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author | zimmerma <zimmerma@280ebfd0-de03-0410-8827-d642c229c3f4> | 2007-07-24 13:42:35 +0000 |
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committer | zimmerma <zimmerma@280ebfd0-de03-0410-8827-d642c229c3f4> | 2007-07-24 13:42:35 +0000 |
commit | 465f8c1db1fd6f73b0f5e17ef68a44e430e95023 (patch) | |
tree | bf22097e06725a500df3858612dc208ee5800eb4 | |
parent | aa2d3678e4f217941e5d1bd7c00168963bb3a811 (diff) | |
download | mpfr-465f8c1db1fd6f73b0f5e17ef68a44e430e95023.tar.gz |
implemented asymptotic expansion for large argument in j0/j1/jn
git-svn-id: svn://scm.gforge.inria.fr/svn/mpfr/trunk@4690 280ebfd0-de03-0410-8827-d642c229c3f4
-rw-r--r-- | algorithms.tex | 50 | ||||
-rw-r--r-- | jn.c | 227 | ||||
-rw-r--r-- | tests/data/j0 | 2 | ||||
-rw-r--r-- | tests/tj0.c | 1 | ||||
-rw-r--r-- | tests/tj1.c | 1 |
5 files changed, 280 insertions, 1 deletions
diff --git a/algorithms.tex b/algorithms.tex index bf0f702d4..ef191c6f4 100644 --- a/algorithms.tex +++ b/algorithms.tex @@ -3043,7 +3043,9 @@ is an exponent loss in the final subtraction $r = \circ(v_{n+1}-w)$. The Bessel function $J_n(z)$ of first kind and integer order $n$ is defined as follows \cite[Eq.~(9.1.10)]{AbSt73}: -\[ J_n(z) = (z/2)^n \sum_{k=0}^{\infty} \frac{(-z^2/4)^k}{k! (k+n)!}. \] +\begin{equation} \label{Jn_0} +J_n(z) = (z/2)^n \sum_{k=0}^{\infty} \frac{(-z^2/4)^k}{k! (k+n)!}. +\end{equation} It is real for all real $z$, tends to $0$ by oscillating around $0$ when $z \rightarrow \pm\infty$, and tends to $0$ when $z \rightarrow 0$, except $J_0$ which tends to $1$. @@ -3094,6 +3096,52 @@ Together with $n! \geq \sqrt{2 \pi n} (n/e)^n$, which follows from example from \cite[Eq.~6.1.38]{AbSt73}, this gives: \[ |J_n(z)| \leq \frac{1}{\sqrt{2 \pi n}} \left( \frac{ze}{2n} \right)^n. \] +\paragraph{Large argument.} + +For large argument $z$, formula (\ref{Jn_0}) requires at least +$k \approx z/2$ terms before starting to converge. If $k \leq z/2$, it is +better to use formula 9.2.5 from \cite{AbSt73}, which +provides at least $2$ bits per term: +\[ J_n(z) = \sqrt{\frac{2}{\pi z}} [P(n,z) \cos \chi - Q(n,z) \sin \chi], \] +where $\chi = z - (n/2 + 1/4) \pi$, and $P(n,z)$ and $Q(n,z)$ are two +diverging series: +\begin{small} +\[ P(n,z) \approx \sum_{k=0}^{\infty} (-1)^k \frac{\Gamma(1/2+n+2k) (2z)^{-2k}} + {(2k)! \Gamma(1/2+n-2k)}, \quad + Q(n,z) \approx \sum_{k=0}^{\infty} (-1)^k \frac{\Gamma(1/2+n+2k+1) (2z)^{-2k-1}} + {(2k+1)! \Gamma(1/2+n-2k-1)}. \] +\end{small} +% P:=proc(n,z) add((-1)^k*GAMMA(1/2+n+2*k)/(2*k)!/GAMMA(1/2+n-2*k)/ +% (2*z)^(2*k),k=0..4) end: +% Q:=proc(n,z) add((-1)^k*GAMMA(1/2+n+2*k+1)/(2*k+1)!/GAMMA(1/2+n-2*k-1)/ +% (2*z)^(2*k+1),k=0..4) end: +% J:=proc(n,z) sqrt(2/Pi/z)*(P(n,z)*cos(z-(n/2+1/4)*Pi)-Q(n,z)* +% sin(z-(n/2+1/4)*Pi)) end: +If $n$ is real and non-negative --- which is the case here ---, +the remainder of $P(n,z)$ after $k$ terms does not exceed the $(k+1)$th term +and is of the same sign, provided $k > n/2 - 1/4$; the same holds for +$Q(n,z)$ as long as $k > n/2 - 3/4$ \cite[9.2.10]{AbSt73}. + +If we first approximate $\chi = z - (n/2 + 1/4) \pi$ with working precision +$w$, and then approximate $\cos \chi$ and $\sin \chi$, there will be a huge +relative error if $z > 2^w$. Instead, we use the fact that for $n$ even, +\[ P(n,z) \cos \chi - Q(n,z) \sin \chi = \frac{1}{\sqrt{2}} (-1)^{n/2} + [P (\sin z + \cos z) + Q (\cos z - \sin z)], \] +and for $n$ odd, +\[ P(n,z) \cos \chi - Q(n,z) \sin \chi = \frac{1}{\sqrt{2}} (-1)^{(n-1)/2} + [P (\sin z - \cos z) + Q (\cos z + \sin z)], \] +where $\cos z$ and $\sin z$ are computed accurately with +\texttt{mpfr\_sin\_cos}, which uses in turn \texttt{mpfr\_remainder}. + +If we consider $P(n,z)$ and $Q(n,z)$ together as one single series, +its term of index $k$ +behaves like $\Gamma(1/2+n+k)/k!