QuantFuncMathCore.cxx

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// @(#)root/mathcore:$Id$
// Authors: L. Moneta, A. Zsenei   08/2005


#include "Math/Math.h"
#include "Math/QuantFuncMathCore.h"
#include "SpecFuncCephes.h"
#include <limits>


namespace ROOT {
namespace Math {



   double beta_quantile_c(double z, double a, double b) {
      // use Cephes and proprety of icomplete beta function
      if ( z < 0.5)
         return 1. - ROOT::Math::Cephes::incbi(b,a,z);
      else
         return ROOT::Math::Cephes::incbi(a,b,1.0-z);

   }


   double beta_quantile(double z, double a, double b ) {
      // use Cephes function
      return ROOT::Math::Cephes::incbi(a,b,z);

   }


   double cauchy_quantile_c(double z, double b) {
      // inverse of Caucy is simply the tan(PI(z-0.5))
      if (z == 0) return std::numeric_limits<double>::infinity();
      if (z == 1) return - std::numeric_limits<double>::infinity();
      if (z < 0.5)
         // use fact that tan(PI(0.5-z)) = 1/tan(PI*z)
         return  b / std::tan( M_PI * z );
      else
         return  b * std::tan( M_PI * (0.5 -  z ) );
   }



   double cauchy_quantile(double z, double b) {
      // inverse of Caucy is simply the tan(PI(z-0.5))
      if (z == 0) return - std::numeric_limits<double>::infinity();
      if (z == 1) return + std::numeric_limits<double>::infinity();
      if (z < 0.5)
         // use fact that tan(PI(0.5-z)) = 1/tan(PI*z)
         return  - b / std::tan( M_PI * z );
      else
         return  b * std::tan( M_PI * ( z -  0.5 ) );

   }



   double chisquared_quantile_c(double z, double r) {
      // use Cephes igami which return inverse of complemented regularized gamma
      return 2.* ROOT::Math::Cephes::igami( 0.5 *r, z);

   }


   double chisquared_quantile(double z, double r) {
      // if possible, should use MathMore function ROOT::Math::chisquared_quantile for z close to zero
      // otherwise will always return zero for z  value smaller than eps
      return 2.* ROOT::Math::Cephes::igami( 0.5 *r, 1. - z);
   }


   double exponential_quantile_c(double z, double lambda) {

      return - std::log(z)/ lambda;

   }



   double exponential_quantile(double z, double lambda) {
      // use log1p for avoid errors at small z
      return - ROOT::Math::log1p(-z)/lambda;

   }


   double fdistribution_quantile_c(double z, double n, double m) {
      // use cephes incbi function and use propreties of incomplete beta for case <> 0.5
      if (n == 0) return 0;  // is value of cdf for n = 0
      if (z < 0.5) {
         double y =  ROOT::Math::Cephes::incbi( .5*m, .5*n, z);
         return m/(n * y) - m/n;
      }
      else {
         double y =  ROOT::Math::Cephes::incbi( .5*n, .5*m, 1.0 - z);
         // will lose precision for y approx to 1
         return  m * y /(n * ( 1. - y) );
      }
   }

   double fdistribution_quantile(double z, double n, double m) {
      // use cephes incbi function
      if (n == 0) return 0;  // is value of cdf for n = 0
      double y =  ROOT::Math::Cephes::incbi( .5*n, .5*m, z);
      // will lose precision for y approx to 1
      return  m * y /(n * ( 1. - y) );
   }


   double gamma_quantile_c(double z, double alpha, double theta) {

      return theta * ROOT::Math::Cephes::igami( alpha, z);

   }

   double gamma_quantile(double z, double alpha, double theta) {
      // if possible, should use MathMore function ROOT::Math::gamma_quantile for z close to zero
      // otherwise will always return zero for z  value smaller than eps
      return theta * ROOT::Math::Cephes::igami( alpha, 1.- z);
   }



   double normal_quantile_c(double z, double sigma) {
      // use cephes and fact that ntri(1.-x) = - ndtri(x)
      return - sigma * ROOT::Math::Cephes::ndtri(z);

   }



   double normal_quantile(double z, double sigma) {
      // use cephes ndtri function
      return  sigma * ROOT::Math::Cephes::ndtri(z);

