--- /dev/null
+/* mpfr_jn_asympt, mpfr_yn_asympt -- shared code for mpfr_jn and mpfr_yn
+
+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. */
+
+#ifdef MPFR_JN
+# define FUNCTION mpfr_jn_asympt
+#else
+# ifdef MPFR_YN
+# define FUNCTION mpfr_yn_asympt
+# else
+# error "neither MPFR_JN nor MPFR_YN is defined"
+# endif
+#endif
+
+/* Implements asymptotic expansion for jn or yn (formulae 9.2.5 and 9.2.6
+ from Abramowitz & Stegun).
+ Assumes |z| > p log(2)/2, where p is the target precision
+ (z can be negative only for jn).
+ Return 0 if the expansion does not converge enough (the value 0 as inexact
+ flag should not happen for normal input).
+*/
+static int
+FUNCTION (mpfr_ptr res, long n, mpfr_srcptr z, mp_rnd_t r)
+{
+ mpfr_t s, c, P, Q, t, iz, err_t, err_s, err_u;
+ mp_prec_t w;
+ long k;
+ int inex, stop, diverge = 0;
+ mp_exp_t err2, err;
+ MPFR_ZIV_DECL (loop);
+
+ mpfr_init (c);
+
+ w = MPFR_PREC(res) + MPFR_INT_CEIL_LOG2(MPFR_PREC(res)) + 4;
+
+ MPFR_ZIV_INIT (loop, w);
+ for (;;)
+ {
+ mpfr_set_prec (c, w);
+ mpfr_init2 (s, w);
+ mpfr_init2 (P, w);
+ mpfr_init2 (Q, w);
+ mpfr_init2 (t, w);
+ mpfr_init2 (iz, w);
+ mpfr_init2 (err_t, 31);
+ mpfr_init2 (err_s, 31);
+ mpfr_init2 (err_u, 31);
+
+ /* Approximate sin(z) and cos(z). In the following, err <= k means that
+ the approximate value y and the true value x are related by
+ y = x * (1 + u)^k with |u| <= 2^(-w), following Higham's method. */
+ mpfr_sin_cos (s, c, z, GMP_RNDN);
+ if (MPFR_IS_NEG(z))
+ mpfr_neg (s, s, GMP_RNDN); /* compute jn/yn(|z|), fix sign later */
+ /* The absolute error on s/c is bounded by 1/2 ulp(1/2) <= 2^(-w-1). */
+ mpfr_add (t, s, c, GMP_RNDN);
+ mpfr_sub (c, s, c, GMP_RNDN);
+ mpfr_swap (s, t);
+ /* now s approximates sin(z)+cos(z), and c approximates sin(z)-cos(z),
+ with total absolute error bounded by 2^(1-w). */
+
+ /* precompute 1/(8|z|) */
+ mpfr_si_div (iz, MPFR_IS_POS(z) ? 1 : -1, z, GMP_RNDN); /* err <= 1 */
+ mpfr_div_2ui (iz, iz, 3, GMP_RNDN);
+
+ /* compute P and Q */
+ mpfr_set_ui (P, 1, GMP_RNDN);
+ mpfr_set_ui (Q, 0, GMP_RNDN);
+ mpfr_set_ui (t, 1, GMP_RNDN); /* current term */
+ mpfr_set_ui (err_t, 0, GMP_RNDN); /* error on t */
+ mpfr_set_ui (err_s, 0, GMP_RNDN); /* error on P and Q (sum of errors) */
+ for (k = 1, stop = 0; stop < 4; k++)
+ {
+ /* compute next term: t(k)/t(k-1) = (2n+2k-1)(2n-2k+1)/(8kz) */
+ mpfr_mul_si (t, t, 2 * (n + k) - 1, GMP_RNDN); /* err <= err_k + 1 */
+ mpfr_mul_si (t, t, 2 * (n - k) + 1, GMP_RNDN); /* err <= err_k + 2 */
+ mpfr_div_ui (t, t, k, GMP_RNDN); /* err <= err_k + 3 */
+ mpfr_mul (t, t, iz, GMP_RNDN); /* err <= err_k + 5 */
+ /* the relative error on t is bounded by (1+u)^(5k)-1, which is
+ bounded by 6ku for 6ku <= 0.02: first |5 log(1+u)| <= |5.5u|
+ for |u| <= 0.15, then |exp(5.5u)-1| <= 6u for |u| <= 0.02. */
