/*
* 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 pweibull(double x, double shape, double scale, int lower_tail, int log_p)
{
#ifdef IEEE_754
if (ISNAN(x) || ISNAN(shape) || ISNAN(scale))
return x + shape + scale;
#endif
if(shape <= 0 || scale <= 0) ML_ERR_return_NAN;
if (x <= 0)
return R_DT_0;
x = -pow(x / scale, shape);
return lower_tail
? (log_p ? R_Log1_Exp(x) : -expm1(x))
: R_D_exp(x);
}
double pweibull(double x, double shape, double scale, int lower_tail, int log_p)
{
#ifdef IEEE_754
if (ISNAN(x) || ISNAN(shape) || ISNAN(scale))
return x + shape + scale;
#endif
if(shape <= 0 || scale <= 0) ML_ERR_return_NAN;
if (x <= 0)
return R_DT_0;
x = -pow(x / scale, shape);
if (lower_tail)
return (log_p
/* log(1 - exp(x)) for x < 0 : */
? R_Log1_Exp(x) : -expm1(x));
/* else: !lower_tail */
return R_D_exp(x);
}
double pexp(double x, double scale, int lower_tail, int log_p)
{
#ifdef IEEE_754
if (ISNAN(x) || ISNAN(scale))
return x + scale;
if (scale < 0) ML_ERR_return_NAN;
#else
if (scale <= 0) ML_ERR_return_NAN;
#endif
if (x <= 0.)
return R_DT_0;
/* same as weibull( shape = 1): */
x = -(x / scale);
return lower_tail
? (log_p ? R_Log1_Exp(x) : -expm1(x))
: R_D_exp(x);
}
double pgamma_raw (double x, double alph, int lower_tail, int log_p)
{
/* Here, assume that (x,alph) are not NA & alph > 0 . */
double res;
#ifdef DEBUG_p
REprintf("pgamma_raw(x=%.14g, alph=%.14g, low=%d, log=%d)\n",
x, alph, lower_tail, log_p);
#endif
R_P_bounds_01(x, 0., ML_POSINF);
if (x < 1) {
res = pgamma_smallx (x, alph, lower_tail, log_p);
} else if (x <= alph - 1 && x < 0.8 * (alph + 50)) {
/* incl. large alph compared to x */
double sum = pd_upper_series (x, alph, log_p);/* = x/alph + o(x/alph) */
double d = dpois_wrap (alph, x, log_p);
#ifdef DEBUG_p
REprintf(" alph 'large': sum=pd_upper*()= %.12g, d=dpois_w(*)= %.12g\n",
sum, d);
#endif
if (!lower_tail)
res = log_p
? R_Log1_Exp (d + sum)
: 1 - d * sum;
else
res = log_p ? sum + d : sum * d;
} else if (alph - 1 < x && alph < 0.8 * (x + 50)) {
/* incl. large x compared to alph */
double sum;
double d = dpois_wrap (alph, x, log_p);
#ifdef DEBUG_p
REprintf(" x 'large': d=dpois_w(*)= %.14g ", d);
#endif
if (alph < 1) {
if (x * DBL_EPSILON > 1 - alph)
sum = R_D__1;
else {
double f = pd_lower_cf (alph, x - (alph - 1)) * x / alph;
/* = [alph/(x - alph+1) + o(alph/(x-alph+1))] * x/alph = 1 + o(1) */
sum = log_p ? log (f) : f;
}
} else {
sum = pd_lower_series (x, alph - 1);/* = (alph-1)/x + o((alph-1)/x) */
sum = log_p ? log1p (sum) : 1 + sum;
}
#ifdef DEBUG_p
REprintf(", sum= %.14g\n", sum);
#endif
if (!lower_tail)
res = log_p ? sum + d : sum * d;
else
res = log_p
? R_Log1_Exp (d + sum)
: 1 - d * sum;
} else { /* x >= 1 and x fairly near alph. */
#ifdef DEBUG_p
REprintf(" using ppois_asymp()\n");
#endif
res = ppois_asymp (alph - 1, x, !lower_tail, log_p);
}
/*
* We lose a fair amount of accuracy to underflow in the cases
* where the final result is very close to DBL_MIN. In those
* cases, simply redo via log space.
*/
if (!log_p && res < DBL_MIN / DBL_EPSILON) {
/* with(.Machine, double.xmin / double.eps) #|-> 1.002084e-292 */
#ifdef DEBUG_p
REprintf(" very small res=%.14g; -> recompute via log\n", res);
#endif
return exp (pgamma_raw (x, alph, lower_tail, 1));
} else
return res;
}
/*
* Compute the log of a difference from logs of terms, i.e.,
*
* log (exp (logx) - exp (logy))
*
* without causing overflows and without throwing away large handfuls
* of accuracy.
