/* mpfr_digamma -- digamma function of a floating-point number
Copyright 2009-2017 Free Software Foundation, Inc.
Contributed by the AriC and Caramba projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU MPFR Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 3 of the License, or (at your
option) any later version.
The GNU MPFR Library 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 Lesser General Public
License for more details.
You should have received a copy of the GNU Lesser General Public License
along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
#include "mpfr-impl.h"
/* Put in s an approximation of digamma(x).
Assumes x >= 2.
Assumes s does not overlap with x.
Returns an integer e such that the error is bounded by 2^e ulps
of the result s.
*/
static mpfr_exp_t
mpfr_digamma_approx (mpfr_ptr s, mpfr_srcptr x)
{
mpfr_prec_t p = MPFR_PREC (s);
mpfr_t t, u, invxx;
mpfr_exp_t e, exps, f, expu;
mpz_t *INITIALIZED(B); /* variable B declared as initialized */
unsigned long n0, n; /* number of allocated B[] */
MPFR_ASSERTN(MPFR_IS_POS(x) && (MPFR_EXP(x) >= 2));
mpfr_init2 (t, p);
mpfr_init2 (u, p);
mpfr_init2 (invxx, p);
mpfr_log (s, x, MPFR_RNDN); /* error <= 1/2 ulp */
mpfr_ui_div (t, 1, x, MPFR_RNDN); /* error <= 1/2 ulp */
mpfr_div_2exp (t, t, 1, MPFR_RNDN); /* exact */
mpfr_sub (s, s, t, MPFR_RNDN);
/* error <= 1/2 + 1/2*2^(EXP(olds)-EXP(s)) + 1/2*2^(EXP(t)-EXP(s)).
For x >= 2, log(x) >= 2*(1/(2x)), thus olds >= 2t, and olds - t >= olds/2,
thus 0 <= EXP(olds)-EXP(s) <= 1, and EXP(t)-EXP(s) <= 0, thus
error <= 1/2 + 1/2*2 + 1/2 <= 2 ulps. */
e = 2; /* initial error */
mpfr_mul (invxx, x, x, MPFR_RNDZ); /* invxx = x^2 * (1 + theta)
for |theta| <= 2^(-p) */
mpfr_ui_div (invxx, 1, invxx, MPFR_RNDU); /* invxx = 1/x^2 * (1 + theta)^2 */
/* in the following we note err=xxx when the ratio between the approximation
and the exact result can be written (1 + theta)^xxx for |theta| <= 2^(-p),
following Higham's method */
B = mpfr_bernoulli_internal ((mpz_t *) 0, 0);
mpfr_set_ui (t, 1, MPFR_RNDN); /* err = 0 */
for (n = 1;; n++)
{
/* compute next Bernoulli number */
B = mpfr_bernoulli_internal (B, n);
/* The main term is Bernoulli[2n]/(2n)/x^(2n) = B[n]/(2n+1)!(2n)/x^(2n)
= B[n]*t[n]/(2n) where t[n]/t[n-1] = 1/(2n)/(2n+1)/x^2. */
mpfr_mul (t, t, invxx, MPFR_RNDU); /* err = err + 3 */
mpfr_div_ui (t, t, 2 * n, MPFR_RNDU); /* err = err + 1 */
mpfr_div_ui (t, t, 2 * n + 1, MPFR_RNDU); /* err = err + 1 */
/* we thus have err = 5n here */
mpfr_div_ui (u, t, 2 * n, MPFR_RNDU); /* err = 5n+1 */
mpfr_mul_z (u, u, B[n], MPFR_RNDU); /* err = 5n+2, and the
absolute error is bounded
by 10n+4 ulp(u) [Rule 11] */
/* if the terms 'u' are decreasing by a factor two at least,
then the error coming from those is bounded by
sum((10n+4)/2^n, n=1..infinity) = 24 */
exps = mpfr_get_exp (s);
expu = mpfr_get_exp (u);
if (expu < exps - (mpfr_exp_t) p)
break;
mpfr_sub (s, s, u, MPFR_RNDN); /* error <= 24 + n/2 */
if (mpfr_get_exp (s) < exps)
e <<= exps - mpfr_get_exp (s);
e ++; /* error in mpfr_sub */
f = 10 * n + 4;
while (expu < exps)
{
f = (1 + f) / 2;
expu ++;
}
e += f; /* total rouding error coming from 'u' term */
}
n0 = ++n;
while (n--)
mpz_clear (B[n]);
(*__gmp_free_func) (B, n0 * sizeof (mpz_t));
mpfr_clear (t);
mpfr_clear (u);
mpfr_clear (invxx);
f = 0;
while (e > 1)
{
f++;
e = (e + 1) / 2;
/* Invariant: 2^f * e does not decrease */
}
return f;
}
/* Use the reflection formula Digamma(1-x) = Digamma(x) + Pi * cot(Pi*x),
i.e., Digamma(x) = Digamma(1-x) - Pi * cot(Pi*x).
