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- /* cdf/tdistinv.c
- *
- * Copyright (C) 2007 Brian Gough
- * Copyright (C) 2002 Jason H. Stover.
- *
- * 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., 59 Temple Place, Suite 330, Boston, MA 02111-1307, USA.
- */
- #include "gsl__config.h"
- #include <math.h>
- #include "gsl_cdf.h"
- #include "gsl_math.h"
- #include "gsl_randist.h"
- #include "gsl_sf_gamma.h"
- #include <stdio.h>
- static double
- inv_cornish_fisher (double z, double nu)
- {
- double a = 1 / (nu - 0.5);
- double b = 48.0 / (a * a);
- double cf1 = z * (3 + z * z);
- double cf2 = z * (945 + z * z * (360 + z * z * (63 + z * z * 4)));
- double y = z - cf1 / b + cf2 / (10 * b * b);
- double t = GSL_SIGN (z) * sqrt (nu * expm1 (a * y * y));
- return t;
- }
- double
- gsl_cdf_tdist_Pinv (const double P, const double nu)
- {
- double x, ptail;
- if (P == 1.0)
- {
- return GSL_POSINF;
- }
- else if (P == 0.0)
- {
- return GSL_NEGINF;
- }
- if (nu == 1.0)
- {
- x = tan (M_PI * (P - 0.5));
- }
- else if (nu == 2.0)
- {
- double a = 2 * P - 1;
- x = a / sqrt (2 * (1 - a * a));
- }
- ptail = (P < 0.5) ? P : 1 - P;
- if (sqrt (M_PI * nu / 2) * ptail > pow (0.05, nu / 2))
- {
- double xg = gsl_cdf_ugaussian_Pinv (P);
- x = inv_cornish_fisher (xg, nu);
- }
- else
- {
- /* Use an asymptotic expansion of the tail of integral */
- double beta = gsl_sf_beta (0.5, nu / 2);
- if (P < 0.5)
- {
- x = -sqrt (nu) * pow (beta * nu * P, -1.0 / nu);
- }
- else
- {
- x = sqrt (nu) * pow (beta * nu * (1 - P), -1.0 / nu);
- }
- /* Correct nu -> nu/(1+nu/x^2) in the leading term to account
- for higher order terms. This avoids overestimating x, which
- makes the iteration unstable due to the rapidly decreasing
- tails of the distribution. */
- x /= sqrt (1 + nu / (x * x));
- }
- {
- double dP, phi;
- unsigned int n = 0;
- start:
- dP = P - gsl_cdf_tdist_P (x, nu);
- phi = gsl_ran_tdist_pdf (x, nu);
- if (dP == 0.0 || n++ > 32)
- goto end;
- {
- double lambda = dP / phi;
- double step0 = lambda;
- double step1 = ((nu + 1) * x / (x * x + nu)) * (lambda * lambda / 4.0);
- double step = step0;
- if (fabs (step1) < fabs (step0))
- {
- step += step1;
- }
- if (P > 0.5 && x + step < 0)
- x /= 2;
- else if (P < 0.5 && x + step > 0)
- x /= 2;
- else
- x += step;
- if (fabs (step) > 1e-10 * fabs (x))
- goto start;
- }
-
- end:
- if (fabs(dP) > GSL_SQRT_DBL_EPSILON * P)
- {
- GSL_ERROR_VAL("inverse failed to converge", GSL_EFAILED, GSL_NAN);
- }
-
- return x;
- }
- }
- double
- gsl_cdf_tdist_Qinv (const double Q, const double nu)
- {
- double x, qtail;
- if (Q == 0.0)
- {
- return GSL_POSINF;
- }
- else if (Q == 1.0)
- {
- return GSL_NEGINF;
- }
- if (nu == 1.0)
- {
- x = tan (M_PI * (0.5 - Q));
- }
- else if (nu == 2.0)
- {
- double a = 2 * (1 - Q) - 1;
- x = a / sqrt (2 * (1 - a * a));
- }
- qtail = (Q < 0.5) ? Q : 1 - Q;
- if (sqrt (M_PI * nu / 2) * qtail > pow (0.05, nu / 2))
- {
- double xg = gsl_cdf_ugaussian_Qinv (Q);
- x = inv_cornish_fisher (xg, nu);
- }
- else
- {
- /* Use an asymptotic expansion of the tail of integral */
- double beta = gsl_sf_beta (0.5, nu / 2);
- if (Q < 0.5)
- {
- x = sqrt (nu) * pow (beta * nu * Q, -1.0 / nu);
- }
- else
- {
- x = -sqrt (nu) * pow (beta * nu * (1 - Q), -1.0 / nu);
- }
- /* Correct nu -> nu/(1+nu/x^2) in the leading term to account
- for higher order terms. This avoids overestimating x, which
- makes the iteration unstable due to the rapidly decreasing
- tails of the distribution. */
- x /= sqrt (1 + nu / (x * x));
- }
- {
- double dQ, phi;
- unsigned int n = 0;
- start:
- dQ = Q - gsl_cdf_tdist_Q (x, nu);
- phi = gsl_ran_tdist_pdf (x, nu);
- if (dQ == 0.0 || n++ > 32)
- goto end;
- {
- double lambda = - dQ / phi;
- double step0 = lambda;
- double step1 = ((nu + 1) * x / (x * x + nu)) * (lambda * lambda / 4.0);
- double step = step0;
- if (fabs (step1) < fabs (step0))
- {
- step += step1;
- }
- if (Q < 0.5 && x + step < 0)
- x /= 2;
- else if (Q > 0.5 && x + step > 0)
- x /= 2;
- else
- x += step;
- if (fabs (step) > 1e-10 * fabs (x))
- goto start;
- }
- }
- end:
- return x;
- }
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