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#include <Python.h>
#include <numpy/npy_math.h>
#include <stdio.h>
#include <math.h>
#include "sf_error.h"
#include "../cephes.h"
#undef fabs
#include "misc.h"
/* Limits after which to issue warnings about non-convergence */
#define ALLOWED_ATOL (1e-306)
#define ALLOWED_RTOL (1e-6)
/*
Inverse of the (regularised) incomplete Gamma integral.
Given a, find x such that igam(a, x) = y.
For y not small, we just use igami(a, 1-y) (inverse of the complemented
incomplete Gamma integral). For y small, however, 1-y is about 1, and we
lose digits.
*/
extern double MACHEP;
static double
gammainc(double x, double params[2])
{
return cephes_igam(params[0], x) - params[1];
}
double
gammaincinv(double a, double y)
{
double lo = 0.0, hi;
double flo = -y, fhi = 0.25 - y;
double params[2];
double best_x, best_f, errest;
fsolve_result_t r;
if (a <= 0.0 || y <= 0.0 || y >= 0.25) {
return cephes_igami(a, 1-y);
}
/* Note: flo and fhi must have different signs (and be != 0),
* otherwise fsolve terminates with an error.
*/
params[0] = a;
params[1] = y;
hi = cephes_igami(a, 0.75);
/* I found Newton to be unreliable. Also, after we generate a small
interval by bisection above, false position will do a large step
from an interval of width ~1e-4 to ~1e-14 in one step (a=10, x=0.05,
but similiar for other values).
*/
r = false_position(&lo, &flo, &hi, &fhi,
(objective_function)gammainc, params,
2*MACHEP, 2*MACHEP, 1e-2*a,
&best_x, &best_f, &errest);
if (!(r == FSOLVE_CONVERGED || r == FSOLVE_EXACT) &&
errest > ALLOWED_ATOL + ALLOWED_RTOL*fabs(best_x)) {
sf_error("gammaincinv", SF_ERROR_NO_RESULT,
"failed to converge at (a, y) = (%.20g, %.20g): got %g +- %g, code %d\n",
a, y, best_x, errest, r);
best_x = NPY_NAN;
}
return best_x;
}
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