *              dx
     *
     * is the logarithmic derivative of the gamma function.
     * For integer x,
     *                   n-1
     *                    -
     * psi(n) = -EUL  +   >  1/k.
     *                    -
     *                   k=1
     *
     * If x is negative, it is transformed to a positive argument by the
     * reflection formula  psi(1-x) = psi(x) + pi cot(pi x).
     * For general positive x, the argument is made greater than 10
     * using the recurrence  psi(x+1) = psi(x) + 1/x.
     * Then the following asymptotic expansion is applied:
     *
     *                           inf.   B
     *                            -      2k
     * psi(x) = log(x) - 1/2x -   >   -------
     *                            -        2k
     *                           k=1   2k x
     *
     * where the B2k are Bernoulli numbers.
     *
     * ACCURACY (float):
     *    Relative error (except absolute when |psi| < 1):
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      0,30        30000       1.3e-15     1.4e-16
     *    IEEE      -30,0       40000       1.5e-15     2.2e-16
     *
     * ACCURACY (double):
     *    Absolute error,  relative when |psi| > 1 :
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      -33,0        30000      8.2e-7      1.2e-7
     *    IEEE      0,33        100000      7.3e-7      7.7e-8
     *
     * ERROR MESSAGES:
     *     message         condition      value returned
     * psi singularity    x integer <=0      INFINITY
     */
 
    Scalar p, q, nz, s, w, y;
    bool negative = false;
 
    const Scalar nan = NumTraits<Scalar>::quiet_NaN();
    const Scalar m_pi = Scalar(EIGEN_PI);
 
    const Scalar zero = Scalar(0);
    const Scalar one = Scalar(1);
    const Scalar half = Scalar(0.5);
    nz = zero;
 
    if (x <= zero) {
      negative = true;
      q = x;
      p = numext::floor(q);
      if (p == q) {
        return nan;
      }
      /* Remove the zeros of tan(m_pi x)
       * by subtracting the nearest integer from x
       */
      nz = q - p;
      if (nz != half) {
        if (nz > half) {
          p += one;
          nz = q - p;
        }
        nz = m_pi / numext::tan(m_pi * nz);
      } else {
        nz = zero;
      }
      x = one - x;
    }
 
    /* use the recurrence psi(x+1) = psi(x) + 1/x. */
    s = x;
    w = zero;
    while (s < Scalar(10)) {
      w += one / s;
      s += one;
    }
 
    y = digamma_impl_maybe_poly<Scalar>::run(s);
 
    y = numext::log(s) - (half / s) - y - w;
 
    return (negative) ? y - nz : y;
  }

References EIGEN_PI, Eigen::numext::floor(), Eigen::numext::log(), Mesh_Parameters::nz, p, Eigen::numext::q, Eigen::internal::digamma_impl_maybe_poly< Scalar >::run(), s, Eigen::numext::tan(), w, Eigen::numext::x, Eigen::internal::y, and zero().

Referenced by Eigen::internal::igammac_cf_impl< Scalar, mode >::run(), and Eigen::internal::igamma_series_impl< Scalar, mode >::run().

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The documentation for this struct was generated from the following file:

• SpecialFunctionsImpl.h