| /* 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; |
| } |