*/
double logspace_sub (double logx, double logy)
{
return logx + R_Log1_Exp(logy - logx);
}
double qt(double p, double ndf, int lower_tail, int log_p)
{
const static double eps = 1.e-12;
double P, q;
Rboolean neg;
#ifdef IEEE_754
if (ISNAN(p) || ISNAN(ndf))
return p + ndf;
#endif
R_Q_P01_boundaries(p, ML_NEGINF, ML_POSINF);
if (ndf <= 0) ML_ERR_return_NAN;
if (ndf < 1) { /* based on qnt */
const static double accu = 1e-13;
const static double Eps = 1e-11; /* must be > accu */
double ux, lx, nx, pp;
int iter = 0;
p = R_DT_qIv(p);
/* Invert pt(.) :
* 1. finding an upper and lower bound */
if(p > 1 - DBL_EPSILON) return ML_POSINF;
pp = fmin2(1 - DBL_EPSILON, p * (1 + Eps));
for(ux = 1.; ux < DBL_MAX && pt(ux, ndf, TRUE, FALSE) < pp; ux *= 2);
pp = p * (1 - Eps);
for(lx =-1.; lx > -DBL_MAX && pt(lx, ndf, TRUE, FALSE) > pp; lx *= 2);
/* 2. interval (lx,ux) halving
regula falsi failed on qt(0.1, 0.1)
*/
do {
nx = 0.5 * (lx + ux);
if (pt(nx, ndf, TRUE, FALSE) > p) ux = nx; else lx = nx;
} while ((ux - lx) / fabs(nx) > accu && ++iter < 1000);
if(iter >= 1000) ML_ERROR(ME_PRECISION, "qt");
return 0.5 * (lx + ux);
}
/* Old comment:
* FIXME: "This test should depend on ndf AND p !!
* ----- and in fact should be replaced by
* something like Abramowitz & Stegun 26.7.5 (p.949)"
*
* That would say that if the qnorm value is x then
* the result is about x + (x^3+x)/4df + (5x^5+16x^3+3x)/96df^2
* The differences are tiny even if x ~ 1e5, and qnorm is not
* that accurate in the extreme tails.
*/
if (ndf > 1e20) return qnorm(p, 0., 1., lower_tail, log_p);
P = R_D_qIv(p); /* if exp(p) underflows, we fix below */
neg = (!lower_tail || P < 0.5) && (lower_tail || P > 0.5);
if(neg)
P = 2 * (log_p ? (lower_tail ? P : -expm1(p)) : R_D_Lval(p));
else
P = 2 * (log_p ? (lower_tail ? -expm1(p) : P) : R_D_Cval(p));
/* 0 <= P <= 1 ; P = 2*min(P', 1 - P') in all cases */
/* Use this if(log_p) only : */
#define P_is_exp_2p (lower_tail == neg) /* both TRUE or FALSE == !xor */
if (fabs(ndf - 2) < eps) { /* df ~= 2 */
if(P > DBL_MIN) {
if(3* P < DBL_EPSILON) /* P ~= 0 */
q = 1 / sqrt(P);
else if (P > 0.9) /* P ~= 1 */
q = (1 - P) * sqrt(2 /(P * (2 - P)));
else /* eps/3 <= P <= 0.9 */
q = sqrt(2 / (P * (2 - P)) - 2);
}
else { /* P << 1, q = 1/sqrt(P) = ... */
if(log_p)
q = P_is_exp_2p ? exp(- p/2) / M_SQRT2 : 1/sqrt(-expm1(p));
else
q = ML_POSINF;
}
}
else if (ndf < 1 + eps) { /* df ~= 1 (df < 1 excluded above): Cauchy */
if(P > 0)
q = 1/tan(P * M_PI_2);/* == - tan((P+1) * M_PI_2) -- suffers for P ~= 0 */
else { /* P = 0, but maybe = 2*exp(p) ! */
if(log_p) /* 1/tan(e) ~ 1/e */
q = P_is_exp_2p ? M_1_PI * exp(-p) : -1./(M_PI * expm1(p));
else
q = ML_POSINF;
}
}
else { /*-- usual case; including, e.g., df = 1.1 */
double x = 0., y, log_P2 = 0./* -Wall */,
a = 1 / (ndf - 0.5),
b = 48 / (a * a),
c = ((20700 * a / b - 98) * a - 16) * a + 96.36,
d = ((94.5 / (b + c) - 3) / b + 1) * sqrt(a * M_PI_2) * ndf;
Rboolean P_ok1 = P > DBL_MIN || !log_p, P_ok = P_ok1;
if(P_ok1) {
y = pow(d * P, 2 / ndf);
P_ok = (y >= DBL_EPSILON);
}
if(!P_ok) { /* log_p && P very small */
log_P2 = P_is_exp_2p ? p : R_Log1_Exp(p); /* == log(P / 2) */
x = (log(d) + M_LN2 + log_P2) / ndf;
y = exp(2 * x);
}
if ((ndf < 2.1 && P > 0.5) || y > 0.05 + a) { /* P > P0(df) */
/* Asymptotic inverse expansion about normal */
if(P_ok)
x = qnorm(0.5 * P, 0., 1., /*lower_tail*/TRUE, /*log_p*/FALSE);
else /* log_p && P underflowed */
x = qnorm(log_P2, 0., 1., lower_tail, /*log_p*/ TRUE);
y = x * x;
if (ndf < 5)
c += 0.3 * (ndf - 4.5) * (x + 0.6);
c = (((0.05 * d * x - 5) * x - 7) * x - 2) * x + b + c;
y = (((((0.4 * y + 6.3) * y + 36) * y + 94.5) / c
- y - 3) / b + 1) * x;
y = expm1(a * y * y);
q = sqrt(ndf * y);
} else { /* re-use 'y' from above */
if(!P_ok && x < - M_LN2 * DBL_MANT_DIG) {/* 0.5* log(DBL_EPSILON) */
/* y above might have underflown */
q = sqrt(ndf) * exp(-x);
}
else {
y = ((1 / (((ndf + 6) / (ndf * y) - 0.089 * d - 0.822)
* (ndf + 2) * 3) + 0.5 / (ndf + 4))
* y - 1) * (ndf + 1) / (ndf + 2) + 1 / y;
q = sqrt(ndf * y);
}
}
/* Now apply 2-term Taylor expansion improvement (1-term = Newton):
* as by Hill (1981) [ref.above] */
/* FIXME: This can be far from optimal when log_p = TRUE
* but is still needed, e.g. for qt(-2, df=1.01, log=TRUE).