   }




   double lognormal_quantile_c(double z, double m, double s) {
      // if y is log normal, u = exp(y) is log-normal distributed
      double y = - s * ROOT::Math::Cephes::ndtri(z) + m;
      return std::exp(y);
   }



   double lognormal_quantile(double z, double m, double s) {
      // if y is log normal, u = exp(y) is log-normal distributed
      double y = s * ROOT::Math::Cephes::ndtri(z) + m;
      return std::exp(y);

   }


//   double tdistribution_quantile_c(double z, double r) {

//     return gsl_cdf_tdist_Qinv(z, r);

//   }



//   double tdistribution_quantile(double z, double r) {

//     return gsl_cdf_tdist_Pinv(z, r);

//   }



   double uniform_quantile_c(double z, double a, double b) {

      return a * z  + b * (1.0 - z);

   }



   double uniform_quantile(double z, double a, double b) {

      return b * z + a * (1.0 - z);

   }

   double landau_quantile(double z, double xi) {
      // LANDAU quantile : algorithm from CERNLIB G110 ranlan
      // with scale parameter xi
      // Converted by Rene Brun from CERNLIB routine ranlan(G110),
      // Moved and adapted to QuantFuncMathCore by B. List 29.4.2010

      static double f[982] = {
          0       , 0       , 0       ,0        ,0        ,-2.244733,
         -2.204365,-2.168163,-2.135219,-2.104898,-2.076740,-2.050397,
         -2.025605,-2.002150,-1.979866,-1.958612,-1.938275,-1.918760,
         -1.899984,-1.881879,-1.864385,-1.847451,-1.831030,-1.815083,
         -1.799574,-1.784473,-1.769751,-1.755383,-1.741346,-1.727620,
         -1.714187,-1.701029,-1.688130,-1.675477,-1.663057,-1.650858,
         -1.638868,-1.627078,-1.615477,-1.604058,-1.592811,-1.581729,
         -1.570806,-1.560034,-1.549407,-1.538919,-1.528565,-1.518339,
         -1.508237,-1.498254,-1.488386,-1.478628,-1.468976,-1.459428,
         -1.449979,-1.440626,-1.431365,-1.422195,-1.413111,-1.404112,
         -1.395194,-1.386356,-1.377594,-1.368906,-1.360291,-1.351746,
         -1.343269,-1.334859,-1.326512,-1.318229,-1.310006,-1.301843,
         -1.293737,-1.285688,-1.277693,-1.269752,-1.261863,-1.254024,
         -1.246235,-1.238494,-1.230800,-1.223153,-1.215550,-1.207990,
         -1.200474,-1.192999,-1.185566,-1.178172,-1.170817,-1.163500,
         -1.156220,-1.148977,-1.141770,-1.134598,-1.127459,-1.120354,
         -1.113282,-1.106242,-1.099233,-1.092255,
         -1.085306,-1.078388,-1.071498,-1.064636,-1.057802,-1.050996,
         -1.044215,-1.037461,-1.030733,-1.024029,-1.017350,-1.010695,
         -1.004064, -.997456, -.990871, -.984308, -.977767, -.971247,
          -.964749, -.958271, -.951813, -.945375, -.938957, -.932558,
          -.926178, -.919816, -.913472, -.907146, -.900838, -.894547,
          -.888272, -.882014, -.875773, -.869547, -.863337, -.857142,
          -.850963, -.844798, -.838648, -.832512, -.826390, -.820282,
          -.814187, -.808106, -.802038, -.795982, -.789940, -.783909,
          -.777891, -.771884, -.765889, -.759906, -.753934, -.747973,
          -.742023, -.736084, -.730155, -.724237, -.718328, -.712429,
          -.706541, -.700661, -.694791, -.688931, -.683079, -.677236,
          -.671402, -.665576, -.659759, -.653950, -.648149, -.642356,
          -.636570, -.630793, -.625022, -.619259, -.613503, -.607754,
          -.602012, -.596276, -.590548, -.584825, -.579109, -.573399,
          -.567695, -.561997, -.556305, -.550618, -.544937, -.539262,
          -.533592, -.527926, -.522266, -.516611, -.510961, -.505315,
          -.499674, -.494037, -.488405, -.482777,
          -.477153, -.471533, -.465917, -.460305, -.454697, -.449092,
          -.443491, -.437893, -.432299, -.426707, -.421119, -.415534,
          -.409951, -.404372, -.398795, -.393221, -.387649, -.382080,
          -.376513, -.370949, -.365387, -.359826, -.354268, -.348712,
          -.343157, -.337604, -.332053, -.326503, -.320955, -.315408,
          -.309863, -.304318, -.298775, -.293233, -.287692, -.282152,
          -.276613, -.271074, -.265536, -.259999, -.254462, -.248926,
          -.243389, -.237854, -.232318, -.226783, -.221247, -.215712,
          -.210176, -.204641, -.199105, -.193568, -.188032, -.182495,
          -.176957, -.171419, -.165880, -.160341, -.154800, -.149259,
