double qpois(double p, double lambda, int lower_tail, int log_p)
{
double mu, sigma, gamma, z, y;
#ifdef IEEE_754
if (ISNAN(p) || ISNAN(lambda))
return p + lambda;
#endif
if(!R_FINITE(lambda))
ML_ERR_return_NAN;
if(lambda < 0) ML_ERR_return_NAN;
R_Q_P01_check(p);
if(lambda == 0) return 0;
if(p == R_DT_0) return 0;
if(p == R_DT_1) return ML_POSINF;
mu = lambda;
sigma = sqrt(lambda);
/* gamma = sigma; PR#8058 should be kurtosis which is mu^-0.5 */
gamma = 1.0/sigma;
/* Note : "same" code in qpois.c, qbinom.c, qnbinom.c --
* FIXME: This is far from optimal [cancellation for p ~= 1, etc]: */
if(!lower_tail || log_p) {
p = R_DT_qIv(p); /* need check again (cancellation!): */
if (p == 0.) return 0;
if (p == 1.) return ML_POSINF;
}
/* temporary hack --- FIXME --- */
if (p + 1.01*DBL_EPSILON >= 1.) return ML_POSINF;
/* y := approx.value (Cornish-Fisher expansion) : */
z = qnorm(p, 0., 1., /*lower_tail*/TRUE, /*log_p*/FALSE);
#ifdef HAVE_NEARBYINT
y = nearbyint(mu + sigma * (z + gamma * (z*z - 1) / 6));
#else
y = round(mu + sigma * (z + gamma * (z*z - 1) / 6));
#endif
z = ppois(y, lambda, /*lower_tail*/TRUE, /*log_p*/FALSE);
/* fuzz to ensure left continuity; 1 - 1e-7 may lose too much : */
p *= 1 - 64*DBL_EPSILON;
/* If the mean is not too large a simple search is OK */
if(lambda < 1e5) return do_search(y, &z, p, lambda, 1);
/* Otherwise be a bit cleverer in the search */
{
double incr = floor(y * 0.001), oldincr;
do {
oldincr = incr;
y = do_search(y, &z, p, lambda, incr);
incr = fmax2(1, floor(incr/100));
} while(oldincr > 1 && incr > lambda*1e-15);
return y;
}
}
double qunif(double p, double a, double b, int lower_tail, int log_p)
{
#ifdef IEEE_754
if (ISNAN(p) || ISNAN(a) || ISNAN(b))
return p + a + b;
#endif
R_Q_P01_check(p);
if (b <= a ) ML_ERR_return_NAN;
return a + R_DT_qIv(p) * (b - a);
}
double qwilcox(double x, double m, double n, int lower_tail, int log_p)
{
double c, p;
#ifdef IEEE_754
if (ISNAN(x) || ISNAN(m) || ISNAN(n))
return(x + m + n);
#endif
if(!R_FINITE(x) || !R_FINITE(m) || !R_FINITE(n))
ML_ERR_return_NAN;
R_Q_P01_check(x);
m = R_forceint(m);
n = R_forceint(n);
if (m <= 0 || n <= 0)
ML_ERR_return_NAN;
if (x == R_DT_0)
return(0);
if (x == R_DT_1)
return(m * n);
if(log_p || !lower_tail)
x = R_DT_qIv(x); /* lower_tail,non-log "p" */
int mm = (int) m, nn = (int) n;
w_init_maybe(mm, nn);
c = choose(m + n, n);
p = 0;
int q = 0;
if (x <= 0.5) {
x = x - 10 * DBL_EPSILON;
for (;;) {
p += cwilcox(q, mm, nn) / c;
if (p >= x)
break;
q++;
}
}
else {
x = 1 - x + 10 * DBL_EPSILON;
for (;;) {
p += cwilcox(q, mm, nn) / c;
if (p > x) {
q = (int) (m * n - q);
break;
}
q++;
}
}
return(q);
}
double qsignrank(double x, double n, int lower_tail, int log_p)
{
double f, p;
#ifdef IEEE_754
if (ISNAN(x) || ISNAN(n))
return(x + n);
#endif
if (!R_FINITE(x) || !R_FINITE(n))
ML_ERR_return_NAN;
R_Q_P01_check(x);
n = floor(n + 0.5);
if (n <= 0)
ML_ERR_return_NAN;
if (x == R_DT_0)
return(0);
if (x == R_DT_1)
return(n * (n + 1) / 2);
if(log_p || !lower_tail)
