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/* mpfr_y0, mpfr_y1, mpfr_yn -- Bessel functions of 2nd kind, integer order.
http://www.opengroup.org/onlinepubs/009695399/functions/y0.html
Copyright 2007, 2008, 2009 Free Software Foundation, Inc.
Contributed by the Arenaire and Cacao 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 2.1 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.LIB. If not, write to
the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston,
MA 02110-1301, USA. */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
static int mpfr_yn_asympt (mpfr_ptr, long, mpfr_srcptr, mp_rnd_t);
int
mpfr_y0 (mpfr_ptr res, mpfr_srcptr z, mp_rnd_t r)
{
return mpfr_yn (res, 0, z, r);
}
int
mpfr_y1 (mpfr_ptr res, mpfr_srcptr z, mp_rnd_t r)
{
return mpfr_yn (res, 1, z, r);
}
/* compute in s an approximation of S1 = sum((n-k)!/k!*y^k,k=0..n)
return e >= 0 the exponent difference between the maximal value of |s|
during the for loop and the final value of |s|.
*/
static mp_exp_t
mpfr_yn_s1 (mpfr_ptr s, mpfr_srcptr y, unsigned long n)
{
unsigned long k;
mpz_t f;
mp_exp_t e, emax;
mpz_init_set_ui (f, 1);
/* we compute n!*S1 = sum(a[k]*y^k,k=0..n) where a[k] = n!*(n-k)!/k!,
a[0] = (n!)^2, a[1] = n!*(n-1)!, ..., a[n-1] = n, a[n] = 1 */
mpfr_set_ui (s, 1, GMP_RNDN); /* a[n] */
emax = MPFR_EXP(s);
for (k = n; k-- > 0;)
{
/* a[k]/a[k+1] = (n-k)!/k!/(n-(k+1))!*(k+1)! = (k+1)*(n-k) */
mpfr_mul (s, s, y, GMP_RNDN);
mpz_mul_ui (f, f, n - k);
mpz_mul_ui (f, f, k + 1);
/* invariant: f = a[k] */
mpfr_add_z (s, s, f, GMP_RNDN);
e = MPFR_EXP(s);
if (e > emax)
emax = e;
}
/* now we have f = (n!)^2 */
mpz_sqrt (f, f);
mpfr_div_z (s, s, f, GMP_RNDN);
mpz_clear (f);
return emax - MPFR_EXP(s);
}
/* compute in s an approximation of
S3 = c*sum((h(k)+h(n+k))*y^k/k!/(n+k)!,k=0..infinity)
where h(k) = 1 + 1/2 + ... + 1/k
k=0: h(n)
k=1: 1+h(n+1)
k=2: 3/2+h(n+2)
Returns e such that the error is bounded by 2^e ulp(s).
*/
static mp_exp_t
mpfr_yn_s3 (mpfr_ptr s, mpfr_srcptr y, mpfr_srcptr c, unsigned long n)
{
unsigned long k, zz;
mpfr_t t, u;
mpz_t p, q; /* p/q will store h(k)+h(n+k) */
mp_exp_t exps, expU;
zz = mpfr_get_ui (y, GMP_RNDU); /* y = z^2/4 */
MPFR_ASSERTN (zz < ULONG_MAX - 2);
zz += 2; /* z^2 <= 2^zz */
mpz_init_set_ui (p, 0);
mpz_init_set_ui (q, 1);
/* initialize p/q to h(n) */
for (k = 1; k <= n; k++)
{
/* p/q + 1/k = (k*p+q)/(q*k) */
mpz_mul_ui (p, p, k);
mpz_add (p, p, q);
mpz_mul_ui (q, q, k);
}
mpfr_init2 (t, MPFR_PREC(s));
mpfr_init2 (u, MPFR_PREC(s));
mpfr_fac_ui (t, n, GMP_RNDN);
