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/* cdf/gammainv.c
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*
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* Copyright (C) 2003, 2007 Brian Gough
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*
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* This program is free software; you can redistribute it and/or modify
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* it under the terms of the GNU General Public License as published by
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* the Free Software Foundation; either version 3 of the License, or (at
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* your option) any later version.
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*
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* This program is distributed in the hope that it will be useful, but
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* WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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* General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License
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* along with this program; if not, write to the Free Software Foundation,
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* Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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*/
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#include <config.h>
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#include <math.h>
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#include <gsl/gsl_cdf.h>
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#include <gsl/gsl_math.h>
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#include <gsl/gsl_randist.h>
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#include <gsl/gsl_sf_gamma.h>
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#include <stdio.h>
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double
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gsl_cdf_gamma_Pinv (const double P, const double a, const double b)
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{
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double x;
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if (P == 1.0)
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{
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return GSL_POSINF;
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}
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else if (P == 0.0)
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{
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return 0.0;
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}
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/* Consider, small, large and intermediate cases separately. The
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boundaries at 0.05 and 0.95 have not been optimised, but seem ok
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for an initial approximation.
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BJG: These approximations aren't really valid, the relevant
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criterion is P*gamma(a+1) < 1. Need to rework these routines and
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use a single bisection style solver for all the inverse
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functions.
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*/
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if (P < 0.05)
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{
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double x0 = exp ((gsl_sf_lngamma (a) + log (P)) / a);
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x = x0;
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}
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else if (P > 0.95)
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{
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double x0 = -log1p (-P) + gsl_sf_lngamma (a);
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x = x0;
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}
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else
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{
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double xg = gsl_cdf_ugaussian_Pinv (P);
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double x0 = (xg < -0.5*sqrt (a)) ? a : sqrt (a) * xg + a;
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x = x0;
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}
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/* Use Lagrange's interpolation for E(x)/phi(x0) to work backwards
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to an improved value of x (Abramowitz & Stegun, 3.6.6)
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where E(x)=P-integ(phi(u),u,x0,x) and phi(u) is the pdf.
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*/
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{
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double lambda, dP, phi;
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unsigned int n = 0;
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start:
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dP = P - gsl_cdf_gamma_P (x, a, 1.0);
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phi = gsl_ran_gamma_pdf (x, a, 1.0);
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if (dP == 0.0 || n++ > 32)
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goto end;
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lambda = dP / GSL_MAX (2 * fabs (dP / x), phi);
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{
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double step0 = lambda;
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double step1 = -((a - 1) / x - 1) * lambda * lambda / 4.0;
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double step = step0;
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if (fabs (step1) < 0.5 * fabs (step0))
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step += step1;
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if (x + step > 0)
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x += step;
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else
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{
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x /= 2.0;
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}
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if (fabs (step0) > 1e-10 * x || fabs(step0 * phi) > 1e-10 * P)
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goto start;
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}
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end:
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if (fabs(dP) > GSL_SQRT_DBL_EPSILON * P)
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{
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GSL_ERROR_VAL("inverse failed to converge", GSL_EFAILED, GSL_NAN);
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}
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return b * x;
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}
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}
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double
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gsl_cdf_gamma_Qinv (const double Q, const double a, const double b)
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{
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double x;
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if (Q == 1.0)
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{
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return 0.0;
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}
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else if (Q == 0.0)
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{
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return GSL_POSINF;
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}
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/* Consider, small, large and intermediate cases separately. The
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boundaries at 0.05 and 0.95 have not been optimised, but seem ok
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for an initial approximation. */
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if (Q < 0.05)
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{
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double x0 = -log (Q) + gsl_sf_lngamma (a);
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x = x0;
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}
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else if (Q > 0.95)
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{
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double x0 = exp ((gsl_sf_lngamma (a) + log1p (-Q)) / a);
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x = x0;
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}
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else
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{
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double xg = gsl_cdf_ugaussian_Qinv (Q);
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double x0 = (xg < -0.5*sqrt (a)) ? a : sqrt (a) * xg + a;
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x = x0;
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}
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/* Use Lagrange's interpolation for E(x)/phi(x0) to work backwards
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to an improved value of x (Abramowitz & Stegun, 3.6.6)
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where E(x)=P-integ(phi(u),u,x0,x) and phi(u) is the pdf.
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*/
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{
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double lambda, dQ, phi;
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unsigned int n = 0;
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start:
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dQ = Q - gsl_cdf_gamma_Q (x, a, 1.0);
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phi = gsl_ran_gamma_pdf (x, a, 1.0);
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if (dQ == 0.0 || n++ > 32)
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goto end;
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lambda = -dQ / GSL_MAX (2 * fabs (dQ / x), phi);
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{
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double step0 = lambda;
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double step1 = -((a - 1) / x - 1) * lambda * lambda / 4.0;
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double step = step0;
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if (fabs (step1) < 0.5 * fabs (step0))
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step += step1;
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if (x + step > 0)
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x += step;
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else
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{
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x /= 2.0;
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}
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if (fabs (step0) > 1e-10 * x)
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goto start;
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}
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}
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end:
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return b * x;
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}
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