x = R_DT_qIv(x); /* lower_tail,non-log "p" */
int nn = (int) n;
w_init_maybe(nn);
f = exp(- n * M_LN2);
p = 0;
int q = 0;
if (x <= 0.5) {
x = x - 10 * DBL_EPSILON;
for (;;) {
p += csignrank(q, nn) * f;
if (p >= x)
break;
q++;
}
}
else {
x = 1 - x + 10 * DBL_EPSILON;
for (;;) {
p += csignrank(q, nn) * f;
if (p > x) {
q = (int)(n * (n + 1) / 2 - q);
break;
}
q++;
}
}
return(q);
}
double qnbinom(double p, double size, double prob, int lower_tail, int log_p)
{
double P, Q, mu, sigma, gamma, z, y;
#ifdef IEEE_754
if (ISNAN(p) || ISNAN(size) || ISNAN(prob))
return p + size + prob;
#endif
if (prob <= 0 || prob > 1 || size <= 0) ML_ERR_return_NAN;
/* FIXME: size = 0 is well defined ! */
if (prob == 1) return 0;
R_Q_P01_boundaries(p, 0, ML_POSINF);
Q = 1.0 / prob;
P = (1.0 - prob) * Q;
mu = size * P;
sigma = sqrt(size * P * Q);
gamma = (Q + P)/sigma;
/* Note : "same" code in qpois.c, qbinom.c, qnbinom.c --
* FIXME: This is far from optimal [cancellation for p ~= 1, etc]: */
if(!lower_tail || log_p) {
p = R_DT_qIv(p); /* need check again (cancellation!): */
if (p == R_DT_0) return 0;
if (p == R_DT_1) return ML_POSINF;
}
/* temporary hack --- FIXME --- */
if (p + 1.01*DBL_EPSILON >= 1.) return ML_POSINF;
/* y := approx.value (Cornish-Fisher expansion) : */
z = qnorm(p, 0., 1., /*lower_tail*/TRUE, /*log_p*/FALSE);
y = floor(mu + sigma * (z + gamma * (z*z - 1) / 6) + 0.5);
z = pnbinom(y, size, prob, /*lower_tail*/TRUE, /*log_p*/FALSE);
/* fuzz to ensure left continuity: */
p *= 1 - 64*DBL_EPSILON;
/* If the C-F value is not too large a simple search is OK */
if(y < 1e5) return do_search(y, &z, p, size, prob, 1);
/* Otherwise be a bit cleverer in the search */
{
double incr = floor(y * 0.001), oldincr;
do {
oldincr = incr;
y = do_search(y, &z, p, size, prob, incr);
incr = fmax2(1, floor(incr/100));
} while(oldincr > 1 && incr > y*1e-15);
return y;
}
}
double qcauchy(double p, double location, double scale,
int lower_tail, int log_p)
{
#ifdef IEEE_754
if (ISNAN(p) || ISNAN(location) || ISNAN(scale))
return p + location + scale;
#endif
if(!R_FINITE(p) || !R_FINITE(location) || !R_FINITE(scale))
ML_ERR_return_NAN;
R_Q_P01_check(p);
if (scale <= 0) ML_ERR_return_NAN;
return location + scale * tan(M_PI * (R_DT_qIv(p) - 0.5));
}
double qnt(double p, double df, double ncp, int lower_tail, int log_p)
{
const static double accu = 1e-13;
const static double Eps = 1e-11; /* must be > accu */
double ux, lx, nx, pp;
#ifdef IEEE_754
if (ISNAN(p) || ISNAN(df) || ISNAN(ncp))
return p + df + ncp;
#endif
if (!R_FINITE(df)) ML_ERR_return_NAN;
/* Was
* df = floor(df + 0.5);
* if (df < 1 || ncp < 0) ML_ERR_return_NAN;
*/
if (df <= 0.0) ML_ERR_return_NAN;
if(ncp == 0.0 && df >= 1.0) return qt(p, df, lower_tail, log_p);
R_Q_P01_boundaries(p, ML_NEGINF, ML_POSINF);
p = R_DT_qIv(p);
/* Invert pnt(.) :
* 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 = fmax2(1., ncp);
ux < DBL_MAX && pnt(ux, df, ncp, TRUE, FALSE) < pp;
ux *= 2);
pp = p * (1 - Eps);
for(lx = fmin2(-1., -ncp);
