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Diffstat (limited to 'Build/source/libs/mpfr/mpfr-3.1.3/src/jyn_asympt.c')
-rw-r--r-- | Build/source/libs/mpfr/mpfr-3.1.3/src/jyn_asympt.c | 269 |
1 files changed, 269 insertions, 0 deletions
diff --git a/Build/source/libs/mpfr/mpfr-3.1.3/src/jyn_asympt.c b/Build/source/libs/mpfr/mpfr-3.1.3/src/jyn_asympt.c new file mode 100644 index 00000000000..b630c9bf2b7 --- /dev/null +++ b/Build/source/libs/mpfr/mpfr-3.1.3/src/jyn_asympt.c @@ -0,0 +1,269 @@ +/* mpfr_jn_asympt, mpfr_yn_asympt -- shared code for mpfr_jn and mpfr_yn + +Copyright 2007-2015 Free Software Foundation, Inc. +Contributed by the AriC and Caramel 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. */ + +#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, 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_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, 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_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_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, MPFR_RNDN); + else + mpfr_sub (P, P, t, MPFR_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, 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_EXP(c); + if (MPFR_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_EXP(c); + if (MPFR_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_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, 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_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_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; +} |