* Probably also improvable when lower_tail = FALSE */
if(P_ok1) {
int it=0;
while(it++ < 10 && (y = dt(q, ndf, FALSE)) > 0 &&
R_FINITE(x = (pt(q, ndf, FALSE, FALSE) - P/2) / y) &&
fabs(x) > 1e-14*fabs(q))
/* Newton (=Taylor 1 term):
* q += x;
* Taylor 2-term : */
q += x * (1. + x * q * (ndf + 1) / (2 * (q * q + ndf)));
}
}
if(neg) q = -q;
return q;
}
// Returns both qbeta() and its "mirror" 1-qbeta(). Useful notably when qbeta() ~= 1
attribute_hidden void
qbeta_raw(double alpha, double p, double q, int lower_tail, int log_p,
int swap_01, // {TRUE, NA, FALSE}: if NA, algorithm decides swap_tail
double log_q_cut, /* if == Inf: return log(qbeta(..));
otherwise, if finite: the bound for
switching to log(x)-scale; see use_log_x */
int n_N, // number of "unconstrained" Newton steps before switching to constrained
double *qb) // = qb[0:1] = { qbeta(), 1 - qbeta() }
{
Rboolean
swap_choose = (swap_01 == MLOGICAL_NA),
swap_tail,
log_, give_log_q = (log_q_cut == ML_POSINF),
use_log_x = give_log_q, // or u < log_q_cut below
warned = FALSE, add_N_step = TRUE;
int i_pb, i_inn;
double a, la, logbeta, g, h, pp, p_, qq, r, s, t, w, y = -1.;
volatile double u, xinbta;
// Assuming p >= 0, q >= 0 here ...
// Deal with boundary cases here:
if(alpha == R_DT_0) {
#define return_q_0 \
if(give_log_q) { qb[0] = ML_NEGINF; qb[1] = 0; } \
else { qb[0] = 0; qb[1] = 1; } \
return
return_q_0;
}
if(alpha == R_DT_1) {
#define return_q_1 \
if(give_log_q) { qb[0] = 0; qb[1] = ML_NEGINF; } \
else { qb[0] = 1; qb[1] = 0; } \
return
return_q_1;
}
// check alpha {*before* transformation which may all accuracy}:
if((log_p && alpha > 0) ||
(!log_p && (alpha < 0 || alpha > 1))) { // alpha is outside
R_ifDEBUG_printf("qbeta(alpha=%g, %g, %g, .., log_p=%d): %s%s\n",
alpha, p,q, log_p, "alpha not in ",
log_p ? "[-Inf, 0]" : "[0,1]");
// ML_ERR_return_NAN :
ML_ERROR(ME_DOMAIN, "");
qb[0] = qb[1] = ML_NAN; return;
}
// p==0, q==0, p = Inf, q = Inf <==> treat as one- or two-point mass
if(p == 0 || q == 0 || !R_FINITE(p) || !R_FINITE(q)) {
// We know 0 < T(alpha) < 1 : pbeta() is constant and trivial in {0, 1/2, 1}
R_ifDEBUG_printf(
"qbeta(%g, %g, %g, lower_t=%d, log_p=%d): (p,q)-boundary: trivial\n",
alpha, p,q, lower_tail, log_p);
if(p == 0 && q == 0) { // point mass 1/2 at each of {0,1} :
if(alpha < R_D_half) { return_q_0; }
if(alpha > R_D_half) { return_q_1; }
// else: alpha == "1/2"
#define return_q_half \
if(give_log_q) qb[0] = qb[1] = -M_LN2; \
else qb[0] = qb[1] = 0.5; \
return
return_q_half;
} else if (p == 0 || p/q == 0) { // point mass 1 at 0 - "flipped around"
return_q_0;
} else if (q == 0 || q/p == 0) { // point mass 1 at 0 - "flipped around"
return_q_1;
}
// else: p = q = Inf : point mass 1 at 1/2
return_q_half;
}
/* initialize */
p_ = R_DT_qIv(alpha);/* lower_tail prob (in any case) */
// Conceptually, 0 < p_ < 1 (but can be 0 or 1 because of cancellation!)
logbeta = lbeta(p, q);
swap_tail = (swap_choose) ? (p_ > 0.5) : swap_01;
// change tail; default (swap_01 = NA): afterwards 0 < a <= 1/2
if(swap_tail) { /* change tail, swap p <-> q :*/
a = R_DT_CIv(alpha); // = 1 - p_ < 1/2
/* la := log(a), but without numerical cancellation: */
la = R_DT_Clog(alpha);
pp = q; qq = p;
}
else {
a = p_;
la = R_DT_log(alpha);
pp = p; qq = q;
}
/* calculate the initial approximation */
/* Desired accuracy for Newton iterations (below) should depend on (a,p)
* This is from Remark .. on AS 109, adapted.
* However, it's not clear if this is "optimal" for IEEE double prec.