lx > -DBL_MAX && pnt(lx, df, ncp, TRUE, FALSE) > pp;
lx *= 2);
/* 2. interval (lx,ux) halving : */
do {
nx = 0.5 * (lx + ux);
if (pnt(nx, df, ncp, TRUE, FALSE) > p) ux = nx; else lx = nx;
}
while ((ux - lx) / fabs(nx) > accu);
return 0.5 * (lx + ux);
}
double qnbeta(double p, double a, double b, double ncp,
int lower_tail, int log_p)
{
const static double accu = 1e-15;
const static double Eps = 1e-14; /* must be > accu */
double ux, lx, nx, pp;
#ifdef IEEE_754
if (ISNAN(p) || ISNAN(a) || ISNAN(b) || ISNAN(ncp))
return p + a + b + ncp;
#endif
if (!R_FINITE(a)) ML_ERR_return_NAN;
if (ncp < 0. || a <= 0. || b <= 0.) ML_ERR_return_NAN;
R_Q_P01_boundaries(p, 0, 1);
p = R_DT_qIv(p);
/* Invert pnbeta(.) :
* 1. finding an upper and lower bound */
if(p > 1 - DBL_EPSILON) return 1.0;
pp = fmin2(1 - DBL_EPSILON, p * (1 + Eps));
for(ux = 0.5;
ux < 1 - DBL_EPSILON && pnbeta(ux, a, b, ncp, TRUE, FALSE) < pp;
ux = 0.5*(1+ux));
pp = p * (1 - Eps);
for(lx = 0.5;
lx > DBL_MIN && pnbeta(lx, a, b, ncp, TRUE, FALSE) > pp;
lx *= 0.5);
/* 2. interval (lx,ux) halving : */
do {
nx = 0.5 * (lx + ux);
if (pnbeta(nx, a, b, ncp, TRUE, FALSE) > p) ux = nx; else lx = nx;
}
while ((ux - lx) / nx > accu);
return 0.5 * (ux + lx);
}
// 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 qbinom(double p, double n, double pr, int lower_tail, int log_p)
{
double q, mu, sigma, gamma, z, y;
#ifdef IEEE_754
if (ISNAN(p) || ISNAN(n) || ISNAN(pr))
return p + n + pr;
#endif
if(!R_FINITE(n) || !R_FINITE(pr))
ML_ERR_return_NAN;
/* if log_p is true, p = -Inf is a legitimate value */
if(!R_FINITE(p) && !log_p)
ML_ERR_return_NAN;
if(n != floor(n + 0.5)) ML_ERR_return_NAN;
if (pr < 0 || pr > 1 || n < 0)
ML_ERR_return_NAN;
R_Q_P01_boundaries(p, 0, n);
if (pr == 0. || n == 0) return 0.;
q = 1 - pr;
if(q == 0.) return n; /* covers the full range of the distribution */
mu = n * pr;
sigma = sqrt(n * pr * q);
gamma = (q - pr) / sigma;
#ifdef DEBUG_qbinom
REprintf("qbinom(p=%7g, n=%g, pr=%7g, l.t.=%d, log=%d): sigm=%g, gam=%g\n",
p,n,pr, lower_tail, log_p, sigma, gamma);
#endif
/* Note : "same" code in qpois.c, qbinom.c, qnbinom.c --
* FIXME: This is far from optimal [cancellation for p ~= 1, etc]: */
if(!lower_tail || log_p) {
p = R_DT_qIv(p); /* need check again (cancellation!): */
if (p == 0.) return 0.;
if (p == 1.) return n;
}
/* temporary hack --- FIXME --- */
if (p + 1.01*DBL_EPSILON >= 1.) return n;
/* y := approx.value (Cornish-Fisher expansion) : */
z = qnorm(p, 0., 1., /*lower_tail*/TRUE, /*log_p*/FALSE);
y = floor(mu + sigma * (z + gamma * (z*z - 1) / 6) + 0.5);
if(y > n) /* way off */ y = n;
#ifdef DEBUG_qbinom
REprintf(" new (p,1-p)=(%7g,%7g), z=qnorm(..)=%7g, y=%5g\n", p, 1-p, z, y);
#endif
z = pbinom(y, n, pr, /*lower_tail*/TRUE, /*log_p*/FALSE);
/* fuzz to ensure left continuity: */
p *= 1 - 64*DBL_EPSILON;
if(n < 1e5) return do_search(y, &z, p, n, pr, 1);
/* Otherwise be a bit cleverer in the search */
{
double incr = floor(n * 0.001), oldincr;
do {
oldincr = incr;
y = do_search(y, &z, p, n, pr, incr);