Assume x < 1/2. */
static int
mpfr_digamma_reflection (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_prec_t p = MPFR_PREC(y) + 10, q;
mpfr_t t, u, v;
mpfr_exp_t e1, expv;
int inex;
MPFR_ZIV_DECL (loop);
/* we want that 1-x is exact with precision q: if 0 < x < 1/2, then
q = PREC(x)-EXP(x) is ok, otherwise if -1 <= x < 0, q = PREC(x)-EXP(x)
is ok, otherwise for x < -1, PREC(x) is ok if EXP(x) <= PREC(x),
otherwise we need EXP(x) */
if (MPFR_EXP(x) < 0)
q = MPFR_PREC(x) + 1 - MPFR_EXP(x);
else if (MPFR_EXP(x) <= MPFR_PREC(x))
q = MPFR_PREC(x) + 1;
else
q = MPFR_EXP(x);
mpfr_init2 (u, q);
MPFR_ASSERTN(mpfr_ui_sub (u, 1, x, MPFR_RNDN) == 0);
/* if x is half an integer, cot(Pi*x) = 0, thus Digamma(x) = Digamma(1-x) */
mpfr_mul_2exp (u, u, 1, MPFR_RNDN);
inex = mpfr_integer_p (u);
mpfr_div_2exp (u, u, 1, MPFR_RNDN);
if (inex)
{
inex = mpfr_digamma (y, u, rnd_mode);
goto end;
}
mpfr_init2 (t, p);
mpfr_init2 (v, p);
MPFR_ZIV_INIT (loop, p);
for (;;)
{
mpfr_const_pi (v, MPFR_RNDN); /* v = Pi*(1+theta) for |theta|<=2^(-p) */
mpfr_mul (t, v, x, MPFR_RNDN); /* (1+theta)^2 */
e1 = MPFR_EXP(t) - (mpfr_exp_t) p + 1; /* bound for t: err(t) <= 2^e1 */
mpfr_cot (t, t, MPFR_RNDN);
/* cot(t * (1+h)) = cot(t) - theta * (1 + cot(t)^2) with |theta|<=t*h */
if (MPFR_EXP(t) > 0)
e1 = e1 + 2 * MPFR_EXP(t) + 1;
else
e1 = e1 + 1;
/* now theta * (1 + cot(t)^2) <= 2^e1 */
e1 += (mpfr_exp_t) p - MPFR_EXP(t); /* error is now 2^e1 ulps */
mpfr_mul (t, t, v, MPFR_RNDN);
e1 ++;
mpfr_digamma (v, u, MPFR_RNDN); /* error <= 1/2 ulp */
expv = MPFR_EXP(v);
mpfr_sub (v, v, t, MPFR_RNDN);
if (MPFR_EXP(v) < MPFR_EXP(t))
e1 += MPFR_EXP(t) - MPFR_EXP(v); /* scale error for t wrt new v */
/* now take into account the 1/2 ulp error for v */
if (expv - MPFR_EXP(v) - 1 > e1)
e1 = expv - MPFR_EXP(v) - 1;
else
e1 ++;
e1 ++; /* rounding error for mpfr_sub */
if (MPFR_CAN_ROUND (v, p - e1, MPFR_PREC(y), rnd_mode))
break;
MPFR_ZIV_NEXT (loop, p);
mpfr_set_prec (t, p);
mpfr_set_prec (v, p);
}
MPFR_ZIV_FREE (loop);
inex = mpfr_set (y, v, rnd_mode);
mpfr_clear (t);
mpfr_clear (v);
end:
mpfr_clear (u);
return inex;
}
/* we have x >= 1/2 here */
static int
mpfr_digamma_positive (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_prec_t p = MPFR_PREC(y) + 10, q;
mpfr_t t, u, x_plus_j;
int inex;
mpfr_exp_t errt, erru, expt;
unsigned long j = 0, min;
MPFR_ZIV_DECL (loop);
/* compute a precision q such that x+1 is exact */
if (MPFR_PREC(x) < MPFR_EXP(x))
q = MPFR_EXP(x);
else
q = MPFR_PREC(x) + 1;
mpfr_init2 (x_plus_j, q);
mpfr_init2 (t, p);
mpfr_init2 (u, p);
MPFR_ZIV_INIT (loop, p);
for(;;)
{
/* Lower bound for x+j in mpfr_digamma_approx call: since the smallest
term of the divergent series for Digamma(x) is about exp(-2*Pi*x), and
we want it to be less than 2^(-p), this gives x > p*log(2)/(2*Pi)
i.e., x >= 0.1103 p.
To be safe, we ensure x >= 0.25 * p.