          -.143717, -.138173, -.132629, -.127083, -.121537, -.115989,
          -.110439, -.104889, -.099336, -.093782, -.088227, -.082670,
          -.077111, -.071550, -.065987, -.060423, -.054856, -.049288,
          -.043717, -.038144, -.032569, -.026991, -.021411, -.015828,
          -.010243, -.004656,  .000934,  .006527,  .012123,  .017722,
           .023323,  .028928,  .034535,  .040146,  .045759,  .051376,
           .056997,  .062620,  .068247,  .073877,
           .079511,  .085149,  .090790,  .096435,  .102083,  .107736,
           .113392,  .119052,  .124716,  .130385,  .136057,  .141734,
           .147414,  .153100,  .158789,  .164483,  .170181,  .175884,
           .181592,  .187304,  .193021,  .198743,  .204469,  .210201,
           .215937,  .221678,  .227425,  .233177,  .238933,  .244696,
           .250463,  .256236,  .262014,  .267798,  .273587,  .279382,
           .285183,  .290989,  .296801,  .302619,  .308443,  .314273,
           .320109,  .325951,  .331799,  .337654,  .343515,  .349382,
           .355255,  .361135,  .367022,  .372915,  .378815,  .384721,
           .390634,  .396554,  .402481,  .408415,  .414356,  .420304,
           .426260,  .432222,  .438192,  .444169,  .450153,  .456145,
           .462144,  .468151,  .474166,  .480188,  .486218,  .492256,
           .498302,  .504356,  .510418,  .516488,  .522566,  .528653,
           .534747,  .540850,  .546962,  .553082,  .559210,  .565347,
           .571493,  .577648,  .583811,  .589983,  .596164,  .602355,
           .608554,  .614762,  .620980,  .627207,  .633444,  .639689,
           .645945,  .652210,  .658484,  .664768,
           .671062,  .677366,  .683680,  .690004,  .696338,  .702682,
           .709036,  .715400,  .721775,  .728160,  .734556,  .740963,
           .747379,  .753807,  .760246,  .766695,  .773155,  .779627,
           .786109,  .792603,  .799107,  .805624,  .812151,  .818690,
           .825241,  .831803,  .838377,  .844962,  .851560,  .858170,
           .864791,  .871425,  .878071,  .884729,  .891399,  .898082,
           .904778,  .911486,  .918206,  .924940,  .931686,  .938446,
           .945218,  .952003,  .958802,  .965614,  .972439,  .979278,
           .986130,  .992996,  .999875, 1.006769, 1.013676, 1.020597,
          1.027533, 1.034482, 1.041446, 1.048424, 1.055417, 1.062424,
          1.069446, 1.076482, 1.083534, 1.090600, 1.097681, 1.104778,
          1.111889, 1.119016, 1.126159, 1.133316, 1.140490, 1.147679,
          1.154884, 1.162105, 1.169342, 1.176595, 1.183864, 1.191149,
          1.198451, 1.205770, 1.213105, 1.220457, 1.227826, 1.235211,
          1.242614, 1.250034, 1.257471, 1.264926, 1.272398, 1.279888,
          1.287395, 1.294921, 1.302464, 1.310026, 1.317605, 1.325203,
          1.332819, 1.340454, 1.348108, 1.355780,
          1.363472, 1.371182, 1.378912, 1.386660, 1.394429, 1.402216,
          1.410024, 1.417851, 1.425698, 1.433565, 1.441453, 1.449360,
          1.457288, 1.465237, 1.473206, 1.481196, 1.489208, 1.497240,
          1.505293, 1.513368, 1.521465, 1.529583, 1.537723, 1.545885,
          1.554068, 1.562275, 1.570503, 1.578754, 1.587028, 1.595325,
          1.603644, 1.611987, 1.620353, 1.628743, 1.637156, 1.645593,
          1.654053, 1.662538, 1.671047, 1.679581, 1.688139, 1.696721,
          1.705329, 1.713961, 1.722619, 1.731303, 1.740011, 1.748746,
          1.757506, 1.766293, 1.775106, 1.783945, 1.792810, 1.801703,
          1.810623, 1.819569, 1.828543, 1.837545, 1.846574, 1.855631,
          1.864717, 1.873830, 1.882972, 1.892143, 1.901343, 1.910572,
          1.919830, 1.929117, 1.938434, 1.947781, 1.957158, 1.966566,
          1.976004, 1.985473, 1.994972, 2.004503, 2.014065, 2.023659,
          2.033285, 2.042943, 2.052633, 2.062355, 2.072110, 2.081899,
          2.091720, 2.101575, 2.111464, 2.121386, 2.131343, 2.141334,
          2.151360, 2.161421, 2.171517, 2.181648, 2.191815, 2.202018,
          2.212257, 2.222533, 2.232845, 2.243195,
          2.253582, 2.264006, 2.274468, 2.284968, 2.295507, 2.306084,
          2.316701, 2.327356, 2.338051, 2.348786, 2.359562, 2.370377,
          2.381234, 2.392131, 2.403070, 2.414051, 2.425073, 2.436138,
          2.447246, 2.458397, 2.469591, 2.480828, 2.492110, 2.503436,