mpfr_div (t, c, t, GMP_RNDN); /* c/n! */
mpfr_mul_z (u, t, p, GMP_RNDN);
mpfr_div_z (s, u, q, GMP_RNDN);
exps = MPFR_EXP (s);
expU = exps;
for (k = 1; ;k ++)
{
/* update t */
mpfr_mul (t, t, y, GMP_RNDN);
mpfr_div_ui (t, t, k, GMP_RNDN);
mpfr_div_ui (t, t, n + k, GMP_RNDN);
/* update p/q:
p/q + 1/k + 1/(n+k) = [p*k*(n+k) + q*(n+k) + q*k]/(q*k*(n+k)) */
mpz_mul_ui (p, p, k);
mpz_mul_ui (p, p, n + k);
mpz_addmul_ui (p, q, n + 2 * k);
mpz_mul_ui (q, q, k);
mpz_mul_ui (q, q, n + k);
mpfr_mul_z (u, t, p, GMP_RNDN);
mpfr_div_z (u, u, q, GMP_RNDN);
exps = MPFR_EXP (u);
if (exps > expU)
expU = exps;
mpfr_add (s, s, u, GMP_RNDN);
exps = MPFR_EXP (s);
if (exps > expU)
expU = exps;
if (MPFR_EXP (u) + (mp_exp_t) MPFR_PREC (u) < MPFR_EXP (s) &&
zz / (2 * k) < k + n)
break;
}
mpfr_clear (t);
mpfr_clear (u);
mpz_clear (p);
mpz_clear (q);
exps = expU - MPFR_EXP (s);
/* the error is bounded by (6k^2+33/2k+11) 2^exps ulps
<= 8*(k+2)^2 2^exps ulps */
return 3 + 2 * MPFR_INT_CEIL_LOG2(k + 2) + exps;
}
int
mpfr_yn (mpfr_ptr res, long n, mpfr_srcptr z, mp_rnd_t r)
{
int inex;
unsigned long absn;
MPFR_LOG_FUNC (("x[%#R]=%R n=%d rnd=%d", z, z, n, r),
("y[%#R]=%R", res, res));
absn = SAFE_ABS (unsigned long, n);
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (z)))
{
if (MPFR_IS_NAN (z))
{
MPFR_SET_NAN (res); /* y(n,NaN) = NaN */
MPFR_RET_NAN;
}
/* y(n,z) tends to zero when z goes to +Inf, oscillating around
0. We choose to return +0 in that case. */
else if (MPFR_IS_INF (z))
{
if (MPFR_SIGN(z) > 0)
return mpfr_set_ui (res, 0, r);
else /* y(n,-Inf) = NaN */
{
MPFR_SET_NAN (res);
MPFR_RET_NAN;
}
}
else /* y(n,z) tends to -Inf for n >= 0 or n even, to +Inf otherwise,
when z goes to zero */
{
MPFR_SET_INF(res);
if (n >= 0 || (n & 1) == 0)
MPFR_SET_NEG(res);
else
MPFR_SET_POS(res);
MPFR_RET(0);
}
}
/* for z < 0, y(n,z) is imaginary except when j(n,|z|) = 0, which we
assume does not happen for a rational z. */
if (MPFR_SIGN(z) < 0)
{
MPFR_SET_NAN (res);
MPFR_RET_NAN;
}
/* now z is not singular, and z > 0 */
/* Deal with tiny arguments. We have:
y0(z) = 2 log(z)/Pi + 2 (euler - log(2))/Pi + O(log(z)*z^2), more
precisely for 0 <= z <= 1/2, with g(z) = 2/Pi + 2(euler-log(2))/Pi/log(z),
g(z) - 0.41*z^2 < y0(z)/log(z) < g(z)
thus since log(z) is negative:
g(z)*log(z) < y0(z) < (g(z) - z^2/2)*log(z)
and since |g(z)| >= 0.63 for 0 <= z <= 1/2, the relative error on
y0(z)/log(z) is bounded by 0.41*z^2/0.63 <= 0.66*z^2.
Note: we use both the main term in log(z) and the constant term, because
otherwise the relative error would be only in 1/log(|log(z)|).