* acu = fmax2(acu_min, pow(10., -25. - 5./(pp * pp) - 1./(a * a)));
* NEW: 'acu' accuracy NOT for squared adjustment, but simple;
* ---- i.e., "new acu" = sqrt(old acu)
*/
double acu = fmax2(acu_min, pow(10., -13. - 2.5/(pp * pp) - 0.5/(a * a)));
// try to catch "extreme left tail" early
double tx, u0 = (la + log(pp) + logbeta) / pp; // = log(x_0)
static const double
log_eps_c = M_LN2 * (1. - DBL_MANT_DIG);// = log(DBL_EPSILON) = -36.04..
r = pp*(1.-qq)/(pp+1.);
t = 0.2;
// FIXME: Factor 0.2 is a bit arbitrary; '1' is clearly much too much.
R_ifDEBUG_printf(
"qbeta(%g, %g, %g, lower_t=%d, log_p=%d):%s\n"
" swap_tail=%d, la=%g, u0=%g (bnd: %g (%g)) ",
alpha, p,q, lower_tail, log_p,
(log_p && (p_ == 0. || p_ == 1.)) ? (p_==0.?" p_=0":" p_=1") : "",
swap_tail, la, u0,
(t*log_eps_c - log(fabs(pp*(1.-qq)*(2.-qq)/(2.*(pp+2.)))))/2.,
t*log_eps_c - log(fabs(r))
);
if(M_LN2 * DBL_MIN_EXP < u0 && // cannot allow exp(u0) = 0 ==> exp(u1) = exp(u0) = 0
u0 < -0.01 && // (must: u0 < 0, but too close to 0 <==> x = exp(u0) = 0.99..)
// qq <= 2 && // <--- "arbitrary"
// u0 < t*log_eps_c - log(fabs(r)) &&
u0 < (t*log_eps_c - log(fabs(pp*(1.-qq)*(2.-qq)/(2.*(pp+2.)))))/2.)
{
// TODO: maybe jump here from below, when initial u "fails" ?
// L_tail_u:
// MM's one-step correction (cheaper than 1 Newton!)
r = r*exp(u0);// = r*x0
if(r > -1.) {
u = u0 - log1p(r)/pp;
R_ifDEBUG_printf("u1-u0=%9.3g --> choosing u = u1\n", u-u0);
} else {
u = u0;
R_ifDEBUG_printf("cannot cheaply improve u0\n");
}
tx = xinbta = exp(u);
use_log_x = TRUE; // or (u < log_q_cut) ??
goto L_Newton;
}
// y := y_\alpha in AS 64 := Hastings(1955) approximation of qnorm(1 - a) :
r = sqrt(-2 * la);
y = r - (const1 + const2 * r) / (1. + (const3 + const4 * r) * r);
if (pp > 1 && qq > 1) { // use Carter(1947), see AS 109, remark '5.'
r = (y * y - 3.) / 6.;
s = 1. / (pp + pp - 1.);
t = 1. / (qq + qq - 1.);
h = 2. / (s + t);
w = y * sqrt(h + r) / h - (t - s) * (r + 5. / 6. - 2. / (3. * h));
R_ifDEBUG_printf("p,q > 1 => w=%g", w);
if(w > 300) { // exp(w+w) is huge or overflows
t = w+w + log(qq) - log(pp); // = argument of log1pexp(.)
u = // log(xinbta) = - log1p(qq/pp * exp(w+w)) = -log(1 + exp(t))
(t <= 18) ? -log1p(exp(t)) : -t - exp(-t);
xinbta = exp(u);
} else {
xinbta = pp / (pp + qq * exp(w + w));
u = // log(xinbta)
- log1p(qq/pp * exp(w+w));
}
} else { // use the original AS 64 proposal, Scheffé-Tukey (1944) and Wilson-Hilferty
r = qq + qq;
/* A slightly more stable version of t := \chi^2_{alpha} of AS 64
* t = 1. / (9. * qq); t = r * R_pow_di(1. - t + y * sqrt(t), 3); */
t = 1. / (3. * sqrt(qq));
t = r * R_pow_di(1. + t*(-t + y), 3);// = \chi^2_{alpha} of AS 64
s = 4. * pp + r - 2.;// 4p + 2q - 2 = numerator of new t = (...) / chi^2
R_ifDEBUG_printf("min(p,q) <= 1: t=%g", t);
if (t == 0 || (t < 0. && s >= t)) { // cannot use chisq approx
// x0 = 1 - { (1-a)*q*B(p,q) } ^{1/q} {AS 65}
// xinbta = 1. - exp((log(1-a)+ log(qq) + logbeta) / qq);
double l1ma;/* := log(1-a), directly from alpha (as 'la' above):
* FIXME: not worth it? log1p(-a) always the same ?? */
if(swap_tail)
l1ma = R_DT_log(alpha);
else
l1ma = R_DT_Clog(alpha);
R_ifDEBUG_printf(" t <= 0 : log1p(-a)=%.15g, better l1ma=%.15g\n", log1p(-a), l1ma);
double xx = (l1ma + log(qq) + logbeta) / qq;
if(xx <= 0.) {
xinbta = -expm1(xx);
u = R_Log1_Exp (xx);// = log(xinbta) = log(1 - exp(...A...))
} else { // xx > 0 ==> 1 - e^xx < 0 .. is nonsense
R_ifDEBUG_printf(" xx=%g > 0: xinbta:= 1-e^xx < 0\n", xx);
xinbta = 0; u = ML_NEGINF; /// FIXME can do better?
