/*
* Abramowitz and Stegun 6.5.29 [right]
*/
static double
pgamma_smallx (double x, double alph, int lower_tail, int log_p)
{
double sum = 0, c = alph, n = 0, term;
#ifdef DEBUG_p
REprintf (" pg_smallx(x=%.12g, alph=%.12g): ", x, alph);
#endif
/*
* Relative to 6.5.29 all terms have been multiplied by alph
* and the first, thus being 1, is omitted.
*/
do {
n++;
c *= -x / n;
term = c / (alph + n);
sum += term;
} while (fabs (term) > DBL_EPSILON * fabs (sum));
#ifdef DEBUG_p
REprintf ("%5.0f terms --> conv.sum=%g;", n, sum);
#endif
if (lower_tail) {
double f1 = log_p ? log1p (sum) : 1 + sum;
double f2;
if (alph > 1) {
f2 = dpois_raw (alph, x, log_p);
f2 = log_p ? f2 + x : f2 * exp (x);
} else if (log_p)
f2 = alph * log (x) - lgamma1p (alph);
else
f2 = pow (x, alph) / exp (lgamma1p (alph));
#ifdef DEBUG_p
REprintf (" (f1,f2)= (%g,%g)\n", f1,f2);
#endif
return log_p ? f1 + f2 : f1 * f2;
} else {
double lf2 = alph * log (x) - lgamma1p (alph);
#ifdef DEBUG_p
REprintf (" 1:%.14g 2:%.14g\n", alph * log (x), lgamma1p (alph));
REprintf (" sum=%.14g log(1+sum)=%.14g lf2=%.14g\n",
sum, log1p (sum), lf2);
#endif
if (log_p)
return R_Log1_Exp (log1p (sum) + lf2);
else {
double f1m1 = sum;
double f2m1 = expm1 (lf2);
return -(f1m1 + f2m1 + f1m1 * f2m1);
}
}
} /* pgamma_smallx() */
double
lgamma_r (double x, int *signp)
{
*signp = +1;
if (gnm_isnan (x))
return gnm_nan;
if (x > 0) {
gnm_float f = 1;
while (x < 10) {
f *= x;
x++;
}
return (M_LN_SQRT_2PI + (x - 0.5) * gnm_log(x) -
x + lgammacor(x)) - gnm_log (f);
} else {
gnm_float axm2 = gnm_fmod (-x, 2.0);
gnm_float y = gnm_sinpi (axm2) / M_PIgnum;
*signp = axm2 > 1.0 ? +1 : -1;
return y == 0
? gnm_nan
: - gnm_log (gnm_abs (y)) - lgamma1p (-x);
}
}
gnm_float stirlerr(gnm_float n)
{
#define S0 GNM_const(0.083333333333333333333) /* 1/12 */
#define S1 GNM_const(0.00277777777777777777778) /* 1/360 */
#define S2 GNM_const(0.00079365079365079365079365) /* 1/1260 */
#define S3 GNM_const(0.000595238095238095238095238) /* 1/1680 */
#define S4 GNM_const(0.0008417508417508417508417508)/* 1/1188 */
/*
error for 0, 0.5, 1.0, 1.5, ..., 14.5, 15.0.