          2.514807, 2.526222, 2.537684, 2.549190, 2.560743, 2.572343,
          2.583989, 2.595682, 2.607423, 2.619212, 2.631050, 2.642936,
          2.654871, 2.666855, 2.678890, 2.690975, 2.703110, 2.715297,
          2.727535, 2.739825, 2.752168, 2.764563, 2.777012, 2.789514,
          2.802070, 2.814681, 2.827347, 2.840069, 2.852846, 2.865680,
          2.878570, 2.891518, 2.904524, 2.917588, 2.930712, 2.943894,
          2.957136, 2.970439, 2.983802, 2.997227, 3.010714, 3.024263,
          3.037875, 3.051551, 3.065290, 3.079095, 3.092965, 3.106900,
          3.120902, 3.134971, 3.149107, 3.163312, 3.177585, 3.191928,
          3.206340, 3.220824, 3.235378, 3.250005, 3.264704, 3.279477,
          3.294323, 3.309244, 3.324240, 3.339312, 3.354461, 3.369687,
          3.384992, 3.400375, 3.415838, 3.431381, 3.447005, 3.462711,
          3.478500, 3.494372, 3.510328, 3.526370,
          3.542497, 3.558711, 3.575012, 3.591402, 3.607881, 3.624450,
          3.641111, 3.657863, 3.674708, 3.691646, 3.708680, 3.725809,
          3.743034, 3.760357, 3.777779, 3.795300, 3.812921, 3.830645,
          3.848470, 3.866400, 3.884434, 3.902574, 3.920821, 3.939176,
          3.957640, 3.976215, 3.994901, 4.013699, 4.032612, 4.051639,
          4.070783, 4.090045, 4.109425, 4.128925, 4.148547, 4.168292,
          4.188160, 4.208154, 4.228275, 4.248524, 4.268903, 4.289413,
          4.310056, 4.330832, 4.351745, 4.372794, 4.393982, 4.415310,
          4.436781, 4.458395, 4.480154, 4.502060, 4.524114, 4.546319,
          4.568676, 4.591187, 4.613854, 4.636678, 4.659662, 4.682807,
          4.706116, 4.729590, 4.753231, 4.777041, 4.801024, 4.825179,
          4.849511, 4.874020, 4.898710, 4.923582, 4.948639, 4.973883,
          4.999316, 5.024942, 5.050761, 5.076778, 5.102993, 5.129411,
          5.156034, 5.182864, 5.209903, 5.237156, 5.264625, 5.292312,
          5.320220, 5.348354, 5.376714, 5.405306, 5.434131, 5.463193,
          5.492496, 5.522042, 5.551836, 5.581880, 5.612178, 5.642734,
          5.673552, 5.704634, 5.735986, 5.767610,
          5.799512, 5.831694, 5.864161, 5.896918, 5.929968, 5.963316,
          5.996967, 6.030925, 6.065194, 6.099780, 6.134687, 6.169921,
          6.205486, 6.241387, 6.277630, 6.314220, 6.351163, 6.388465,
          6.426130, 6.464166, 6.502578, 6.541371, 6.580553, 6.620130,
          6.660109, 6.700495, 6.741297, 6.782520, 6.824173, 6.866262,
          6.908795, 6.951780, 6.995225, 7.039137, 7.083525, 7.128398,
          7.173764, 7.219632, 7.266011, 7.312910, 7.360339, 7.408308,
          7.456827, 7.505905, 7.555554, 7.605785, 7.656608, 7.708035,
          7.760077, 7.812747, 7.866057, 7.920019, 7.974647, 8.029953,
          8.085952, 8.142657, 8.200083, 8.258245, 8.317158, 8.376837,
          8.437300, 8.498562, 8.560641, 8.623554, 8.687319, 8.751955,
          8.817481, 8.883916, 8.951282, 9.019600, 9.088889, 9.159174,
          9.230477, 9.302822, 9.376233, 9.450735, 9.526355, 9.603118,
          9.681054, 9.760191, 9.840558, 9.922186,10.005107,10.089353,
         10.174959,10.261958,10.350389,10.440287,10.531693,10.624646,
         10.719188,10.815362,10.913214,11.012789,11.114137,11.217307,
         11.322352,11.429325,11.538283,11.649285,
         11.762390,11.877664,11.995170,12.114979,12.237161,12.361791,
         12.488946,12.618708,12.751161,12.886394,13.024498,13.165570,
         13.309711,13.457026,13.607625,13.761625,13.919145,14.080314,
         14.245263,14.414134,14.587072,14.764233,14.945778,15.131877,
         15.322712,15.518470,15.719353,15.925570,16.137345,16.354912,
         16.578520,16.808433,17.044929,17.288305,17.538873,17.796967,
         18.062943,18.337176,18.620068,18.912049,19.213574,19.525133,
         19.847249,20.180480,20.525429,20.882738,21.253102,21.637266,
         22.036036,22.450278,22.880933,23.329017,23.795634,24.281981,
         24.789364,25.319207,25.873062,26.452634,27.059789,27.696581,
         28.365274,29.068370,29.808638,30.589157,31.413354,32.285060,
         33.208568,34.188705,35.230920,36.341388,37.527131,38.796172,
         40.157721,41.622399,43.202525,44.912465,46.769077,48.792279,
         51.005773,53.437996,56.123356,59.103894 };