}
} else {
t = s / t;
R_ifDEBUG_printf(" t > 0 or s < t < 0: new t = %g ( > 1 ?)\n", t);
if (t <= 1.) { // cannot use chisq, either
u = (la + log(pp) + logbeta) / pp;
xinbta = exp(u);
} else { // (1+x0)/(1-x0) = t, solved for x0 :
xinbta = 1. - 2. / (t + 1.);
u = log1p(-2. / (t + 1.));
}
}
}
// Problem: If initial u is completely wrong, we make a wrong decision here
if(swap_choose &&
(( swap_tail && u >= -exp( log_q_cut)) || // ==> "swap back"
(!swap_tail && u >= -exp(4*log_q_cut) && pp / qq < 1000.))) { // ==> "swap now" (much less easily)
// "revert swap" -- and use_log_x
swap_tail = !swap_tail;
R_ifDEBUG_printf(" u = %g (e^u = xinbta = %.16g) ==> ", u, xinbta);
if(swap_tail) {
a = R_DT_CIv(alpha); // needed ?
la = R_DT_Clog(alpha);
pp = q; qq = p;
}
else {
a = p_;
la = R_DT_log(alpha);
pp = p; qq = q;
}
R_ifDEBUG_printf("\"%s\"; la = %g\n",
(swap_tail ? "swap now" : "swap back"), la);
// we could redo computations above, but this should be stable
u = R_Log1_Exp(u);
xinbta = exp(u);
/* Careful: "swap now" should not fail if
1) the above initial xinbta is "completely wrong"
2) The correction step can go outside (u_n > 0 ==> e^u > 1 is illegal)
e.g., for
qbeta(0.2066, 0.143891, 0.05)
*/
}
if(!use_log_x)
use_log_x = (u < log_q_cut);//(per default) <==> xinbta = e^u < 4.54e-5
Rboolean
bad_u = !R_FINITE(u),
bad_init = bad_u || xinbta > p_hi;
R_ifDEBUG_printf(" -> u = %g, e^u = xinbta = %.16g, (Newton acu=%g%s)\n",
u, xinbta, acu,
(bad_u ? ", ** bad u **" :
(use_log_x ? ", on u = log(x) scale" : "")));
double u_n = 1.; // -Wall
tx = xinbta; // keeping "original initial x" (for now)
if(bad_u || u < log_q_cut) { /* e.g.
qbeta(0.21, .001, 0.05)
try "left border" quickly, i.e.,
try at smallest positive number: */
w = pbeta_raw(DBL_very_MIN, pp, qq, TRUE, log_p);
if(w > (log_p ? la : a)) {
R_ifDEBUG_printf(" quantile is left of smallest positive number; \"convergence\"\n");
if(log_p || fabs(w - a) < fabs(0 - a)) { // DBL_very_MIN is better than 0
tx = DBL_very_MIN;
u_n = DBL_log_v_MIN;// = log(DBL_very_MIN)
} else {
tx = 0.;
u_n = ML_NEGINF;
}
use_log_x = log_p; add_N_step = FALSE; goto L_return;
}
else {
R_ifDEBUG_printf(" pbeta(smallest pos.) = %g <= %g --> continuing\n",
w, (log_p ? la : a));
if(u < DBL_log_v_MIN) {
u = DBL_log_v_MIN;// = log(DBL_very_MIN)
xinbta = DBL_very_MIN;
}
}
}
/* Sometimes the approximation is negative (and == 0 is also not "ok") */
if (bad_init && !(use_log_x && tx > 0)) {
if(u == ML_NEGINF) {
R_ifDEBUG_printf(" u = -Inf;");
u = M_LN2 * DBL_MIN_EXP;
xinbta = DBL_MIN;
} else {
R_ifDEBUG_printf(" bad_init: u=%g, xinbta=%g;", u,xinbta);
xinbta = (xinbta > 1.1) // i.e. "way off"
? 0.5 // otherwise, keep the respective boundary:
: ((xinbta < p_lo) ? exp(u) : p_hi);
if(bad_u)
u = log(xinbta);
// otherwise: not changing "potentially better" u than the above
}
R_ifDEBUG_printf(" -> (partly)new u=%g, xinbta=%g\n", u,xinbta);
}
L_Newton:
/* --------------------------------------------------------------------
* Solve for x by a modified Newton-Raphson method, using pbeta_raw()
*/
r = 1 - pp;
t = 1 - qq;
double wprev = 0., prev = 1., adj = 1.; // -Wall
if(use_log_x) { // find log(xinbta) -- work in u := log(x) scale
// if(bad_init && tx > 0) xinbta = tx;// may have been better
for (i_pb=0; i_pb < 1000; i_pb++) {
// using log_p == TRUE unconditionally here
// FIXME: if exp(u) = xinbta underflows to 0, like different formula pbeta_log(u, *)
y = pbeta_raw(xinbta, pp, qq, /*lower_tail = */ TRUE, TRUE);
/* w := Newton step size for L(u) = log F(e^u) =!= 0; u := log(x)
* = (L(.) - la) / L'(.); L'(u)= (F'(e^u) * e^u ) / F(e^u)
* = (L(.) - la)*F(.) / {F'(e^u) * e^u } =
* = (L(.) - la) * e^L(.) * e^{-log F'(e^u) - u}
* = ( y - la) * e^{ y - u -log F'(e^u)}
and -log F'(x)= -log f(x) = + logbeta + (1-p) log(x) + (1-q) log(1-x)
= logbeta + (1-p) u + (1-q) log(1-e^u)
*/
w = (y == ML_NEGINF) // y = -Inf well possible: we are on log scale!