incr = fmax2(1, floor(incr/100));
} while(oldincr > 1 && incr > n*1e-15);
return y;
}
}
double qbinom(double p, double n, double pr, int lower_tail, int log_p)
{
double q, mu, sigma, gamma, z, y;
#ifdef IEEE_754
if (ISNAN(p) || ISNAN(n) || ISNAN(pr))
return p + n + pr;
#endif
if(!R_FINITE(p) || !R_FINITE(n) || !R_FINITE(pr))
ML_ERR_return_NAN;
R_Q_P01_check(p);
if(n != floor(n + 0.5)) ML_ERR_return_NAN;
if (pr < 0 || pr > 1 || n < 0)
ML_ERR_return_NAN;
if (pr == 0. || n == 0) return 0.;
if (p == R_DT_0) return 0.;
if (p == R_DT_1) return n;
q = 1 - pr;
if(q == 0.) return n; /* covers the full range of the distribution */
mu = n * pr;
sigma = sqrt(n * pr * q);
gamma = (q - pr) / sigma;
#ifdef DEBUG_qbinom
REprintf("qbinom(p=%7g, n=%g, pr=%7g, l.t.=%d, log=%d): sigm=%g, gam=%g\n",
p,n,pr, lower_tail, log_p, sigma, gamma);
#endif
/* Note : "same" code in qpois.c, qbinom.c, qnbinom.c --
* FIXME: This is far from optimal [cancellation for p ~= 1, etc]: */
if(!lower_tail || log_p) {
p = R_DT_qIv(p); /* need check again (cancellation!): */
if (p == 0.) return 0.;
if (p == 1.) return n;
}
/* temporary hack --- FIXME --- */
if (p + 1.01*DBL_EPSILON >= 1.) return n;
/* y := approx.value (Cornish-Fisher expansion) : */
z = qnorm(p, 0., 1., /*lower_tail*/TRUE, /*log_p*/FALSE);
y = floor(mu + sigma * (z + gamma * (z*z - 1) / 6) + 0.5);
if(y > n) /* way off */ y = n;
#ifdef DEBUG_qbinom
REprintf(" new (p,1-p)=(%7g,%7g), z=qnorm(..)=%7g, y=%5g\n", p, 1-p, z, y);
#endif
z = pbinom(y, n, pr, /*lower_tail*/TRUE, /*log_p*/FALSE);
/* fuzz to ensure left continuity: */
p *= 1 - 64*DBL_EPSILON;
/*-- Fixme, here y can be way off --
should use interval search instead of primitive stepping down or up */
#ifdef maybe_future
if((lower_tail && z >= p) || (!lower_tail && z <= p)) {
#else
if(z >= p) {
#endif
/* search to the left */
#ifdef DEBUG_qbinom
REprintf("\tnew z=%7g >= p = %7g --> search to left (y--) ..\n", z,p);
#endif
for(;;) {
if(y == 0 ||
(z = pbinom(y - 1, n, pr, /*l._t.*/TRUE, /*log_p*/FALSE)) < p)
return y;
y = y - 1;
}
}
else { /* search to the right */
#ifdef DEBUG_qbinom
REprintf("\tnew z=%7g < p = %7g --> search to right (y++) ..\n", z,p);
#endif
for(;;) {
y = y + 1;
if(y == n ||
(z = pbinom(y, n, pr, /*l._t.*/TRUE, /*log_p*/FALSE)) >= p)
return y;
}
}
}
double qnorm5(double p, double mu, double sigma, int lower_tail, int log_p)
{
double p_, q, r, val;
#ifdef IEEE_754
if (ISNAN(p) || ISNAN(mu) || ISNAN(sigma))
return p + mu + sigma;
#endif
R_Q_P01_boundaries(p, ML_NEGINF, ML_POSINF);
if(sigma < 0) ML_ERR_return_NAN;
if(sigma == 0) return mu;
p_ = R_DT_qIv(p);/* real lower_tail prob. p */
q = p_ - 0.5;
#ifdef DEBUG_qnorm
REprintf("qnorm(p=%10.7g, m=%g, s=%g, l.t.= %d, log= %d): q = %g\n",
p,mu,sigma, lower_tail, log_p, q);
#endif
/*-- use AS 241 --- */
/* double ppnd16_(double *p, long *ifault)*/
/* ALGORITHM AS241 APPL. STATIST. (1988) VOL. 37, NO. 3
Produces the normal deviate Z corresponding to a given lower
tail area of P; Z is accurate to about 1 part in 10**16.
