/* specfunc/lambert.c

 * 

 * Copyright (C) 2007 Brian Gough

 * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001 Gerard Jungman

 *

 * This program is free software; you can redistribute it and/or modify

 * it under the terms of the GNU General Public License as published by

 * the Free Software Foundation; either version 3 of the License, or (at

 * your option) any later version.

 *

 * This program is distributed in the hope that it will be useful, but

 * WITHOUT ANY WARRANTY; without even the implied warranty of

 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU

 * General Public License for more details.

 *

 * You should have received a copy of the GNU General Public License

 * along with this program; if not, write to the Free Software

 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.

 */

/* Author:  G. Jungman */

#include <config.h>

#include <math.h>

#include <gsl/gsl_math.h>

#include <gsl/gsl_errno.h>

#include <gsl/gsl_sf_lambert.h>

/* Started with code donated by K. Briggs; added

 * error estimates, GSL foo, and minor tweaks.

 * Some Lambert-ology from

 *  [Corless, Gonnet, Hare, and Jeffrey, "On Lambert's W Function".]

 */

/* Halley iteration (eqn. 5.12, Corless et al) */

static int

halley_iteration(

  double x,

  double w_initial,

  unsigned int max_iters,

  gsl_sf_result * result

  )

{

  double w = w_initial;

  unsigned int i;

  for(i=0; i

    double tol;

    const double e = exp(w);

    const double p = w + 1.0;

    double t = w*e - x;

    /* printf("FOO: %20.16g  %20.16g\n", w, t); */

    if (w > 0) {

      t = (t/p)/e;  /* Newton iteration */

    } else {

      t /= e*p - 0.5*(p + 1.0)*t/p;  /* Halley iteration */

    };

    w -= t;

    tol = 10 * GSL_DBL_EPSILON * GSL_MAX_DBL(fabs(w), 1.0/(fabs(p)*e));

    if(fabs(t) < tol)

    {

      result->val = w;

      result->err = 2.0*tol;

      return GSL_SUCCESS;

    }

  }

  /* should never get here */

  result->val = w;

  result->err = fabs(w);

  return GSL_EMAXITER;

}

/* series which appears for q near zero;

 * only the argument is different for the different branches

 */

static double

series_eval(double r)

{

  static const double c[12] = {

    -1.0,

     2.331643981597124203363536062168,

    -1.812187885639363490240191647568,

     1.936631114492359755363277457668,

    -2.353551201881614516821543561516,

     3.066858901050631912893148922704,

    -4.175335600258177138854984177460,

     5.858023729874774148815053846119,

    -8.401032217523977370984161688514,

     12.250753501314460424,

    -18.100697012472442755,

     27.029044799010561650

  };

  const double t_8 = c[8] + r*(c[9] + r*(c[10] + r*c[11]));

  const double t_5 = c[5] + r*(c[6] + r*(c[7]  + r*t_8));

  const double t_1 = c[1] + r*(c[2] + r*(c[3]  + r*(c[4] + r*t_5)));

  return c[0] + r*t_1;

}

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

int

gsl_sf_lambert_W0_e(double x, gsl_sf_result * result)

{

  const double one_over_E = 1.0/M_E;

  const double q = x + one_over_E;

  if(x == 0.0) {

    result->val = 0.0;

    result->err = 0.0;

    return GSL_SUCCESS;

  }

  else if(q < 0.0) {

    /* Strictly speaking this is an error. But because of the

     * arithmetic operation connecting x and q, I am a little

     * lenient in case of some epsilon overshoot. The following

     * answer is quite accurate in that case. Anyway, we have

     * to return GSL_EDOM.

     */

    result->val = -1.0;

    result->err =  sqrt(-q);

    return GSL_EDOM;

  }

  else if(q == 0.0) {

    result->val = -1.0;

    result->err =  GSL_DBL_EPSILON; /* cannot error is zero, maybe q == 0 by "accident" */

    return GSL_SUCCESS;

  }

  else if(q < 1.0e-03) {

    /* series near -1/E in sqrt(q) */

    const double r = sqrt(q);

    result->val = series_eval(r);

    result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);

    return GSL_SUCCESS;

  }

  else {

    static const unsigned int MAX_ITERS = 10;

    double w;

    if (x < 1.0) {

      /* obtain initial approximation from series near x=0;

       * no need for extra care, since the Halley iteration

       * converges nicely on this branch

       */

      const double p = sqrt(2.0 * M_E * q);

      w = -1.0 + p*(1.0 + p*(-1.0/3.0 + p*11.0/72.0)); 

    }

    else {

      /* obtain initial approximation from rough asymptotic */

      w = log(x);

      if(x > 3.0) w -= log(w);

    }

    return halley_iteration(x, w, MAX_ITERS, result);

  }

}

int

gsl_sf_lambert_Wm1_e(double x, gsl_sf_result * result)

{

  if(x > 0.0) {

    return gsl_sf_lambert_W0_e(x, result);

  }

  else if(x == 0.0) {

    result->val = 0.0;

    result->err = 0.0;

    return GSL_SUCCESS;

  }

  else {

    static const unsigned int MAX_ITERS = 32;

    const double one_over_E = 1.0/M_E;

    const double q = x + one_over_E;

    double w;

    if (q < 0.0) {

      /* As in the W0 branch above, return some reasonable answer anyway. */

      result->val = -1.0; 

      result->err =  sqrt(-q);

      return GSL_EDOM;

    }

    if(x < -1.0e-6) {

      /* Obtain initial approximation from series about q = 0,

       * as long as we're not very close to x = 0.

       * Use full series and try to bail out if q is too small,

       * since the Halley iteration has bad convergence properties

       * in finite arithmetic for q very small, because the

       * increment alternates and p is near zero.

       */

      const double r = -sqrt(q);

      w = series_eval(r);

      if(q < 3.0e-3) {

        /* this approximation is good enough */

        result->val = w;

        result->err = 5.0 * GSL_DBL_EPSILON * fabs(w);

        return GSL_SUCCESS;

      }

    }

    else {

      /* Obtain initial approximation from asymptotic near zero. */

      const double L_1 = log(-x);

      const double L_2 = log(-L_1);

      w = L_1 - L_2 + L_2/L_1;

    }

    return halley_iteration(x, w, MAX_ITERS, result);

  }

}

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

#include "eval.h"

double gsl_sf_lambert_W0(double x)

{

  EVAL_RESULT(gsl_sf_lambert_W0_e(x, &result));

}

double gsl_sf_lambert_Wm1(double x)

{

  EVAL_RESULT(gsl_sf_lambert_Wm1_e(x, &result));

}