? 0. : (y - la) * exp(y - u + logbeta + r * u + t * R_Log1_Exp(u));
if(!R_FINITE(w))
break;
if (i_pb >= n_N && w * wprev <= 0.)
prev = fmax2(fabs(adj),fpu);
R_ifDEBUG_printf("N(i=%2d): u=%#20.16g, pb(e^u)=%#12.6g, w=%#15.9g, %s prev=%11g,",
i_pb, u, y, w, (w * wprev <= 0.) ? "new" : "old", prev);
g = 1;
for (i_inn=0; i_inn < 1000; i_inn++) {
adj = g * w;
// take full Newton steps at the beginning; only then safe guard:
if (i_pb < n_N || fabs(adj) < prev) {
u_n = u - adj; // u_{n+1} = u_n - g*w
if (u_n <= 0.) { // <==> 0 < xinbta := e^u <= 1
if (prev <= acu || fabs(w) <= acu) {
/* R_ifDEBUG_printf(" -adj=%g, %s <= acu ==> convergence\n", */
/* -adj, (prev <= acu) ? "prev" : "|w|"); */
R_ifDEBUG_printf(" it{in}=%d, -adj=%g, %s <= acu ==> convergence\n",
i_inn, -adj, (prev <= acu) ? "prev" : "|w|");
goto L_converged;
}
// if (u_n != ML_NEGINF && u_n != 1)
break;
}
}
g /= 3;
}
// (cancellation in (u_n -u) => may differ from adj:
double D = fmin2(fabs(adj), fabs(u_n - u));
/* R_ifDEBUG_printf(" delta(u)=%g\n", u_n - u); */
R_ifDEBUG_printf(" it{in}=%d, delta(u)=%9.3g, D/|.|=%.3g\n",
i_inn, u_n - u, D/fabs(u_n + u));
if (D <= 4e-16 * fabs(u_n + u))
goto L_converged;
u = u_n;
xinbta = exp(u);
wprev = w;
} // for(i )
} else
for (i_pb=0; i_pb < 1000; i_pb++) {
y = pbeta_raw(xinbta, pp, qq, /*lower_tail = */ TRUE, log_p);
// delta{y} : d_y = y - (log_p ? la : a);
#ifdef IEEE_754
if(!R_FINITE(y) && !(log_p && y == ML_NEGINF))// y = -Inf is ok if(log_p)
#else
if (errno)
#endif
{ // ML_ERR_return_NAN :
ML_ERROR(ME_DOMAIN, "");
qb[0] = qb[1] = ML_NAN; return;
}
/* w := Newton step size (F(.) - a) / F'(.) or,
* -- log: (lF - la) / (F' / F) = exp(lF) * (lF - la) / F'
*/
w = log_p
? (y - la) * exp(y + logbeta + r * log(xinbta) + t * log1p(-xinbta))
: (y - a) * exp( logbeta + r * log(xinbta) + t * log1p(-xinbta));
if (i_pb >= n_N && w * wprev <= 0.)
prev = fmax2(fabs(adj),fpu);
R_ifDEBUG_printf("N(i=%2d): x0=%#17.15g, pb(x0)=%#17.15g, w=%#17.15g, %s prev=%g,",
i_pb, xinbta, y, w, (w * wprev <= 0.) ? "new" : "old", prev);
g = 1;
for (i_inn=0; i_inn < 1000;i_inn++) {
adj = g * w;
// take full Newton steps at the beginning; only then safe guard:
if (i_pb < n_N || fabs(adj) < prev) {
tx = xinbta - adj; // x_{n+1} = x_n - g*w
if (0. <= tx && tx <= 1.) {
if (prev <= acu || fabs(w) <= acu) {
R_ifDEBUG_printf(" it{in}=%d, delta(x)=%g, %s <= acu ==> convergence\n",
i_inn, -adj, (prev <= acu) ? "prev" : "|w|");
goto L_converged;
}
if (tx != 0. && tx != 1)
break;
}
}
g /= 3;
}
R_ifDEBUG_printf(" it{in}=%d, delta(x)=%g\n", i_inn, tx - xinbta);
if (fabs(tx - xinbta) <= 4e-16 * (tx + xinbta)) // "<=" : (.) == 0
goto L_converged;
xinbta = tx;
if(tx == 0) // "we have lost"
break;
wprev = w;
}
/*-- NOT converged: Iteration count --*/
warned = TRUE;
ML_ERROR(ME_PRECISION, "qbeta");
L_converged:
log_ = log_p || use_log_x; // only for printing
R_ifDEBUG_printf(" %s: Final delta(y) = %g%s\n",
warned ? "_NO_ convergence" : "converged",
y - (log_ ? la : a), (log_ ? " (log_)" : ""));
if((log_ && y == ML_NEGINF) || (!log_ && y == 0)) {
// stuck at left, try if smallest positive number is "better"
w = pbeta_raw(DBL_very_MIN, pp, qq, TRUE, log_);
if(log_ || fabs(w - a) <= fabs(y - a)) {
tx = DBL_very_MIN;
u_n = DBL_log_v_MIN;// = log(DBL_very_MIN)
}
add_N_step = FALSE; // not trying to do better anymore
}
else if(!warned && (log_ ? fabs(y - la) > 3 : fabs(y - a) > 1e-4)) {
if(!(log_ && y == ML_NEGINF &&