+ mpfr_mul_ui (err_t, t, 6 * k, MPFR_IS_POS(t) ? GMP_RNDU : GMP_RNDD);
+ mpfr_abs (err_t, err_t, GMP_RNDN); /* exact */
+ /* the absolute error on t is bounded by err_t * 2^(-w) */
+ mpfr_abs (err_u, t, GMP_RNDU);
+ mpfr_mul_2ui (err_u, err_u, w, GMP_RNDU); /* t * 2^w */
+ mpfr_add (err_u, err_u, err_t, GMP_RNDU); /* max|t| * 2^w */
+ if (stop >= 2)
+ {
+ /* take into account the neglected terms: t * 2^w */
+ mpfr_div_2ui (err_s, err_s, w, GMP_RNDU);
+ if (MPFR_IS_POS(t))
+ mpfr_add (err_s, err_s, t, GMP_RNDU);
+ else
+ mpfr_sub (err_s, err_s, t, GMP_RNDU);
+ mpfr_mul_2ui (err_s, err_s, w, GMP_RNDU);
+ stop ++;
+ }
+ /* if k is odd, add to Q, otherwise to P */
+ else if (k & 1)
+ {
+ /* if k = 1 mod 4, add, otherwise subtract */
+ if ((k & 2) == 0)
+ mpfr_add (Q, Q, t, GMP_RNDN);
+ else
+ mpfr_sub (Q, Q, t, GMP_RNDN);
+ /* check if the next term is smaller than ulp(Q): if EXP(err_u)
+ <= EXP(Q), since the current term is bounded by
+ err_u * 2^(-w), it is bounded by ulp(Q) */
+ if (MPFR_EXP(err_u) <= MPFR_EXP(Q))
+ stop ++;
+ else
+ stop = 0;
+ }
+ else
+ {
+ /* if k = 0 mod 4, add, otherwise subtract */
+ if ((k & 2) == 0)
+ mpfr_add (P, P, t, GMP_RNDN);
+ else
+ mpfr_sub (P, P, t, GMP_RNDN);
+ /* check if the next term is smaller than ulp(P) */
+ if (MPFR_EXP(err_u) <= MPFR_EXP(P))
+ stop ++;
+ else
+ stop = 0;
+ }
+ mpfr_add (err_s, err_s, err_t, GMP_RNDU);
+ /* the sum of the rounding errors on P and Q is bounded by
+ err_s * 2^(-w) */
+
+ /* stop when start to diverge */
+ if (stop < 2 &&
+ ((MPFR_IS_POS(z) && mpfr_cmp_ui (z, (k + 1) / 2) < 0) ||
+ (MPFR_IS_NEG(z) && mpfr_cmp_si (z, - ((k + 1) / 2)) > 0)))
+ {
+ /* if we have to stop the series because it diverges, then
+ increasing the precision will most probably fail, since
+ we will stop to the same point, and thus compute a very
+ similar approximation */
+ diverge = 1;
+ stop = 2; /* force stop */
+ }
+ }
+ /* the sum of the total errors on P and Q is bounded by err_s * 2^(-w) */
+
+ /* Now combine: the sum of the rounding errors on P and Q is bounded by
+ err_s * 2^(-w), and the absolute error on s/c is bounded by 2^(1-w) */
+ if ((n & 1) == 0) /* n even: P * (sin + cos) + Q (cos - sin) for jn
+ Q * (sin + cos) + P (sin - cos) for yn */
+ {
+#ifdef MPFR_JN
+ mpfr_mul (c, c, Q, GMP_RNDN); /* Q * (sin - cos) */
+ mpfr_mul (s, s, P, GMP_RNDN); /* P * (sin + cos) */
+#else
+ mpfr_mul (c, c, P, GMP_RNDN); /* P * (sin - cos) */
+ mpfr_mul (s, s, Q, GMP_RNDN); /* Q * (sin + cos) */
+#endif
+ err = MPFR_EXP(c);
+ if (MPFR_EXP(s) > err)
+ err = MPFR_EXP(s);
+#ifdef MPFR_JN
+ mpfr_sub (s, s, c, GMP_RNDN);
+#else
+ mpfr_add (s, s, c, GMP_RNDN);
+#endif
+ }
+ else /* n odd: P * (sin - cos) + Q (cos + sin) for jn,
+ Q * (sin - cos) - P (cos + sin) for yn */
+ {
+#ifdef MPFR_JN
+ mpfr_mul (c, c, P, GMP_RNDN); /* P * (sin - cos) */