(original fortran code used PARAMETER(..) for the coefficients
and provided hash codes for checking them...)
*/
if (fabs(q) <= .425) {/* 0.075 <= p <= 0.925 */
r = .180625 - q * q;
val =
q * (((((((r * 2509.0809287301226727 +
33430.575583588128105) * r + 67265.770927008700853) * r +
45921.953931549871457) * r + 13731.693765509461125) * r +
1971.5909503065514427) * r + 133.14166789178437745) * r +
3.387132872796366608)
/ (((((((r * 5226.495278852854561 +
28729.085735721942674) * r + 39307.89580009271061) * r +
21213.794301586595867) * r + 5394.1960214247511077) * r +
687.1870074920579083) * r + 42.313330701600911252) * r + 1.);
}
else { /* closer than 0.075 from {0,1} boundary */
/* r = min(p, 1-p) < 0.075 */
if (q > 0)
r = R_DT_CIv(p);/* 1-p */
else
r = p_;/* = R_DT_Iv(p) ^= p */
r = sqrt(- ((log_p &&
((lower_tail && q <= 0) || (!lower_tail && q > 0))) ?
p : /* else */ log(r)));
/* r = sqrt(-log(r)) <==> min(p, 1-p) = exp( - r^2 ) */
#ifdef DEBUG_qnorm
REprintf("\t close to 0 or 1: r = %7g\n", r);
#endif
if (r <= 5.) { /* <==> min(p,1-p) >= exp(-25) ~= 1.3888e-11 */
r += -1.6;
val = (((((((r * 7.7454501427834140764e-4 +
.0227238449892691845833) * r + .24178072517745061177) *
r + 1.27045825245236838258) * r +
3.64784832476320460504) * r + 5.7694972214606914055) *
r + 4.6303378461565452959) * r +
1.42343711074968357734)
/ (((((((r *
1.05075007164441684324e-9 + 5.475938084995344946e-4) *
r + .0151986665636164571966) * r +
.14810397642748007459) * r + .68976733498510000455) *
r + 1.6763848301838038494) * r +
2.05319162663775882187) * r + 1.);
}
else { /* very close to 0 or 1 */
r += -5.;
val = (((((((r * 2.01033439929228813265e-7 +
2.71155556874348757815e-5) * r +
.0012426609473880784386) * r + .026532189526576123093) *
r + .29656057182850489123) * r +
1.7848265399172913358) * r + 5.4637849111641143699) *
r + 6.6579046435011037772)
/ (((((((r *
2.04426310338993978564e-15 + 1.4215117583164458887e-7)*
r + 1.8463183175100546818e-5) * r +
7.868691311456132591e-4) * r + .0148753612908506148525)
* r + .13692988092273580531) * r +
.59983220655588793769) * r + 1.);
}
if(q < 0.0)
val = -val;
/* return (q >= 0.)? r : -r ;*/
}
return mu + sigma * val;
}
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;
}
double qbeta(double alpha, double p, double q, int lower_tail, int log_p)
{
int swap_tail, i_pb, i_inn;
double a, adj, logbeta, g, h, pp, p_, prev, qq, r, s, t, tx, w, y, yprev;
double acu;
volatile double xinbta;
/* test for admissibility of parameters */
if (isnan(p) || isnan(q) || isnan(alpha)){
return p + q + alpha;
}
if(p < 0. || q < 0.){
report_error("shape parameters for qbeta must be > 0.");
}
R_Q_P01_boundaries(alpha, 0, 1);
p_ = R_DT_qIv(alpha);/* lower_tail prob (in any case) */
if(log_p && (p_ == 0. || p_ == 1.))