// e.g. qbeta(-1e-10, .2, .03, log=TRUE) cannot get accurate ==> do NOT warn
pbeta_raw(DBL_1__eps, // = 1 - eps
pp, qq, TRUE, TRUE) > la + 2))
MATHLIB_WARNING2( // low accuracy for more platform independent output:
"qbeta(a, *) =: x0 with |pbeta(x0,*%s) - alpha| = %.5g is not accurate",
(log_ ? ", log_" : ""), fabs(y - (log_ ? la : a)));
}
L_return:
if(give_log_q) { // ==> use_log_x , too
if(!use_log_x) // (see if claim above is true)
MATHLIB_WARNING(
"qbeta() L_return, u_n=%g; give_log_q=TRUE but use_log_x=FALSE -- please report!",
u_n);
double r = R_Log1_Exp(u_n);
if(swap_tail) {
qb[0] = r; qb[1] = u_n;
} else {
qb[0] = u_n; qb[1] = r;
}
} else {
if(use_log_x) {
if(add_N_step) {
/* add one last Newton step on original x scale, e.g., for
qbeta(2^-98, 0.125, 2^-96) */
xinbta = exp(u_n);
y = pbeta_raw(xinbta, pp, qq, /*lower_tail = */ TRUE, log_p);
w = log_p
? (y - la) * exp(y + logbeta + r * log(xinbta) + t * log1p(-xinbta))
: (y - a) * exp( logbeta + r * log(xinbta) + t * log1p(-xinbta));
tx = xinbta - w;
R_ifDEBUG_printf(
"Final Newton correction(non-log scale): xinbta=%.16g, y=%g, w=%g. => new tx=%.16g\n",
xinbta, y, w, tx);
} else {
if(swap_tail) {
qb[0] = -expm1(u_n); qb[1] = exp (u_n);
} else {
qb[0] = exp (u_n); qb[1] = -expm1(u_n);
}
return;
}
}
if(swap_tail) {
qb[0] = 1 - tx; qb[1] = tx;
} else {
qb[0] = tx; qb[1] = 1 - tx;
}
}
return;
}
double attribute_hidden
pnchisq_raw(double x, double f, double theta /* = ncp */,
double errmax, double reltol, int itrmax,
Rboolean lower_tail, Rboolean log_p)
{
double lam, x2, f2, term, bound, f_x_2n, f_2n;
double l_lam = -1., l_x = -1.; /* initialized for -Wall */
int n;
Rboolean lamSml, tSml, is_r, is_b, is_it;
LDOUBLE ans, u, v, t, lt, lu =-1;
if (x <= 0.) {
if(x == 0. && f == 0.) {
#define _L (-0.5 * theta) // = -lambda
return lower_tail ? R_D_exp(_L) : (log_p ? R_Log1_Exp(_L) : -expm1(_L));
}
/* x < 0 or {x==0, f > 0} */
return R_DT_0;
}
if(!R_FINITE(x)) return R_DT_1;
/* This is principally for use from qnchisq */
#ifndef MATHLIB_STANDALONE
R_CheckUserInterrupt();
#endif
if(theta < 80) { /* use 110 for Inf, as ppois(110, 80/2, lower.tail=FALSE) is 2e-20 */
LDOUBLE ans;
int i;
// Have pgamma(x,s) < x^s / Gamma(s+1) (< and ~= for small x)
// ==> pchisq(x, f) = pgamma(x, f/2, 2) = pgamma(x/2, f/2)
// < (x/2)^(f/2) / Gamma(f/2+1) < eps
// <==> f/2 * log(x/2) - log(Gamma(f/2+1)) < log(eps) ( ~= -708.3964 )
// <==> log(x/2) < 2/f*(log(Gamma(f/2+1)) + log(eps))
// <==> log(x) < log(2) + 2/f*(log(Gamma(f/2+1)) + log(eps))
if(lower_tail && f > 0. &&
log(x) < M_LN2 + 2/f*(lgamma(f/2. + 1) + _dbl_min_exp)) {
// all pchisq(x, f+2*i, lower_tail, FALSE), i=0,...,110 would underflow to 0.
// ==> work in log scale
double lambda = 0.5 * theta;
double sum, sum2, pr = -lambda;
sum = sum2 = ML_NEGINF;
/* we need to renormalize here: the result could be very close to 1 */
for(i = 0; i < 110; pr += log(lambda) - log(++i)) {
sum2 = logspace_add(sum2, pr);
sum = logspace_add(sum, pr + pchisq(x, f+2*i, lower_tail, TRUE));
if (sum2 >= -1e-15) /*<=> EXP(sum2) >= 1-1e-15 */ break;
}
ans = sum - sum2;
#ifdef DEBUG_pnch
REprintf("pnchisq(x=%g, f=%g, th.=%g); th. < 80, logspace: i=%d, ans=(sum=%g)-(sum2=%g)\n",
x,f,theta, i, (double)sum, (double)sum2);
#endif
return (double) (log_p ? ans : EXP(ans));
}
else {
LDOUBLE lambda = 0.5 * theta;
LDOUBLE sum = 0, sum2 = 0, pr = EXP(-lambda); // does this need a feature test?
