function [ n_data, x, fx, branch ] = lambert_w_values ( n_data )

%*****************************************************************************80
%
%% lambert_w_values() returns some values of the Lambert W function.
%
%  Discussion:
%
%    The function W(X) is defined implicitly by:
%
%      W(X) * e^W(X) = X
%
%    The function is defined for -1/e <= x.
%
%    There are two branches, joining at -1/e = x.
%    The lower branch extends from -1/e <= x < 0
%    The upper branch extends from -1/e <= x
%
%    The function is also known as the "Omega" function.
%
%    In Mathematica, the function can be evaluated by:
%      W = ProductLog [ X ]
%
%    In MATLAB,
%      W = lambertw ( b, x )
%    where b = -1 for lower branch, 0 for upper branch.
%
%    In Python,
%      W = scipy.special.lambertw ( x, b )
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    19 June 2023
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    Brian Hayes,
%    "Why W?",
%    The American Scientist,
%    Volume 93, March-April 2005, pages 104-108.
%
%    Eric Weisstein,
%    "Lambert's W-Function",
%    CRC Concise Encyclopedia of Mathematics,
%    CRC Press, 1998.
%
%    Stephen Wolfram,
%    The Mathematica Book,
%    Fourth Edition,
%    Wolfram Media / Cambridge University Press, 1999.
%
%  Input:
%
%    integer n_data: The user sets N_DATA to 0 before the first call.  
%    Thereafter, it should simply be the value returned by the previous call.
%
%  Output:
%
%    integer n_data:  On each call, the routine increments N_DATA by 1, and
%    returns the corresponding data; when there is no more data, the
%    output value of N_DATA will be 0 again.
%
%    real x: the argument of the function.
%
%    real fx: the value of the function.
%
%    integer branch: -1 (lower branch) or 0 (upper branch).
%
  n_max = 41;

  fx_vec = [ ...
   -4.889720169867429, ...
   -3.994308347002122, ...
   -3.439216483280204, ...
   -3.022313245324657, ...
   -2.678346990016661, ...
   -2.376421342062887, ...
   -2.097349210703492, ...
   -1.824388309032984, ...
   -1.531811608389612, ...
   -1.000000000000000, ...
   -0.608341284733432, ...
   -0.471671909743522, ...
   -0.374493134019498, ...
   -0.297083462446424, ...
   -0.231960952986534, ...
   -0.175356500529299, ...
   -0.125066982982524, ...
   -0.079678160511477, ...
   -0.038221241746799, ...
    0.0000000000000000E+00, ...
    0.3517337112491958E+00, ...
    0.5671432904097839E+00, ...
    0.7258613577662263E+00, ...
    0.8526055020137255E+00, ...
    0.9585863567287029E+00, ...
    0.1000000000000000E+01, ...
    0.1049908894964040E+01, ...
    0.1130289326974136E+01, ...
    0.1202167873197043E+01, ...
    0.1267237814307435E+01, ...
    0.1326724665242200E+01, ...
    0.1381545379445041E+01, ...
    0.1432404775898300E+01, ...
    0.1479856830173851E+01, ...
    0.1524345204984144E+01, ...
    0.1566230953782388E+01, ...
    0.1605811996320178E+01, ...
    0.1745528002740699E+01, ...
    0.3385630140290050E+01, ...
    0.5249602852401596E+01, ...
    0.1138335808614005E+02 ];

  x_vec = [ ...
   -0.036787944117144, ...
   -0.073575888234288, ...
   -0.110363832351433, ...
   -0.147151776468577, ...
   -0.183939720585721, ...
   -0.220727664702865, ...
   -0.257515608820010, ...
   -0.294303552937154, ...
   -0.331091497054298, ...
   -0.367879441171442, ...
   -0.331091497054298, ...
   -0.294303552937154, ...
   -0.257515608820010, ...
   -0.220727664702865, ...
   -0.183939720585721, ...
   -0.147151776468577, ...
   -0.110363832351433, ...
   -0.073575888234288, ...
   -0.036787944117144, ...
    0.0000000000000000E+00, ...
    0.5000000000000000E+00, ...
    0.1000000000000000E+01, ...
    0.1500000000000000E+01, ...
    0.2000000000000000E+01, ...
    0.2500000000000000E+01, ...
    0.2718281828459045E+01, ...
    0.3000000000000000E+01, ...
    0.3500000000000000E+01, ...
    0.4000000000000000E+01, ...
    0.4500000000000000E+01, ...
    0.5000000000000000E+01, ...
    0.5500000000000000E+01, ...
    0.6000000000000000E+01, ...
    0.6500000000000000E+01, ...
    0.7000000000000000E+01, ...
    0.7500000000000000E+01, ...
    0.8000000000000000E+01, ...
    0.1000000000000000E+02, ...
    0.1000000000000000E+03, ...
    0.1000000000000000E+04, ...
    0.1000000000000000E+07 ];

  branch_vec = [ ...
    -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, ...
     0,  0,  0,  0,  0,  0,  0,  0,  0,  0, ...
     0,  0,  0,  0,  0,  0,  0,  0,  0,  0, ...
     0,  0,  0,  0,  0,  0,  0,  0,  0,  0, ...
     0 ];

  if ( n_data < 0 )
    n_data = 0;
  end

  n_data = n_data + 1;

  if ( n_max < n_data )
    n_data = 0;
    x = 0.0;
    fx = 0.0;
    branch = 0;
  else
    x = x_vec(n_data);
    fx = fx_vec(n_data);
    branch = branch_vec(n_data);
  end

  return
end