return p_; /* better than NaN or infinite loop;
FIXME: suboptimal, since -Inf < alpha ! */
/* initialize */
logbeta = lbeta(p, q);
/* change tail if necessary; afterwards 0 < a <= 1/2 */
if (p_ <= 0.5) {
a = p_; pp = p; qq = q; swap_tail = 0;
} else { /* change tail, swap p <-> q :*/
a = (!lower_tail && !log_p)? alpha : 1 - p_;
pp = q; qq = p; swap_tail = 1;
}
/* calculate the initial approximation */
/* y := {fast approximation of} qnorm(1 - a) :*/
r = sqrt(-2 * log(a));
y = r - (const1 + const2 * r) / (1. + (const3 + const4 * r) * r);
if (pp > 1 && qq > 1) {
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));
xinbta = pp / (pp + qq * exp(w + w));
} else {
r = qq + qq;
t = 1. / (9. * qq);
t = r * pow(1. - t + y * sqrt(t), 3.0);
if (t <= 0.)
xinbta = 1. - exp((log1p(-a)+ log(qq) + logbeta) / qq);
else {
t = (4. * pp + r - 2.) / t;
if (t <= 1.)
xinbta = exp((log(a * pp) + logbeta) / pp);
else
xinbta = 1. - 2. / (t + 1.);
}
}
/* solve for x by a modified newton-raphson method, */
/* using the function pbeta_raw */
r = 1 - pp;
t = 1 - qq;
yprev = 0.;
adj = 1;
/* Sometimes the approximation is negative! */
if (xinbta < lower)
xinbta = 0.5;
else if (xinbta > upper)
xinbta = 0.5;
/* Desired accuracy 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 = std::max<double>(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)
*/
acu = std::max<double>(acu_min, pow(10., -13 - 2.5/(pp * pp) - 0.5/(a * a)));
tx = prev = 0.; /* keep -Wall happy */
for (i_pb=0; i_pb < 1000; i_pb++) {
y = pbeta_raw(xinbta, pp, qq, /*lower_tail = */ true, false);
if(!std::isfinite(y)){
report_error("algorithm blew up ni qbeta");
}
y = (y - a) *
exp(logbeta + r * log(xinbta) + t * log1p(-xinbta));
if (y * yprev <= 0.)
prev = std::max<double>(fabs(adj),fpu);
g = 1;
for (i_inn=0; i_inn < 1000;i_inn++) {
adj = g * y;
if (fabs(adj) < prev) {
tx = xinbta - adj; /* trial new x */
if (tx >= 0. && tx <= 1) {
if (prev <= acu) goto L_converged;
if (fabs(y) <= acu) goto L_converged;
if (tx != 0. && tx != 1)
break;
}
}
g /= 3;
}
if (fabs(tx - xinbta) < 1e-15*xinbta) goto L_converged;
xinbta = tx;
yprev = y;
}
/*-- NOT converged: Iteration count --*/
report_error("algorithm did not converge in qbeta");
L_converged:
return swap_tail ? 1 - xinbta : xinbta;
}
double qgamma(double p, double alpha, double scale, int lower_tail, int log_p)
/* shape = alpha */
{
#define C7 4.67
#define C8 6.66
#define C9 6.73
#define C10 13.32
#define EPS1 1e-2
#define EPS2 5e-7/* final precision */
#define MAXIT 1000/* was 20 */
#define pMIN 1e-100 /* was 0.000002 = 2e-6 */
#define pMAX (1-1e-12)/* was 0.999998 = 1 - 2e-6 */
const double
i420 = 1./ 420.,
i2520 = 1./ 2520.,
i5040 = 1./ 5040;
double p_, a, b, c, ch, g, p1, v;
double p2, q, s1, s2, s3, s4, s5, s6, t, x;
int i;
