new function lngamma

git-svn-id: svn://scm.gforge.inria.fr/svn/mpfr/trunk@3754 280ebfd0-de03-0410-8827-d642c229c3f4

1 files changed, 453 insertions, 0 deletions

diff --git a/lngamma.c b/lngamma.c
new file mode 100644
index 000000000..e97e8f872
--- /dev/null
+++ b/lngamma.c
@@ -0,0 +1,453 @@
+/* mpfr_lngamma -- lngamma function
+
+Copyright 2005 Free Software Foundation.
+
+This file is part of the MPFR Library.
+
+The 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 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 MPFR Library; see the file COPYING.LIB.  If not, write to
+the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
+MA 02111-1307, USA. */
+
+#include <stdio.h>
+#include <stdlib.h>
+#include "mpfr-impl.h"
+
+/* assuming b[0]...b[2(n-1)] are computed, computes and stores B[2n]*(2n+1)!
+
+   t/(exp(t)-1) = sum(B[j]*t^j/j!, j=0..infinity)
+   thus t = (exp(t)-1) * sum(B[j]*t^j/j!, n=0..infinity).
+   Taking the coefficient of degree n+1 > 1, we get:
+   0 = sum(1/(n+1-k)!*B[k]/k!, k=0..n)
+   which gives:
+   B[n] = -sum(binomial(n+1,k)*B[k], k=0..n-1)/(n+1).
+   
+   Let C[n] = B[n]*(n+1)!.
+   Then C[n] = -sum(binomial(n+1,k)*C[k]*n!/(k+1)!,  k=0..n-1),
+   which proves that the C[n] are integers.
+*/
+static mpz_t*
+bernoulli (mpz_t *b, unsigned long n)
+{
+  if (n == 0)
+    {
+      b = (mpz_t*) malloc (sizeof (mpz_t));
+      mpz_init_set_ui (b[0], 1);
+    }
+  else
+    {
+      mpz_t t;
+      unsigned long k;
+
+      b = (mpz_t*) realloc (b, (n + 1) * sizeof (mpz_t));
+      mpz_init (b[n]);
+      /* b[n] = -sum(binomial(2n+1,2k)*C[k]*(2n)!/(2k+1)!,  k=0..n-1) */
+      mpz_init_set_ui (t, 2 * n + 1);
+      mpz_mul_ui (t, t, 2 * n - 1);
+      mpz_mul_ui (t, t, 2 * n);
+      mpz_mul_ui (t, t, n);
+      mpz_div_ui (t, t, 3); /* exact: t=binomial(2*n+1,2*k)*(2*n)!/(2*k+1)!
+                               for k=n-1 */
+      mpz_mul (b[n], t, b[n-1]);
+      for (k = n - 1; k-- > 0;)
+        {
+          mpz_mul_ui (t, t, 2 * k + 1);
+          mpz_mul_ui (t, t, 2 * k + 2);
+          mpz_mul_ui (t, t, 2 * k + 2);
+          mpz_mul_ui (t, t, 2 * k + 3);
+          mpz_div_ui (t, t, 2 * (n - k) + 1);
+          mpz_div_ui (t, t, 2 * (n - k));
+          mpz_addmul (b[n], t, b[k]);
+        }
+      /* take into account C[1] */
+      mpz_mul_ui (t, t, 2 * n + 1);
+      mpz_div_2exp (t, t, 1);
+      mpz_sub (b[n], b[n], t);
+      mpz_neg (b[n], b[n]);
+      mpz_clear (t);
+    }
+  return b;
+}
+
+/* given a precision p, return alpha, such that the argument reduction
+   will use k = alpha*p*log(2).
+
+   Warning: we should always have alpha >= log(2)/(2Pi) ~ 0.11,
+   and the smallest value of alpha multiplied by the smallest working
+   precision should be >= 4.
+*/
+static double
+mpfr_gamma_alpha (mp_prec_t p)
+{
+  if (p <= 100)
+    return 0.6;
+  else if (p <= 200)
+    return 0.8;
+  else if (p <= 500)
+    return 0.8;
+  else if (p <= 1000)
+    return 1.3;
+  else if (p <= 2000)
+    return 1.7;
+  else if (p <= 5000)
+    return 2.2;
+  else if (p <= 10000)
+    return 3.4;
+  else /* heuristic fit from above */
+    return 0.26 * (double) __gmpfr_ceil_log2 ((double) p);
+}
+
+/* lngamma(x) = log(gamma(x)).
+   We use formula [6.1.40] from Abramowitz&Stegun:
+   lngamma(z) = (z-1/2)*log(z) - z + 1/2*log(2*Pi)
+              + sum (Bernoulli[2n]/(2m)/(2m-1)/z^(2m-1),m=1..infinity)
+   According to [6.1.42], if the sum is truncated after m=n, the error
+   R_n(z) is bounded by |B[2n+2]|*K(z)/(2n+1)/(2n+2)/|z|^(2n+1)
+   where K(z) = max (z^2/(u^2+z^2)) for u >= 0.
+   For z real, |K(z)| <= 1 thus R_n(z) is bounded by the first neglected term.
+ */
+#ifdef IS_GAMMA
+static int mpfr_gamma_aux
+#else
+int mpfr_lngamma
+#endif
