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Diffstat (limited to 'Build/source/libs/mpfr/mpfr-src/src/jyn_asympt.c')
-rw-r--r-- | Build/source/libs/mpfr/mpfr-src/src/jyn_asympt.c | 289 |
1 files changed, 0 insertions, 289 deletions
diff --git a/Build/source/libs/mpfr/mpfr-src/src/jyn_asympt.c b/Build/source/libs/mpfr/mpfr-src/src/jyn_asympt.c deleted file mode 100644 index 9fa603246f1..00000000000 --- a/Build/source/libs/mpfr/mpfr-src/src/jyn_asympt.c +++ /dev/null @@ -1,289 +0,0 @@ -/* mpfr_jn_asympt, mpfr_yn_asympt -- shared code for mpfr_jn and mpfr_yn - -Copyright 2007-2020 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 -https://www.gnu.org/licenses/ or 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). - Note: for MPFR_RNDF, it returns 0 if the expansion failed, and a non-zero - value otherwise (with no other meaning). -*/ -static int -FUNCTION (mpfr_ptr res, long n, mpfr_srcptr z, mpfr_rnd_t r) -{ - mpfr_t s, c, P, Q, t, iz, err_t, err_s, err_u; - mpfr_prec_t w; - long k; - int inex, stop, diverge = 0; - mpfr_exp_t err2, err; - MPFR_ZIV_DECL (loop); - - mpfr_init2 (c, 64); - - /* The terms of the asymptotic expansion grow like mu^(2k)/(8z)^(2k), where - mu = 4n^2, thus we need mu < 8|z| so that it converges, - i.e., n^2/2 < |z| */ - MPFR_ASSERTD (n >= 0); - mpfr_set_ui (c, n, MPFR_RNDU); - mpfr_mul_ui (c, c, n, MPFR_RNDU); - mpfr_div_2ui (c, c, 1, MPFR_RNDU); - if (mpfr_cmpabs (c, z) >= 0) - { - mpfr_clear (c); - return 0; /* asymptotic expansion failed */ - } - - 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, MPFR_RNDN); - if (MPFR_IS_NEG(z)) - mpfr_neg (s, s, MPFR_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, MPFR_RNDN); - mpfr_sub (c, s, c, MPFR_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, MPFR_RNDN); /* err <= 1 */ - mpfr_div_2ui (iz, iz, 3, MPFR_RNDN); - - /* compute P and Q */ - mpfr_set_ui (P, 1, MPFR_RNDN); - mpfr_set_ui (Q, 0, MPFR_RNDN); - mpfr_set_ui (t, 1, MPFR_RNDN); /* current term */ - mpfr_set_ui (err_t, 0, MPFR_RNDN); /* error on t */ - mpfr_set_ui (err_s, 0, MPFR_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_LOG_MSG (("loop (k,stop) = (%ld,%d)\n", k, stop)); - mpfr_mul_si (t, t, 2 * (n + k) - 1, MPFR_RNDN); /* err <= err_k + 1 */ - mpfr_mul_si (t, t, 2 * (n - k) + 1, MPFR_RNDN); /* err <= err_k + 2 */ - mpfr_div_ui (t, t, k, MPFR_RNDN); /* err <= err_k + 3 */ - mpfr_mul (t, t, iz, MPFR_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) ? MPFR_RNDU : MPFR_RNDD); - mpfr_abs (err_t, err_t, MPFR_RNDN); /* exact */ - /* the absolute error on t is bounded by err_t * 2^(-w) */ - mpfr_abs (err_u, t, MPFR_RNDU); - mpfr_mul_2ui (err_u, err_u, w, MPFR_RNDU); /* t * 2^w */ - mpfr_add (err_u, err_u, err_t, MPFR_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, MPFR_RNDU); - if (MPFR_IS_POS(t)) - mpfr_add (err_s, err_s, t, MPFR_RNDU); - else - mpfr_sub (err_s, err_s, t, MPFR_RNDU); - mpfr_mul_2ui (err_s, err_s, w, MPFR_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, MPFR_RNDN); - else - mpfr_sub (Q, Q, t, MPFR_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_GET_EXP (err_u) <= MPFR_GET_EXP (Q)) - stop ++; - else - stop = 0; - } - else - { - /* if k = 0 mod 4, add, otherwise subtract */ - if ((k & 2) == 0) - mpfr_add (P, P, t, MPFR_RNDN); - else - mpfr_sub (P, P, t, MPFR_RNDN); - /* check if the next term is smaller than ulp(P) */ - if (MPFR_GET_EXP (err_u) <= MPFR_GET_EXP (P)) - stop ++; - else - stop = 0; - } - mpfr_add (err_s, err_s, err_t, MPFR_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, MPFR_RNDN); /* Q * (sin - cos) */ - mpfr_mul (s, s, P, MPFR_RNDN); /* P * (sin + cos) */ -#else - mpfr_mul (c, c, P, MPFR_RNDN); /* P * (sin - cos) */ - mpfr_mul (s, s, Q, MPFR_RNDN); /* Q * (sin + cos) */ -#endif - err = MPFR_GET_EXP (c); - if (MPFR_GET_EXP (s) > err) - err = MPFR_EXP (s); -#ifdef MPFR_JN - mpfr_sub (s, s, c, MPFR_RNDN); -#else - mpfr_add (s, s, c, MPFR_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, MPFR_RNDN); /* P * (sin - cos) */ - mpfr_mul (s, s, Q, MPFR_RNDN); /* Q * (sin + cos) */ -#else - mpfr_mul (c, c, Q, MPFR_RNDN); /* Q * (sin - cos) */ - mpfr_mul (s, s, P, MPFR_RNDN); /* P * (sin + cos) */ -#endif - err = MPFR_GET_EXP (c); - if (MPFR_GET_EXP (s) > err) - err = MPFR_EXP (s); -#ifdef MPFR_JN - mpfr_add (s, s, c, MPFR_RNDN); -#else - mpfr_sub (s, c, s, MPFR_RNDN); -#endif - } - if ((n & 2) != 0) - mpfr_neg (s, s, MPFR_RNDN); - if (MPFR_GET_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_GET_EXP (P) >= MPFR_GET_EXP (Q)) - ? MPFR_EXP (P) + 2 : MPFR_EXP (Q) + 2; - /* (|P| + |Q|) * 2^(1 - w) <= 2^(err2 - w) */ - err = MPFR_GET_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, MPFR_RNDN); /* Pi, err <= 1 */ - mpfr_mul (c, c, z, MPFR_RNDN); /* err <= 2 */ - mpfr_si_div (c, MPFR_IS_POS(z) ? 1 : -1, c, MPFR_RNDN); /* err <= 3 */ - mpfr_sqrt (c, c, MPFR_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) ? MPFR_RNDU : MPFR_RNDD); - mpfr_abs (err_t, err_t, MPFR_RNDU); - mpfr_mul_ui (err_t, err_t, 3, MPFR_RNDU); - /* 3*2^(-w)*|old_c|*|s| [see below] is bounded by err_t * 2^(-w) */ - err2 += MPFR_GET_EXP (c); - /* |old_c| * 2^(err2 - w) [see below] is bounded by 2^(err2-w) */ - mpfr_mul (c, c, s, MPFR_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_GET_EXP (err_t) > MPFR_GET_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_GET_EXP (c); - if (MPFR_LIKELY (MPFR_CAN_ROUND (c, w - err, MPFR_PREC(res), r))) - break; - if (diverge != 0) - { - MPFR_ZIV_FREE (loop); - mpfr_clear (c); - return 0; /* means that the asymptotic expansion failed */ - } - 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); - - /* for RNDF, mpfr_set or mpfr_neg may return 0, but if we return 0, it - would mean the asymptotic expansion failed, thus we return 1 instead */ - return (r != MPFR_RNDF) ? inex : 1; -} |