/* test arguments and initialise */
#ifdef IEEE_754
if (ISNAN(p) || ISNAN(alpha) || ISNAN(scale))
return p + alpha + scale;
#endif
R_Q_P01_check(p);
if (alpha <= 0) ML_ERR_return_NAN;
/* FIXME: This (cutoff to {0, +Inf}) is far from optimal when log_p: */
p_ = R_DT_qIv(p);/* lower_tail prob (in any case) */
if (/* 0 <= */ p_ < pMIN) return 0;
if (/* 1 >= */ p_ > pMAX) return BOOM::infinity();
v = 2*alpha;
c = alpha-1;
g = lgammafn(alpha);/* log Gamma(v/2) */
/*----- Phase I : Starting Approximation */
#ifdef DEBUG_qgamma
REprintf("qgamma(p=%7g, alpha=%7g, scale=%7g, l.t.=%2d, log_p=%2d): ",
p,alpha,scale, lower_tail, log_p);
#endif
if(v < (-1.24)*R_DT_log(p)) { /* for small chi-squared */
#ifdef DEBUG_qgamma
REprintf(" small chi-sq.\n");
#endif
/* FIXME: Improve this "if (log_p)" :
* (A*exp(b)) ^ 1/al */
ch = pow(p_* alpha*exp(g+alpha*M_LN2), 1/alpha);
if(ch < EPS2) {/* Corrected according to AS 91; MM, May 25, 1999 */
goto END;
}
} else if(v > 0.32) { /* using Wilson and Hilferty estimate */
x = qnorm(p, 0, 1, lower_tail, log_p);
p1 = 0.222222/v;
ch = v*pow(x*sqrt(p1)+1-p1, 3);
#ifdef DEBUG_qgamma
REprintf(" v > .32: Wilson-Hilferty; x = %7g\n", x);
#endif
/* starting approximation for p tending to 1 */
if( ch > 2.2*v + 6 )
ch = -2*(R_DT_Clog(p) - c*log(0.5*ch) + g);
} else { /* for v <= 0.32 */
ch = 0.4;
a = R_DT_Clog(p) + g + c*M_LN2;
#ifdef DEBUG_qgamma
REprintf(" v <= .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) > EPS1*fabs(ch));
}
#ifdef DEBUG_qgamma
REprintf("\t==> ch = %10g:", ch);
#endif
/*----- Phase II: Iteration
* Call pgamma() [AS 239] and calculate seven term taylor series
*/
for( i=1 ; i <= MAXIT ; i++ ) {
q = ch;
p1 = 0.5*ch;
p2 = p_ - pgamma(p1, alpha, 1, /*lower_tail*/true, /*log_p*/false);
#ifdef IEEE_754
if(!R_FINITE(p2))
#else
if(errno != 0)
#endif
return numeric_limits<double>::quiet_NaN();
t = p2*exp(alpha*M_LN2+g+p1-c*log(ch));
b = t/ch;
a = 0.5*t - b*c;
s1 = (210+a*(140+a*(105+a*(84+a*(70+60*a))))) * i420;
s2 = (420+a*(735+a*(966+a*(1141+1278*a)))) * i2520;
s3 = (210+a*(462+a*(707+932*a))) * i2520;
s4 = (252+a*(672+1182*a)+c*(294+a*(889+1740*a))) * i5040;
s5 = (84+2264*a+c*(1175+606*a)) * i2520;
s6 = (120+c*(346+127*c)) * i5040;
ch += t*(1+0.5*t*s1-b*c*(s1-b*(s2-b*(s3-b*(s4-b*(s5-b*s6))))));
if(fabs(q - ch) < EPS2*ch)
goto END;
}
ML_ERROR(ME_PRECISION);/* no convergence in MAXIT iterations */
END:
return 0.5*scale*ch;
}
double qtukey(double p, double rr, double cc, double df,
int lower_tail, int log_p)
{
const static double eps = 0.0001;
const int maxiter = 50;
double ans = 0.0, valx0, valx1, x0, x1, xabs;
int iter;
#ifdef IEEE_754
if (ISNAN(p) || ISNAN(rr) || ISNAN(cc) || ISNAN(df)) {
ML_ERROR(ME_DOMAIN, "qtukey");
return p + rr + cc + df;
}
#endif
