gamma_inc.c

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                       * ( 3.041323283529310 + lnx)
                       * ( 8.295966556941250 + lnx);
    const double c9 = -2.75573192239859e-6
                       * (-2.8243487670469080 + lnx)
                       * (-2.3798494322701120 + lnx)
                       * (-1.9143674728689960 + lnx)
                       * (-1.3814529102920370 + lnx)
                       * (-0.7294312810261694 + lnx)
                       * ( 0.1299079285269565 + lnx)
                       * ( 1.3873333251885240 + lnx)
                       * ( 3.5857258865210760 + lnx)
                       * ( 9.3214237073814600 + lnx);
    const double c10 = -2.75573192239859e-7
                       * (-2.9540329644556910 + lnx)
                       * (-2.5491366926991850 + lnx)
                       * (-2.1348279229279880 + lnx)
                       * (-1.6741881076349450 + lnx)
                       * (-1.1325949616098420 + lnx)
                       * (-0.4590034650618494 + lnx)
                       * ( 0.4399352987435699 + lnx)
                       * ( 1.7702236517651670 + lnx)
                       * ( 4.1231539047474080 + lnx)
                       * ( 10.342627908148680 + lnx);

    term1 = a*(c1+a*(c2+a*(c3+a*(c4+a*(c5+a*(c6+a*(c7+a*(c8+a*(c9+a*c10)))))))));
  }

  {
    /* Evaluate the sum.
     */
    const int nmax = 5000;
    double t = 1.0;
    int n;
    sum = 1.0;

    for(n=1; n<nmax; n++) {
      t *= -x/(n+1.0);
      sum += (a+1.0)/(a+n+1.0)*t;
      if(fabs(t/sum) < GSL_DBL_EPSILON) break;
    }
    
    if(n == nmax)
      stat_sum = GSL_EMAXITER;
    else
      stat_sum = GSL_SUCCESS;
  }

  term2 = (1.0 - term1) * a/(a+1.0) * x * sum;
  result->val  = term1 + term2;
  result->err  = GSL_DBL_EPSILON * (fabs(term1) + 2.0*fabs(term2));
  result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
  return stat_sum;
}


/* series for small a and x, but not defined for a == 0 */
static int
gamma_inc_series(double a, double x, gsl_sf_result * result)
{
  gsl_sf_result Q;
  gsl_sf_result G;
  const int stat_Q = gamma_inc_Q_series(a, x, &Q);
  const int stat_G = gsl_sf_gamma_e(a, &G);
  result->val = Q.val * G.val;
  result->err = fabs(Q.val * G.err) + fabs(Q.err * G.val);
  result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

  return GSL_ERROR_SELECT_2(stat_Q, stat_G);
}


static int
gamma_inc_a_gt_0(double a, double x, gsl_sf_result * result)
{
  /* x > 0 and a > 0; use result for Q */
  gsl_sf_result Q;
  gsl_sf_result G;
  const int stat_Q = gsl_sf_gamma_inc_Q_e(a, x, &Q);
  const int stat_G = gsl_sf_gamma_e(a, &G);

  result->val = G.val * Q.val;
  result->err = fabs(G.val * Q.err) + fabs(G.err * Q.val);
  result->err += 2.0*GSL_DBL_EPSILON * fabs(result->val);

  return GSL_ERROR_SELECT_2(stat_G, stat_Q);
}


static int
gamma_inc_CF(double a, double x, gsl_sf_result * result)
{
  gsl_sf_result F;
  gsl_sf_result pre;
  const int stat_F = gamma_inc_F_CF(a, x, &F);
  const int stat_E = gsl_sf_exp_e((a-1.0)*log(x) - x, &pre);

  result->val = F.val * pre.val;
  result->err = fabs(F.err * pre.val) + fabs(F.val * pre.err);
  result->err += (2.0 + fabs(a)) * GSL_DBL_EPSILON * fabs(result->val);

  return GSL_ERROR_SELECT_2(stat_F, stat_E);
}


/* evaluate Gamma(0,x), x > 0 */
#define GAMMA_INC_A_0(x, result) gsl_sf_expint_E1_e(x, result)


/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/

int
gsl_sf_gamma_inc_Q_e(const double a, const double x, gsl_sf_result * result)
{
  if(a < 0.0 || x < 0.0) {
    DOMAIN_ERROR(result);
  }
  else if(x == 0.0) {
    result->val = 1.0;
    result->err = 0.0;
    return GSL_SUCCESS;
  }
  else if(a == 0.0)
  {
    result->val = 0.0;
    result->err = 0.0;
    return GSL_SUCCESS;
  }
  else if(x <= 0.5*a) {
    /* If the series is quick, do that. It is
     * robust and simple.
     */
    gsl_sf_result P;
    int stat_P = gamma_inc_P_series(a, x, &P);
    result->val  = 1.0 - P.val;
    result->err  = P.err;
    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return stat_P;
  }
  else if(a >= 1.0e+06 && (x-a)*(x-a) < a) {
    /* Then try the difficult asymptotic regime.
     * This is the only way to do this region.
     */
    return gamma_inc_Q_asymp_unif(a, x, result);
  }
  else if(a < 0.2 && x < 5.0) {
    /* Cancellations at small a must be handled
     * analytically; x should not be too big
     * either since the series terms grow
     * with x and log(x).
     */
    return gamma_inc_Q_series(a, x, result);
  }
  else if(a <= x) {
    if(x <= 1.0e+06) {
      /* Continued fraction is excellent for x >~ a.
       * We do not let x be too large when x > a since
       * it is somewhat pointless to try this there;
       * the function is rapidly decreasing for
       * x large and x > a, and it will just
       * underflow in that region anyway. We 
       * catch that case in the standard
       * large-x method.
       */
      return gamma_inc_Q_CF(a, x, result);
    }
    else {
      return gamma_inc_Q_large_x(a, x, result);
    }
  }
  else {
    if(a < 0.8*x) {
      /* Continued fraction again. The convergence
       * is a little slower here, but that is fine.
       * We have to trade that off against the slow
       * convergence of the series, which is the
       * only other option.
       */
      return gamma_inc_Q_CF(a, x, result);
    }
    else {
      gsl_sf_result P;
      int stat_P = gamma_inc_P_series(a, x, &P);
      result->val  = 1.0 - P.val;
      result->err  = P.err;
      result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
      return stat_P;
    }
  }
}


