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

1 files changed, 227 insertions, 0 deletions

diff --git a/jn.c b/jn.c
index 62075763c..26bdd8aad 100644
--- a/jn.c
+++ b/jn.c
@@ -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;
+}