/* df must be > 1 ; there must be at least two values */
if (df < 2 || rr < 1 || cc < 2) ML_ERR_return_NAN;
R_Q_P01_boundaries(p, 0, ML_POSINF);
p = R_DT_qIv(p); /* lower_tail,non-log "p" */
/* Initial value */
x0 = qinv(p, cc, df);
/* Find prob(value < x0) */
valx0 = ptukey(x0, rr, cc, df, /*LOWER*/TRUE, /*LOG_P*/FALSE) - p;
/* Find the second iterate and prob(value < x1). */
/* If the first iterate has probability value */
/* exceeding p then second iterate is 1 less than */
/* first iterate; otherwise it is 1 greater. */
if (valx0 > 0.0)
x1 = fmax2(0.0, x0 - 1.0);
else
x1 = x0 + 1.0;
valx1 = ptukey(x1, rr, cc, df, /*LOWER*/TRUE, /*LOG_P*/FALSE) - p;
/* Find new iterate */
for(iter=1 ; iter < maxiter ; iter++) {
ans = x1 - ((valx1 * (x1 - x0)) / (valx1 - valx0));
valx0 = valx1;
/* New iterate must be >= 0 */
x0 = x1;
if (ans < 0.0) {
ans = 0.0;
valx1 = -p;
}
/* Find prob(value < new iterate) */
valx1 = ptukey(ans, rr, cc, df, /*LOWER*/TRUE, /*LOG_P*/FALSE) - p;
x1 = ans;
/* If the difference between two successive */
/* iterates is less than eps, stop */
xabs = fabs(x1 - x0);
if (xabs < eps)
return ans;
}
/* The process did not converge in 'maxiter' iterations */
ML_ERROR(ME_NOCONV, "qtukey");
return ans;
}
double qnbinom(double p, double n, double pr, int lower_tail, int log_p)
{
double P, Q, mu, sigma, gamma, z, y;
#ifdef IEEE_754
if (ISNAN(p) || ISNAN(n) || ISNAN(pr))
return p + n + pr;
#endif
R_Q_P01_check(p);
if (pr <= 0 || pr >= 1 || n <= 0) ML_ERR_return_NAN;
if (p == R_DT_0) return 0;
if (p == R_DT_1) return ML_POSINF;
Q = 1.0 / pr;
P = (1.0 - pr) * Q;
mu = n * P;
sigma = sqrt(n * P * Q);
gamma = (Q + P)/sigma;
/* Note : "same" code in qpois.c, qbinom.c, qnbinom.c --
* FIXME: This is far from optimal [cancellation for p ~= 1, etc]: */
if(!lower_tail || log_p) {
p = R_DT_qIv(p); /* need check again (cancellation!): */
if (p == R_DT_0) return 0;
if (p == R_DT_1) return ML_POSINF;
}
/* temporary hack --- FIXME --- */
if (p + 1.01*DBL_EPSILON >= 1.) return ML_POSINF;
/* y := approx.value (Cornish-Fisher expansion) : */
z = qnorm(p, 0., 1., /*lower_tail*/TRUE, /*log_p*/FALSE);
y = floor(mu + sigma * (z + gamma * (z*z - 1) / 6) + 0.5);
z = pnbinom(y, n, pr, /*lower_tail*/TRUE, /*log_p*/FALSE);
/* fuzz to ensure left continuity: */
p *= 1 - 64*DBL_EPSILON;
/*-- Fixme, here y can be way off --
should use interval search instead of primitive stepping down or up */
#ifdef maybe_future
if((lower_tail && z >= p) || (!lower_tail && z <= p)) {
#else
if(z >= p) {
#endif
/* search to the left */
for(;;) {
if(y == 0 ||
(z = pnbinom(y - 1, n, pr, /*l._t.*/TRUE, /*log_p*/FALSE)) < p)
return y;
y = y - 1;
}
}
else { /* search to the right */
for(;;) {
y = y + 1;
if((z = pnbinom(y, n, pr, /*l._t.*/TRUE, /*log_p*/FALSE)) >= p)
return y;
}
}
}