diff options
Diffstat (limited to 'src/yn.c')
| -rw-r--r-- | src/yn.c | 426 |
1 files changed, 426 insertions, 0 deletions
diff --git a/src/yn.c b/src/yn.c new file mode 100644 index 0000000..24d3e17 --- /dev/null +++ b/src/yn.c @@ -0,0 +1,426 @@ +/* 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, 2010, 2011, 2012, 2013 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. */ + +#define MPFR_NEED_LONGLONG_H +#include "mpfr-impl.h" + +static int mpfr_yn_asympt (mpfr_ptr, long, mpfr_srcptr, mpfr_rnd_t); + +int +mpfr_y0 (mpfr_ptr res, mpfr_srcptr z, mpfr_rnd_t r) +{ + return mpfr_yn (res, 0, z, r); +} + +int +mpfr_y1 (mpfr_ptr res, mpfr_srcptr z, mpfr_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 mpfr_exp_t +mpfr_yn_s1 (mpfr_ptr s, mpfr_srcptr y, unsigned long n) +{ + unsigned long k; + mpz_t f; + mpfr_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, MPFR_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, MPFR_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, MPFR_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, MPFR_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 mpfr_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) */ + mpfr_exp_t exps, expU; + + zz = mpfr_get_ui (y, MPFR_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, MPFR_RNDN); + mpfr_div (t, c, t, MPFR_RNDN); /* c/n! */ + mpfr_mul_z (u, t, p, MPFR_RNDN); + mpfr_div_z (s, u, q, MPFR_RNDN); + exps = MPFR_EXP (s); + expU = exps; + for (k = 1; ;k ++) + { + /* update t */ + mpfr_mul (t, t, y, MPFR_RNDN); + mpfr_div_ui (t, t, k, MPFR_RNDN); + mpfr_div_ui (t, t, n + k, MPFR_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, MPFR_RNDN); + mpfr_div_z (u, u, q, MPFR_RNDN); + exps = MPFR_EXP (u); + if (exps > expU) + expU = exps; + mpfr_add (s, s, u, MPFR_RNDN); + exps = MPFR_EXP (s); + if (exps > expU) + expU = exps; + if (MPFR_EXP (u) + (mpfr_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, mpfr_rnd_t r) +{ + int inex; + unsigned long absn; + MPFR_SAVE_EXPO_DECL (expo); + + MPFR_LOG_FUNC + (("n=%ld x[%Pu]=%.*Rg rnd=%d", n, mpfr_get_prec (z), mpfr_log_prec, z, r), + ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (res), mpfr_log_prec, res, inex)); + + 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 || ((unsigned long) n & 1) == 0) + MPFR_SET_NEG(res); + else + MPFR_SET_POS(res); + mpfr_set_divby0 (); + 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 */ + + MPFR_SAVE_EXPO_MARK (expo); + + /* 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) < - (mpfr_exp_t) (MPFR_PREC(res) / 2)) + { + mpfr_t l, h, t, logz; + mpfr_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, MPFR_RNDD); /* lower bound of log(z) */ + mpfr_set (h, logz, MPFR_RNDU); /* exact */ + mpfr_nextabove (h); /* upper bound of log(z) */ + mpfr_const_euler (t, MPFR_RNDD); /* lower bound of euler */ + mpfr_add (l, logz, t, MPFR_RNDD); /* lower bound of log(z) + euler */ + mpfr_nextabove (t); /* upper bound of euler */ + mpfr_add (h, h, t, MPFR_RNDU); /* upper bound of log(z) + euler */ + mpfr_const_log2 (t, MPFR_RNDU); /* upper bound of log(2) */ + mpfr_sub (l, l, t, MPFR_RNDD); /* lower bound of log(z/2) + euler */ + mpfr_nextbelow (t); /* lower bound of log(2) */ + mpfr_sub (h, h, t, MPFR_RNDU); /* upper bound of log(z/2) + euler */ + mpfr_const_pi (t, MPFR_RNDU); /* upper bound of Pi */ + mpfr_div (l, l, t, MPFR_RNDD); /* lower bound of (log(z/2)+euler)/Pi */ + mpfr_nextbelow (t); /* lower bound of Pi */ + mpfr_div (h, h, t, MPFR_RNDD); /* upper bound of (log(z/2)+euler)/Pi */ + mpfr_mul_2ui (l, l, 1, MPFR_RNDD); /* lower bound on g(z)*log(z) */ + mpfr_mul_2ui (h, h, 1, MPFR_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, MPFR_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, MPFR_RNDD); + mpfr_div_2ui (t, t, 1, MPFR_RNDD); + mpfr_mul (t, t, logz, MPFR_RNDU); /* upper bound on z^2/2*log(z) */ + mpfr_add (h, h, t, MPFR_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_equal_p (l, h); + if (ok) + mpfr_set (res, h, r); /* exact */ + mpfr_clear (l); + mpfr_clear (h); + mpfr_clear (t); + mpfr_clear (logz); + if (ok) + goto end; + } + + /* 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 < - (mpfr_exp_t) MPFR_PREC(res)) + { + mpfr_t y; + mpfr_prec_t prec; + mpfr_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, MPFR_RNDU); /* Pi*(1+u)^2, where here and below u + represents a quantity <= 1/2^prec */ + mpfr_mul (y, y, z, MPFR_RNDU); /* Pi*z * (1+u)^4, upper bound */ + MPFR_BLOCK (flags, mpfr_ui_div (y, 2, y, MPFR_RNDZ)); + /* 2/Pi/z * (1+u)^6, lower bound, with possible overflow */ + if (MPFR_OVERFLOW (flags)) + { + mpfr_clear (y); + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_overflow (res, r, -1); + } + mpfr_neg (y, y, MPFR_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 = (mpfr_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) + goto end; + } + + /* 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) + goto end; + } + + /* General case */ + { + mpfr_prec_t prec; + mpfr_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, MPFR_RNDN); + mpfr_div_2ui (y, y, 2, MPFR_RNDN); /* z^2/4 */ + + /* store (z/2)^n temporarily in s2 */ + mpfr_pow_ui (s2, z, absn, MPFR_RNDN); + mpfr_div_2si (s2, s2, absn, MPFR_RNDN); + + /* compute S1 * (z/2)^(-n) */ + if (n == 0) + { + mpfr_set_ui (s1, 0, MPFR_RNDN); + err1 = 0; + } + else + err1 = mpfr_yn_s1 (s1, y, absn - 1); + mpfr_div (s1, s1, s2, MPFR_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, MPFR_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, MPFR_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, MPFR_RNDN); /* z/2 */ + mpfr_log (s2, s2, MPFR_RNDN); /* log(z/2) */ + mpfr_const_euler (s3, MPFR_RNDN); + err2 = MPFR_EXP(s2) > MPFR_EXP(s3) ? MPFR_EXP(s2) : MPFR_EXP(s3); + mpfr_add (s2, s2, s3, MPFR_RNDN); /* log(z/2) + gamma */ + err2 -= MPFR_EXP(s2); + mpfr_mul_2ui (s2, s2, 1, MPFR_RNDN); /* 2*(log(z/2) + gamma) */ + mpfr_jn (s3, absn, z, MPFR_RNDN); /* Jn(z) */ + mpfr_mul (s2, s2, s3, MPFR_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, MPFR_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, MPFR_RNDN); /* error bounded by 1 ulp */ + mpfr_div (s2, s2, y, MPFR_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); + + /* Assume two's complement for the test n & 1 */ + inex = mpfr_set4 (res, s2, r, n >= 0 || (n & 1) == 0 ? + MPFR_SIGN (s2) : - MPFR_SIGN (s2)); + + mpfr_clear (y); + mpfr_clear (s1); + mpfr_clear (s2); + mpfr_clear (s3); + } + + end: + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_check_range (res, inex, r); +} + +#define MPFR_YN +#include "jyn_asympt.c" |