/* we need to renormalize here: the result could be very close to 1 */
for(i = 0; i < 110; pr *= lambda/++i) {
// pr == exp(-lambda) lambda^i / i! == dpois(i, lambda)
sum2 += pr;
// pchisq(*, i, *) is strictly decreasing to 0 for lower_tail=TRUE
// and strictly increasing to 1 for lower_tail=FALSE
sum += pr * pchisq(x, f+2*i, lower_tail, FALSE);
if (sum2 >= 1-1e-15) break;
}
ans = sum/sum2;
#ifdef DEBUG_pnch
REprintf("pnchisq(x=%g, f=%g, theta=%g); theta < 80: i=%d, sum=%g, sum2=%g\n",
x,f,theta, i, (double)sum, (double)sum2);
#endif
return (double) (log_p ? LOG(ans) : ans);
}
} // if(theta < 80)
// else: theta == ncp >= 80 --------------------------------------------
#ifdef DEBUG_pnch
REprintf("pnchisq(x=%g, f=%g, theta=%g >= 80): ",x,f,theta);
#endif
// Series expansion ------- FIXME: log_p=TRUE, lower_tail=FALSE only applied at end
lam = .5 * theta;
lamSml = (-lam < _dbl_min_exp);
if(lamSml) {
/* MATHLIB_ERROR(
"non centrality parameter (= %g) too large for current algorithm",
theta) */
u = 0;
lu = -lam;/* == ln(u) */
l_lam = log(lam);
} else {
u = exp(-lam);
}
/* evaluate the first term */
v = u;
x2 = .5 * x;
f2 = .5 * f;
f_x_2n = f - x;
#ifdef DEBUG_pnch
REprintf("-- v=exp(-th/2)=%g, x/2= %g, f/2= %g\n",v,x2,f2);
#endif
if(f2 * DBL_EPSILON > 0.125 && /* very large f and x ~= f: probably needs */
FABS(t = x2 - f2) < /* another algorithm anyway */
sqrt(DBL_EPSILON) * f2) {
/* evade cancellation error */
/* t = exp((1 - t)*(2 - t/(f2 + 1))) / sqrt(2*M_PI*(f2 + 1));*/
lt = (1 - t)*(2 - t/(f2 + 1)) - M_LN_SQRT_2PI - 0.5 * log(f2 + 1);
#ifdef DEBUG_pnch
REprintf(" (case I) ==> ");
#endif
}
else {
/* Usual case 2: careful not to overflow .. : */
lt = f2*log(x2) -x2 - lgammafn(f2 + 1);
}
#ifdef DEBUG_pnch
REprintf(" lt= %g", lt);
#endif
tSml = (lt < _dbl_min_exp);
if(tSml) {
#ifdef DEBUG_pnch
REprintf(" is very small\n");
#endif
if (x > f + theta + 5* sqrt( 2*(f + 2*theta))) {
/* x > E[X] + 5* sigma(X) */
return R_DT_1; /* FIXME: could be more accurate than 0. */
} /* else */
l_x = log(x);
ans = term = 0.; t = 0;
}
else {
t = EXP(lt);
#ifdef DEBUG_pnch
REprintf(", t=exp(lt)= %g\n", t);
#endif
ans = term = (double) (v * t);
}
for (n = 1, f_2n = f + 2., f_x_2n += 2.; ; n++, f_2n += 2, f_x_2n += 2) {
#ifdef DEBUG_pnch_n
REprintf("\n _OL_: n=%d",n);
#endif
#ifndef MATHLIB_STANDALONE
if(n % 1000) R_CheckUserInterrupt();
#endif
/* f_2n === f + 2*n
* f_x_2n === f - x + 2*n > 0 <==> (f+2n) > x */
if (f_x_2n > 0) {
/* find the error bound and check for convergence */
bound = (double) (t * x / f_x_2n);
#ifdef DEBUG_pnch_n
REprintf("\n L10: n=%d; term= %g; bound= %g",n,term,bound);
#endif
is_r = is_it = FALSE;
/* convergence only if BOTH absolute and relative error < 'bnd' */
if (((is_b = (bound <= errmax)) &&
(is_r = (term <= reltol * ans))) || (is_it = (n > itrmax)))
{
#ifdef DEBUG_pnch
REprintf("BREAK n=%d %s; bound= %g %s, rel.err= %g %s\n",
n, (is_it ? "> itrmax" : ""),
bound, (is_b ? "<= errmax" : ""),
term/ans, (is_r ? "<= reltol" : ""));
#endif
break; /* out completely */
}
}
/* evaluate the next term of the */
/* expansion and then the partial sum */
if(lamSml) {
lu += l_lam - log(n); /* u = u* lam / n */
if(lu >= _dbl_min_exp) {
/* no underflow anymore ==> change regime */
#ifdef DEBUG_pnch_n
REprintf(" n=%d; nomore underflow in u = exp(lu) ==> change\n",
n);
#endif
v = u = EXP(lu); /* the first non-0 'u' */
lamSml = FALSE;
}
} else {
u *= lam / n;
v += u;
}
if(tSml) {
lt += l_x - log(f_2n);/* t <- t * (x / f2n) */
if(lt >= _dbl_min_exp) {
/* no underflow anymore ==> change regime */
#ifdef DEBUG_pnch
REprintf(" n=%d; nomore underflow in t = exp(lt) ==> change\n", n);
#endif
t = EXP(lt); /* the first non-0 't' */
tSml = FALSE;
}
} else {
t *= x / f_2n;
}
if(!lamSml && !tSml) {
term = (double) (v * t);
ans += term;
}
} /* for(n ...) */
if (is_it) {
MATHLIB_WARNING2(_("pnchisq(x=%g, ..): not converged in %d iter."),
x, itrmax);
}
#ifdef DEBUG_pnch
REprintf("\n == L_End: n=%d; term= %g; bound=%g\n",n,term,bound);
#endif
double dans = (double) ans;
return R_DT_val(dans);
}