+ mpfr_mul (s, s, Q, GMP_RNDN); /* Q * (sin + cos) */
+#else
+ mpfr_mul (c, c, Q, GMP_RNDN); /* Q * (sin - cos) */
+ mpfr_mul (s, s, P, GMP_RNDN); /* P * (sin + cos) */
+#endif
+ err = MPFR_EXP(c);
+ if (MPFR_EXP(s) > err)
+ err = MPFR_EXP(s);
+#ifdef MPFR_JN
+ mpfr_add (s, s, c, GMP_RNDN);
+#else
+ mpfr_sub (s, c, s, GMP_RNDN);
+#endif
+ }
+ if ((n & 2) != 0)
+ mpfr_neg (s, s, GMP_RNDN);
+ if (MPFR_EXP(s) > err)
+ err = MPFR_EXP(s);
+ /* the absolute error on s is bounded by P*err(s/c) + Q*err(s/c)
+ + err(P)*(s/c) + err(Q)*(s/c) + 3 * 2^(err - w - 1)
+ <= (|P|+|Q|) * 2^(1-w) + err_s * 2^(1-w) + 2^err * 2^(1-w),
+ since |c|, |old_s| <= 2. */
+ err2 = (MPFR_EXP(P) >= MPFR_EXP(Q)) ? MPFR_EXP(P) + 2 : MPFR_EXP(Q) + 2;
+ /* (|P| + |Q|) * 2^(1 - w) <= 2^(err2 - w) */
+ err = MPFR_EXP(err_s) >= err ? MPFR_EXP(err_s) + 2 : err + 2;
+ /* err_s * 2^(1-w) + 2^old_err * 2^(1-w) <= 2^err * 2^(-w) */
+ err2 = (err >= err2) ? err + 1 : err2 + 1;
+ /* now the absolute error on s is bounded by 2^(err2 - w) */
+
+ /* multiply by sqrt(1/(Pi*z)) */
+ mpfr_const_pi (c, GMP_RNDN); /* Pi, err <= 1 */
+ mpfr_mul (c, c, z, GMP_RNDN); /* err <= 2 */
+ mpfr_si_div (c, MPFR_IS_POS(z) ? 1 : -1, c, GMP_RNDN); /* err <= 3 */
+ mpfr_sqrt (c, c, GMP_RNDN); /* err<=5/2, thus the absolute error is
+ bounded by 3*u*|c| for |u| <= 0.25 */
+ mpfr_mul (err_t, c, s, MPFR_SIGN(c)==MPFR_SIGN(s) ? GMP_RNDU : GMP_RNDD);
+ mpfr_abs (err_t, err_t, GMP_RNDU);
+ mpfr_mul_ui (err_t, err_t, 3, GMP_RNDU);
+ /* 3*2^(-w)*|old_c|*|s| [see below] is bounded by err_t * 2^(-w) */
+ err2 += MPFR_EXP(c);
+ /* |old_c| * 2^(err2 - w) [see below] is bounded by 2^(err2-w) */
+ mpfr_mul (c, c, s, GMP_RNDN); /* the absolute error on c is bounded by
+ 1/2 ulp(c) + 3*2^(-w)*|old_c|*|s|
+ + |old_c| * 2^(err2 - w) */
+ /* compute err_t * 2^(-w) + 1/2 ulp(c) = (err_t + 2^EXP(c)) * 2^(-w) */
+ err = (MPFR_EXP(err_t) > MPFR_EXP(c)) ? MPFR_EXP(err_t) + 1 : MPFR_EXP(c) + 1;
+ /* err_t * 2^(-w) + 1/2 ulp(c) <= 2^(err - w) */
+ /* now err_t * 2^(-w) bounds 1/2 ulp(c) + 3*2^(-w)*|old_c|*|s| */
+ err = (err >= err2) ? err + 1 : err2 + 1;
+ /* the absolute error on c is bounded by 2^(err - w) */
+
+ mpfr_clear (s);
+ mpfr_clear (P);
+ mpfr_clear (Q);
+ mpfr_clear (t);
+ mpfr_clear (iz);
+ mpfr_clear (err_t);
+ mpfr_clear (err_s);
+ mpfr_clear (err_u);
+
+ err -= MPFR_EXP(c);
+ if (MPFR_LIKELY (MPFR_CAN_ROUND (c, w - err, MPFR_PREC(res), r)))
+ break;
+ if (diverge != 0)
+ {
+ mpfr_set (c, z, r); /* will force inex=0 below, which means the
+ asymptotic expansion failed */
+ break;
+ }
+ MPFR_ZIV_NEXT (loop, w);
+ }
+ MPFR_ZIV_FREE (loop);
+
+ inex = (MPFR_IS_POS(z) || ((n & 1) == 0)) ? mpfr_set (res, c, r)
+ : mpfr_neg (res, c, r);
+ mpfr_clear (c);
+
+ return inex;
+}
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