*/
static const gnm_float sferr_halves[31] = {
0.0, /* n=0 - wrong, place holder only */
GNM_const(0.1534264097200273452913848), /* 0.5 */
GNM_const(0.0810614667953272582196702), /* 1.0 */
GNM_const(0.0548141210519176538961390), /* 1.5 */
GNM_const(0.0413406959554092940938221), /* 2.0 */
GNM_const(0.03316287351993628748511048), /* 2.5 */
GNM_const(0.02767792568499833914878929), /* 3.0 */
GNM_const(0.02374616365629749597132920), /* 3.5 */
GNM_const(0.02079067210376509311152277), /* 4.0 */
GNM_const(0.01848845053267318523077934), /* 4.5 */
GNM_const(0.01664469118982119216319487), /* 5.0 */
GNM_const(0.01513497322191737887351255), /* 5.5 */
GNM_const(0.01387612882307074799874573), /* 6.0 */
GNM_const(0.01281046524292022692424986), /* 6.5 */
GNM_const(0.01189670994589177009505572), /* 7.0 */
GNM_const(0.01110455975820691732662991), /* 7.5 */
GNM_const(0.010411265261972096497478567), /* 8.0 */
GNM_const(0.009799416126158803298389475), /* 8.5 */
GNM_const(0.009255462182712732917728637), /* 9.0 */
GNM_const(0.008768700134139385462952823), /* 9.5 */
GNM_const(0.008330563433362871256469318), /* 10.0 */
GNM_const(0.007934114564314020547248100), /* 10.5 */
GNM_const(0.007573675487951840794972024), /* 11.0 */
GNM_const(0.007244554301320383179543912), /* 11.5 */
GNM_const(0.006942840107209529865664152), /* 12.0 */
GNM_const(0.006665247032707682442354394), /* 12.5 */
GNM_const(0.006408994188004207068439631), /* 13.0 */
GNM_const(0.006171712263039457647532867), /* 13.5 */
GNM_const(0.005951370112758847735624416), /* 14.0 */
GNM_const(0.005746216513010115682023589), /* 14.5 */
GNM_const(0.005554733551962801371038690) /* 15.0 */
};
gnm_float nn;
if (n <= 15.0) {
nn = n + n;
if (nn == (int)nn) return(sferr_halves[(int)nn]);
return(lgamma1p (n ) - (n + 0.5)*gnm_log(n) + n - M_LN_SQRT_2PI);
}
nn = n*n;
if (n>500) return((S0-S1/nn)/n);
if (n> 80) return((S0-(S1-S2/nn)/nn)/n);
if (n> 35) return((S0-(S1-(S2-S3/nn)/nn)/nn)/n);
/* 15 < n <= 35 : */
return((S0-(S1-(S2-(S3-S4/nn)/nn)/nn)/nn)/n);
}
// MM_R attribute_hidden
double qchisq_appr(double p, double nu, double g /* = log Gamma(nu/2) */,
logical lower_tail, logical log_p, double tol /* EPS1 */)
{
#define C7 4.67
#define C8 6.66
#define C9 6.73
#define C10 13.32
double alpha, a, c, ch, p1;
double p2, q, t, x;
/* test arguments and initialise */
#ifdef IEEE_754
if (ISNAN(p) || ISNAN(nu))
return p + nu;
#endif
R_Q_P01_check(p);
if (nu <= 0) ML_ERR_return_NAN;
alpha = 0.5 * nu;/* = [pq]gamma() shape */
c = alpha-1;
if(nu < (-1.24)*(p1 = R_DT_log(p))) { /* for small chi-squared */
/* log(alpha) + g = log(alpha) + log(gamma(alpha)) =
* = log(alpha*gamma(alpha)) = lgamma(alpha+1) suffers from
* catastrophic cancellation when alpha << 1
*/
double lgam1pa = (alpha < 0.5) ? lgamma1p(alpha) : (log(alpha) + g);
ch = exp((lgam1pa + p1)/alpha + M_LN2);
#ifdef DEBUG_qgamma
REprintf(" small chi-sq., ch0 = %g\n", ch);
#endif
} else if(nu > 0.32) { /* using Wilson and Hilferty estimate */
x = qnorm(p, 0, 1, lower_tail, log_p);
p1 = 2./(9*nu);
ch = nu*pow(x*sqrt(p1) + 1-p1, 3);
#ifdef DEBUG_qgamma
REprintf(" nu > .32: Wilson-Hilferty; x = %7g\n", x);
#endif
/* approximation for p tending to 1: */
if( ch > 2.2*nu + 6 )
ch = -2*(R_DT_Clog(p) - c*log(0.5*ch) + g);
} else { /* "small nu" : 1.24*(-log(p)) <= nu <= 0.32 */
ch = 0.4;
a = R_DT_Clog(p) + g + c*M_LN2;
#ifdef DEBUG_qgamma
REprintf(" nu <= .32: a = %7g\n", a);
#endif
do {
q = ch;
p1 = 1. / (1+ch*(C7+ch));
p2 = ch*(C9+ch*(C8+ch));
t = -0.5 +(C7+2*ch)*p1 - (C9+ch*(C10+3*ch))/p2;
ch -= (1- exp(a+0.5*ch)*p2*p1)/t;
} while(fabs(q - ch) > tol * fabs(ch));
}
return ch;
}