*/
if (n == 0 && MPFR_EXP(z) < - (mp_exp_t) (MPFR_PREC(res) / 2))
{
mpfr_t l, h, t, logz;
mp_prec_t prec;
int ok, inex2;
prec = MPFR_PREC(res) + 10;
mpfr_init2 (l, prec);
mpfr_init2 (h, prec);
mpfr_init2 (t, prec);
mpfr_init2 (logz, prec);
/* first enclose log(z) + euler - log(2) = log(z/2) + euler */
mpfr_log (logz, z, GMP_RNDD); /* lower bound of log(z) */
mpfr_set (h, logz, GMP_RNDU); /* exact */
mpfr_nextabove (h); /* upper bound of log(z) */
mpfr_const_euler (t, GMP_RNDD); /* lower bound of euler */
mpfr_add (l, logz, t, GMP_RNDD); /* lower bound of log(z) + euler */
mpfr_nextabove (t); /* upper bound of euler */
mpfr_add (h, h, t, GMP_RNDU); /* upper bound of log(z) + euler */
mpfr_const_log2 (t, GMP_RNDU); /* upper bound of log(2) */
mpfr_sub (l, l, t, GMP_RNDD); /* lower bound of log(z/2) + euler */
mpfr_nextbelow (t); /* lower bound of log(2) */
mpfr_sub (h, h, t, GMP_RNDU); /* upper bound of log(z/2) + euler */
mpfr_const_pi (t, GMP_RNDU); /* upper bound of Pi */
mpfr_div (l, l, t, GMP_RNDD); /* lower bound of (log(z/2)+euler)/Pi */
mpfr_nextbelow (t); /* lower bound of Pi */
mpfr_div (h, h, t, GMP_RNDD); /* upper bound of (log(z/2)+euler)/Pi */
mpfr_mul_2ui (l, l, 1, GMP_RNDD); /* lower bound on g(z)*log(z) */
mpfr_mul_2ui (h, h, 1, GMP_RNDU); /* upper bound on g(z)*log(z) */
/* we now have l <= g(z)*log(z) <= h, and we need to add -z^2/2*log(z)
to h */
mpfr_mul (t, z, z, GMP_RNDU); /* upper bound on z^2 */
/* since logz is negative, a lower bound corresponds to an upper bound
for its absolute value */
mpfr_neg (t, t, GMP_RNDD);
mpfr_div_2ui (t, t, 1, GMP_RNDD);
mpfr_mul (t, t, logz, GMP_RNDU); /* upper bound on z^2/2*log(z) */
/* an underflow may happen in the above instructions, clear flag */
mpfr_clear_underflow ();
mpfr_add (h, h, t, GMP_RNDU);
inex = mpfr_prec_round (l, MPFR_PREC(res), r);
inex2 = mpfr_prec_round (h, MPFR_PREC(res), r);
/* we need h=l and inex=inex2 */
ok = (inex == inex2) && (mpfr_cmp (l, h) == 0);
if (ok)
mpfr_set (res, h, r); /* exact */
mpfr_clear (l);
mpfr_clear (h);
mpfr_clear (t);
mpfr_clear (logz);
if (ok)
return inex;
}
/* small argument check for y1(z) = -2/Pi/z + O(log(z)):
for 0 <= z <= 1, |y1(z) + 2/Pi/z| <= 0.25 */
if (n == 1 && MPFR_EXP(z) + 1 < - (mp_exp_t) MPFR_PREC(res))
{
mpfr_t y;
mp_prec_t prec;
mp_exp_t err1;
int ok;
MPFR_BLOCK_DECL (flags);
/* since 2/Pi > 0.5, and |y1(z)| >= |2/Pi/z|, if z <= 2^(-emax-1),
then |y1(z)| > 2^emax */
prec = MPFR_PREC(res) + 10;
mpfr_init2 (y, prec);
mpfr_const_pi (y, GMP_RNDU); /* Pi*(1+u)^2, where here and below u
represents a quantity <= 1/2^prec */
mpfr_mul (y, y, z, GMP_RNDU); /* Pi*z * (1+u)^4, upper bound */
MPFR_BLOCK (flags, mpfr_ui_div (y, 2, y, GMP_RNDZ));
/* 2/Pi/z * (1+u)^6, lower bound, with possible overflow */
if (MPFR_OVERFLOW (flags))
{
mpfr_clear (y);
return mpfr_overflow (res, r, -1);
}
mpfr_neg (y, y, GMP_RNDN);
/* (1+u)^6 can be written 1+7u [for another value of u], thus the
error on 2/Pi/z is less than 7ulp(y). The truncation error is less
than 1/4, thus if ulp(y)>=1/4, the total error is less than 8ulp(y),
otherwise it is less than 1/4+7/8 <= 2. */
if (MPFR_EXP(y) + 2 >= MPFR_PREC(y)) /* ulp(y) >= 1/4 */
err1 = 3;
else /* ulp(y) <= 1/8 */
err1 = (mp_exp_t) MPFR_PREC(y) - MPFR_EXP(y) + 1;