int
gsl_sf_gamma_inc_P_e(const double a, const double x, gsl_sf_result * result)
{
  if(a <= 0.0 || x < 0.0) {
    DOMAIN_ERROR(result);
  }
  else if(x == 0.0) {
    result->val = 0.0;
    result->err = 0.0;
    return GSL_SUCCESS;
  }
  else if(x < 20.0 || x < 0.5*a) {
    /* Do the easy series cases. Robust and quick.
     */
    return gamma_inc_P_series(a, x, result);
  }
  else if(a > 1.0e+06 && (x-a)*(x-a) < a) {
    /* Crossover region. Note that Q and P are
     * roughly the same order of magnitude here,
     * so the subtraction is stable.
     */
    gsl_sf_result Q;
    int stat_Q = gamma_inc_Q_asymp_unif(a, x, &Q);
    result->val  = 1.0 - Q.val;
    result->err  = Q.err;
    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return stat_Q;
  }
  else if(a <= x) {
    /* Q <~ P in this area, so the
     * subtractions are stable.
     */
    gsl_sf_result Q;
    int stat_Q;
    if(a > 0.2*x) {
      stat_Q = gamma_inc_Q_CF(a, x, &Q);
    }
    else {
      stat_Q = gamma_inc_Q_large_x(a, x, &Q);
    }
    result->val  = 1.0 - Q.val;
    result->err  = Q.err;
    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return stat_Q;
  }
  else {
    if((x-a)*(x-a) < a) {
      /* This condition is meant to insure
       * that Q is not very close to 1,
       * so the subtraction is stable.
       */
      gsl_sf_result Q;
      int stat_Q = gamma_inc_Q_CF(a, x, &Q);
      result->val  = 1.0 - Q.val;
      result->err  = Q.err;
      result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
      return stat_Q;
    }
    else {
      return gamma_inc_P_series(a, x, result);
    }
  }
}


int
gsl_sf_gamma_inc_e(const double a, const double x, gsl_sf_result * result)
{  
  if(x < 0.0) {
    DOMAIN_ERROR(result);
  }
  else if(x == 0.0) {
    return gsl_sf_gamma_e(a, result);
  }
  else if(a == 0.0)
  {
    return GAMMA_INC_A_0(x, result);
  }
  else if(a > 0.0)
  {
    return gamma_inc_a_gt_0(a, x, result);
  }
  else if(x > 0.25)
  {
    /* continued fraction seems to fail for x too small; otherwise
       it is ok, independent of the value of |x/a|, because of the
       non-oscillation in the expansion, i.e. the CF is
       un-conditionally convergent for a < 0 and x > 0
     */
    return gamma_inc_CF(a, x, result);
  }
  else if(fabs(a) < 0.5)
  {
    return gamma_inc_series(a, x, result);
  }
  else
  {
    /* a = fa + da; da >= 0 */
    const double fa = floor(a);
    const double da = a - fa;

    gsl_sf_result g_da;
    const int stat_g_da = ( da > 0.0 ? gamma_inc_a_gt_0(da, x, &g_da)
                                     : GAMMA_INC_A_0(x, &g_da));

    double alpha = da;
    double gax = g_da.val;

    /* Gamma(alpha-1,x) = 1/(alpha-1) (Gamma(a,x) - x^(alpha-1) e^-x) */
    do
    {
      const double shift = exp(-x + (alpha-1.0)*log(x));
      gax = (gax - shift) / (alpha - 1.0);
      alpha -= 1.0;
    } while(alpha > a);

    result->val = gax;
    result->err = 2.0*(1.0 + fabs(a))*GSL_DBL_EPSILON*fabs(gax);
    return stat_g_da;
  }

}


/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/

#include "eval.h"

double gsl_sf_gamma_inc_P(const double a, const double x)
{
  EVAL_RESULT(gsl_sf_gamma_inc_P_e(a, x, &result));
}

double gsl_sf_gamma_inc_Q(const double a, const double x)
{
  EVAL_RESULT(gsl_sf_gamma_inc_Q_e(a, x, &result));
}

double gsl_sf_gamma_inc(const double a, const double x)
{
   EVAL_RESULT(gsl_sf_gamma_inc_e(a, x, &result));
}