*/
min = (p + 3) / 4;
if (min < 2)
min = 2;
mpfr_set (x_plus_j, x, MPFR_RNDN);
mpfr_set_ui (u, 0, MPFR_RNDN);
j = 0;
while (mpfr_cmp_ui (x_plus_j, min) < 0)
{
j ++;
mpfr_ui_div (t, 1, x_plus_j, MPFR_RNDN); /* err <= 1/2 ulp */
mpfr_add (u, u, t, MPFR_RNDN);
inex = mpfr_add_ui (x_plus_j, x_plus_j, 1, MPFR_RNDZ);
if (inex != 0) /* we lost one bit */
{
q ++;
mpfr_prec_round (x_plus_j, q, MPFR_RNDZ);
mpfr_nextabove (x_plus_j);
}
/* since all terms are positive, the error is bounded by j ulps */
}
for (erru = 0; j > 1; erru++, j = (j + 1) / 2);
errt = mpfr_digamma_approx (t, x_plus_j);
expt = MPFR_EXP(t);
mpfr_sub (t, t, u, MPFR_RNDN);
if (MPFR_EXP(t) < expt)
errt += expt - MPFR_EXP(t);
if (MPFR_EXP(t) < MPFR_EXP(u))
erru += MPFR_EXP(u) - MPFR_EXP(t);
if (errt > erru)
errt = errt + 1;
else if (errt == erru)
errt = errt + 2;
else
errt = erru + 1;
if (MPFR_CAN_ROUND (t, p - errt, MPFR_PREC(y), rnd_mode))
break;
MPFR_ZIV_NEXT (loop, p);
mpfr_set_prec (t, p);
mpfr_set_prec (u, p);
}
MPFR_ZIV_FREE (loop);
inex = mpfr_set (y, t, rnd_mode);
mpfr_clear (t);
mpfr_clear (u);
mpfr_clear (x_plus_j);
return inex;
}
int
mpfr_digamma (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
int inex;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode),
("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(y), mpfr_log_prec, y, inex));
if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(x)))
{
if (MPFR_IS_NAN(x))
{
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF(x))
{
if (MPFR_IS_POS(x)) /* Digamma(+Inf) = +Inf */
{
MPFR_SET_SAME_SIGN(y, x);
MPFR_SET_INF(y);
MPFR_RET(0);
}
else /* Digamma(-Inf) = NaN */
{
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
}
else /* Zero case */
{
/* the following works also in case of overlap */
MPFR_SET_INF(y);
MPFR_SET_OPPOSITE_SIGN(y, x);
mpfr_set_divby0 ();
MPFR_RET(0);
}
}
/* Digamma is undefined for negative integers */
if (MPFR_IS_NEG(x) && mpfr_integer_p (x))
{
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
/* now x is a normal number */
MPFR_SAVE_EXPO_MARK (expo);
/* for x very small, we have Digamma(x) = -1/x - gamma + O(x), more precisely
-1 < Digamma(x) + 1/x < 0 for -0.2 < x < 0.2, thus:
(i) either x is a power of two, then 1/x is exactly representable, and
as long as 1/2*ulp(1/x) > 1, we can conclude;
(ii) otherwise assume x has <= n bits, and y has <= n+1 bits, then
|y + 1/x| >= 2^(-2n) ufp(y), where ufp means unit in first place.
Since |Digamma(x) + 1/x| <= 1, if 2^(-2n) ufp(y) >= 2, then
|y - Digamma(x)| >= 2^(-2n-1)ufp(y), and rounding -1/x gives the correct result.
If x < 2^E, then y > 2^(-E), thus ufp(y) > 2^(-E-1).
A sufficient condition is thus EXP(x) <= -2 MAX(PREC(x),PREC(Y)). */
if (MPFR_EXP(x) < -2)
{
if (MPFR_EXP(x) <= -2 * (mpfr_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(y)))
{
int signx = MPFR_SIGN(x);
inex = mpfr_si_div (y, -1, x, rnd_mode);
if (inex == 0) /* x is a power of two */
{ /* result always -1/x, except when rounding down */
if (rnd_mode == MPFR_RNDA)
rnd_mode = (signx > 0) ? MPFR_RNDD : MPFR_RNDU;
if (rnd_mode == MPFR_RNDZ)
rnd_mode = (signx > 0) ? MPFR_RNDU : MPFR_RNDD;
if (rnd_mode == MPFR_RNDU)
inex = 1;
else if (rnd_mode == MPFR_RNDD)
{
mpfr_nextbelow (y);
inex = -1;
}
else /* nearest */
inex = 1;
}
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
goto end;
}
}
if (MPFR_IS_NEG(x))
inex = mpfr_digamma_reflection (y, x, rnd_mode);
/* if x < 1/2 we use the reflection formula */
else if (MPFR_EXP(x) < 0)
inex = mpfr_digamma_reflection (y, x, rnd_mode);
else
inex = mpfr_digamma_positive (y, x, rnd_mode);
end:
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inex, rnd_mode);
}