ok = MPFR_CAN_ROUND (y, prec - err1, MPFR_PREC(res), r);
if (ok)
inex = mpfr_set (res, y, r);
mpfr_clear (y);
if (ok)
return inex;
}
/* we can use the asymptotic expansion as soon as z > p log(2)/2,
but to get some margin we use it for z > p/2 */
if (mpfr_cmp_ui (z, MPFR_PREC(res) / 2 + 3) > 0)
{
inex = mpfr_yn_asympt (res, n, z, r);
if (inex != 0)
return inex;
}
/* General case */
{
mp_prec_t prec;
mp_exp_t err1, err2, err3;
mpfr_t y, s1, s2, s3;
MPFR_ZIV_DECL (loop);
mpfr_init (y);
mpfr_init (s1);
mpfr_init (s2);
mpfr_init (s3);
prec = MPFR_PREC(res) + 2 * MPFR_INT_CEIL_LOG2 (MPFR_PREC (res)) + 13;
MPFR_ZIV_INIT (loop, prec);
for (;;)
{
mpfr_set_prec (y, prec);
mpfr_set_prec (s1, prec);
mpfr_set_prec (s2, prec);
mpfr_set_prec (s3, prec);
mpfr_mul (y, z, z, GMP_RNDN);
mpfr_div_2ui (y, y, 2, GMP_RNDN); /* z^2/4 */
/* store (z/2)^n temporarily in s2 */
mpfr_pow_ui (s2, z, absn, GMP_RNDN);
mpfr_div_2si (s2, s2, absn, GMP_RNDN);
/* compute S1 * (z/2)^(-n) */
if (n == 0)
{
mpfr_set_ui (s1, 0, GMP_RNDN);
err1 = 0;
}
else
err1 = mpfr_yn_s1 (s1, y, absn - 1);
mpfr_div (s1, s1, s2, GMP_RNDN); /* (z/2)^(-n) * S1 */
/* See algorithms.tex: the relative error on s1 is bounded by
(3n+3)*2^(e+1-prec). */
err1 = MPFR_INT_CEIL_LOG2 (3 * absn + 3) + err1 + 1;
/* rel_err(s1) <= 2^(err1-prec), thus err(s1) <= 2^err1 ulps */
/* compute (z/2)^n * S3 */
mpfr_neg (y, y, GMP_RNDN); /* -z^2/4 */
err3 = mpfr_yn_s3 (s3, y, s2, absn); /* (z/2)^n * S3 */
/* the error on s3 is bounded by 2^err3 ulps */
/* add s1+s3 */
err1 += MPFR_EXP(s1);
mpfr_add (s1, s1, s3, GMP_RNDN);
/* the error is bounded by 1/2 + 2^err1*2^(- EXP(s1))
+ 2^err3*2^(EXP(s3) - EXP(s1)) */
err3 += MPFR_EXP(s3);
err1 = (err3 > err1) ? err3 + 1 : err1 + 1;
err1 -= MPFR_EXP(s1);
err1 = (err1 >= 0) ? err1 + 1 : 1;
/* now the error on s1 is bounded by 2^err1*ulp(s1) */
/* compute S2 */
mpfr_div_2ui (s2, z, 1, GMP_RNDN); /* z/2 */
mpfr_log (s2, s2, GMP_RNDN); /* log(z/2) */
mpfr_const_euler (s3, GMP_RNDN);
err2 = MPFR_EXP(s2) > MPFR_EXP(s3) ? MPFR_EXP(s2) : MPFR_EXP(s3);
mpfr_add (s2, s2, s3, GMP_RNDN); /* log(z/2) + gamma */
err2 -= MPFR_EXP(s2);
mpfr_mul_2ui (s2, s2, 1, GMP_RNDN); /* 2*(log(z/2) + gamma) */
mpfr_jn (s3, absn, z, GMP_RNDN); /* Jn(z) */
mpfr_mul (s2, s2, s3, GMP_RNDN); /* 2*(log(z/2) + gamma)*Jn(z) */
err2 += 4; /* the error on s2 is bounded by 2^err2 ulps, see
algorithms.tex */
/* add all three sums */
err1 += MPFR_EXP(s1); /* the error on s1 is bounded by 2^err1 */
err2 += MPFR_EXP(s2); /* the error on s2 is bounded by 2^err2 */
mpfr_sub (s2, s2, s1, GMP_RNDN); /* s2 - (s1+s3) */
err2 = (err1 > err2) ? err1 + 1 : err2 + 1;
err2 -= MPFR_EXP(s2);
err2 = (err2 >= 0) ? err2 + 1 : 1;
/* now the error on s2 is bounded by 2^err2*ulp(s2) */
mpfr_const_pi (y, GMP_RNDN); /* error bounded by 1 ulp */
mpfr_div (s2, s2, y, GMP_RNDN); /* error bounded by
2^(err2+1)*ulp(s2) */
err2 ++;
if (MPFR_LIKELY (MPFR_CAN_ROUND (s2, prec - err2, MPFR_PREC(res), r)))
break;
MPFR_ZIV_NEXT (loop, prec);
}
MPFR_ZIV_FREE (loop);
inex = (n >= 0 || (n & 1) == 0)
? mpfr_set (res, s2, r)
: mpfr_neg (res, s2, r);
mpfr_clear (y);
mpfr_clear (s1);
mpfr_clear (s2);
mpfr_clear (s3);
}
return inex;
}
#define MPFR_YN
#include "jyn_asympt.c"