      if (xi <= 0) return 0;
      if (z <= 0) return -std::numeric_limits<double>::infinity();
      if (z >= 1) return std::numeric_limits<double>::infinity();

      double ranlan, u, v;
      u = 1000*z;
      int i = int(u);
      u -= i;
      if (i >= 70 && i < 800) {
         ranlan = f[i-1] + u*(f[i] - f[i-1]);
      } else if (i >= 7 && i <= 980) {
         ranlan =  f[i-1] + u*(f[i]-f[i-1]-0.25*(1-u)*(f[i+1]-f[i]-f[i-1]+f[i-2]));
      } else if (i < 7) {
         v = std::log(z);
         u = 1/v;
         ranlan = ((0.99858950+(3.45213058E1+1.70854528E1*u)*u)/
                   (1         +(3.41760202E1+4.01244582  *u)*u))*
                   (-std::log(-0.91893853-v)-1);
      } else {
         u = 1-z;
         v = u*u;
         if (z <= 0.999) {
            ranlan = (1.00060006+2.63991156E2*u+4.37320068E3*v)/
                    ((1         +2.57368075E2*u+3.41448018E3*v)*u);
         } else {
            ranlan = (1.00001538+6.07514119E3*u+7.34266409E5*v)/
                    ((1         +6.06511919E3*u+6.94021044E5*v)*u);
         }
      }
      return xi*ranlan;
    }

   double landau_quantile_c(double z, double xi) {
      return landau_quantile(1.-z,xi);
   }

} // namespace Math
} // namespace ROOT