/\Gamma(1/2+n-k)/(2z)^k$. +The ratio between the +term of index $k+1$ and that of index $k$ is about $k/(2z)$, thus starts +to diverge when $k \approx 2z$. At that point, the $k$th term is +$\approx e^{-2z}$, thus if $z > p/2 \log 2$, we can use the asymptotic +expansion. + \subsubsection{Bessel function $Y_n(z)$ of second kind} Like $J_n(z)$, $Y_n(z)$ is a solution of the linear differential equation: @@ -26,6 +26,8 @@ MA 02110-1301, USA. */ /* Relations: j(-n,z) = (-1)^n j(n,z) */ +static int mpfr_jn_asympt (mpfr_ptr, long, mpfr_srcptr, mp_rnd_t); + int mpfr_j0 (mpfr_ptr res, mpfr_srcptr z, mp_rnd_t r) { @@ -94,6 +96,16 @@ mpfr_jn (mpfr_ptr res, long n, mpfr_srcptr z, mp_rnd_t r) MPFR_FAST_COMPUTE_IF_SMALL_INPUT (res, z, -2 * MPFR_GET_EXP (z), 3, 0, r, mpfr_div_2ui (res, res, 1, r)); + /* 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 || + mpfr_cmp_si (z, - ((long) MPFR_PREC(res) / 2 + 3)) < 0) + { + inex = mpfr_jn_asympt (res, n, z, r); + if (inex != 0) + return inex; + } + mpfr_init2 (y, 32); /* check underflow case: |j(n,z)| <= 1/sqrt(2 Pi n) (ze/2n)^n @@ -188,3 +200,218 @@ mpfr_jn (mpfr_ptr res, long n, mpfr_srcptr z, mp_rnd_t r) return inex; } + +/* Implements asymptotic expansion (formula 9.2.5 from Abramowitz & Stegun). + Assumes z > p log(2)/2, where p is the target precision. + Return 0 if the expansion does not converge enough (the value 0 as inexact + flag should not happen for normal input). +*/ +static int +mpfr_jn_asympt (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(|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_mul_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_div_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) */ + { + mpfr_mul (c, c, Q, GMP_RNDN); /* Q * (sin - cos) */ + err = MPFR_EXP(c); + mpfr_mul (s, s, P, GMP_RNDN); /* P * (sin + cos) */ + if (MPFR_EXP(s) > err) + err = MPFR_EXP(s); + mpfr_sub (s, s, c, GMP_RNDN); + } + else /* n odd: P * (sin - cos) + Q (cos + sin) */ + { + mpfr_mul (c, c, P, GMP_RNDN); /* P * (sin - cos) */ + err = MPFR_EXP(c); + mpfr_mul (s, s, Q, GMP_RNDN); /* Q * (sin + cos) */ + if (MPFR_EXP(s) > err) + err = MPFR_EXP(s); + mpfr_add (s, s, c, GMP_RNDN); + } + 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) ? mpfr_set (res, c, r) : mpfr_neg (res, c, r); + mpfr_clear (c); + + return inex; +} diff --git a/tests/data/j0 b/tests/data/j0 index 58454ece9..51d81882c 100644 --- a/tests/data/j0 +++ b/tests/data/j0 @@ -1065,3 +1065,5 @@ 53 53 n 0x1F06343D0971FBp-47 0x1F32FAADC9C90Fp-98 53 53 n 0x1F06343D09C6A7p-47 0x112F1357417BEAp-88 53 53 n 0x1F06343D0A11B5p-47 0x1031CA8324C806p-87 +# below are test cases not in Gonnet's testsuite +53 53 n 0x1p1000 0x1DE62DD27F8FBDp-554 diff --git a/tests/tj0.c b/tests/tj0.c index b7dea925d..5c820031f 100644 --- a/tests/tj0.c +++ b/tests/tj0.c @@ -27,6 +27,7 @@ MA 02110-1301, USA. */ #define TEST_FUNCTION mpfr_j0 #define RAND_FUNCTION(x) mpfr_random2(x, MPFR_LIMB_SIZE (x), 8) +#define REDUCE_EMAX 262143 /* otherwise arg. reduction is too expensive */ #include "tgeneric.c" int diff --git a/tests/tj1.c b/tests/tj1.c index d0b68c0be..8dd7b8da6 100644 --- a/tests/tj1.c +++ b/tests/tj1.c @@ -27,6 +27,7 @@ MA 02110-1301, USA. */ #define TEST_FUNCTION mpfr_j1 #define RAND_FUNCTION(x) mpfr_random2(x, MPFR_LIMB_SIZE (x), 8) +#define REDUCE_EMAX 262143 /* otherwise arg. reduction is too expensive */ #include "tgeneric.c" int |