+(mpfr_ptr y, mpfr_srcptr z0, mp_rnd_t rnd)
+{
+  mp_prec_t precy, w; /* working precision */
+  mpfr_t s, t, u, v, z;
+  unsigned long m, k, maxm;
+  mpz_t *B;
+  int inexact, ok, compared;
+  mp_exp_t err_s, err_t;
+  unsigned long Bm = 0; /* number of allocated B[] */
+  double d;
+  MPFR_SAVE_EXPO_DECL (expo);
+
+#ifndef IS_GAMMA
+  /* special cases */
+  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (z0)))
+    {
+      if (MPFR_IS_NAN (z0) || MPFR_IS_NEG (z0))
+        {
+          MPFR_SET_NAN (y);
+          MPFR_RET_NAN;
+        }
+      else /* lngamma(+Inf) = lngamma(+0) = +Inf */
+        {
+          MPFR_SET_INF (y);
+          MPFR_SET_POS (y);
+          MPFR_RET (0);  /* exact */
+        }
+    }
+
+  /* if x < 0 and -2k-1 <= x <= -2k, then lngamma(x) = NaN */
+  if (MPFR_IS_NEG(z0) && ((mpfr_get_si (z0, GMP_RNDZ) % 2) == 0
+                          || mpfr_integer_p (z0)))
+    {
+      MPFR_SET_NAN (y);
+      MPFR_RET_NAN;
+    }
+#endif
+
+  precy = MPFR_PREC(y);
+
+  compared = mpfr_cmp_ui (z0, 1);
+
+#ifndef IS_GAMMA
+  if (compared == 0) /* lngamma(1) = +0 */
+    return mpfr_set_ui (y, 0, GMP_RNDN);
+#endif
+
+  mpfr_init2 (s, MPFR_PREC_MIN);
+  mpfr_init2 (t, MPFR_PREC_MIN);
+  mpfr_init2 (u, MPFR_PREC_MIN);
+  mpfr_init2 (v, MPFR_PREC_MIN);
+  mpfr_init2 (z, MPFR_PREC_MIN);
+
+  MPFR_SAVE_EXPO_MARK (expo);
+
+  if (compared < 0)
+    {
+      mp_exp_t err_u;
+
+      /* use reflection formula:
+         gamma(x) = Pi*(x-1)/sin(Pi*(2-x))/gamma(2-x)
+         thus lngamma(x) = log(Pi*(x-1)/sin(Pi*(2-x))) - lngamma(2-x) */
+
+      w = precy + __gmpfr_ceil_log2 ((double) precy);
+      do
+        {
+          w += __gmpfr_ceil_log2 ((double) w) + 13;
+          MPFR_ASSERTD(w >= 3);
+          mpfr_set_prec (s, w);
+          mpfr_set_prec (t, w);
+          mpfr_set_prec (u, w);
+          mpfr_set_prec (v, w);
+          mpfr_ui_sub (s, 2, z0, GMP_RNDD); /* s = (2-z0) * (1+2u) */
+          mpfr_const_pi (t, GMP_RNDN);      /* t = Pi * (1+u) */
+          mpfr_lngamma (u, s, GMP_RNDN); /* lngamma(2-x) */
+          /* Let s = (2-z0) + h. By construction, -(2-z0)*(2u) <= h <= 0.
+             We have lngamma(s) = lngamma(2-z0) + h*Psi(z), z in [2-z0+h,2-z0].
+             Since 2-z0+h >= 1 and |Psi(x)| <= max(1,log(x)) for x >= 1,
+             the error on u is bounded by
+             ulp(u)/2 + (2-z0)*max(1,log(2-z0))*2^(1-w). */
+          d = (double) MPFR_EXP(s) * 0.694; /* upper bound for log(2-z0) */
+          err_u = MPFR_EXP(s) + __gmpfr_ceil_log2 (d) + 1 - MPFR_EXP(u);
+          err_u = (err_u >= 0) ? err_u + 1 : 0;
+          /* now the error on u is bounded by 2^err_u ulps */
+
+          mpfr_mul (s, s, t, GMP_RNDN); /* Pi*(2-x), (1+u)^4 */
+          err_s = MPFR_EXP(s); /* 2-x <= 2^err_s */
+          mpfr_sin (s, s, GMP_RNDN); /* sin(Pi*(2-x)) */
+          /* the error on s is bounded by 1/2*ulp(s) + [(1+u)^4-1]*(2-x)
+             <= 1/2*ulp(s) + 5*2^(-w)*(2-x) for w >= 3 */
+          err_s += 3 - MPFR_EXP(s);
+          err_s = (err_s >= 0) ? err_s + 1 : 0;
+          /* the error on s is bounded by 2^err_s ulps, thus the relative
+             error is bounded by 2^(err_s+1) */
+          err_s ++; /* relative error */
+
+          mpfr_sub_ui (v, z0, 1, GMP_RNDN); /* v = (x-1) * (1+u) */
+          mpfr_mul (v, v, t, GMP_RNDN); /* v = Pi*(x-1) * (1+u)^3 */
+          mpfr_div (v, v, s, GMP_RNDN); /* Pi*(x-1)/sin(Pi*(2-x)) */
+          /* (1+u)^(4+2^err_s+1) */
+          err_s = (err_s <= 2) ? 3 + (err_s / 2) : err_s + 1;
+          MPFR_ASSERTN(MPFR_IS_POS(v));
+          mpfr_log (v, v, GMP_RNDN);
+          /* log(v*(1+e)) = log(v)+log(1+e) where |e| <= 2^(err_s-w).
+             Since |log(1+e)| <= 2*e for |e| <= 1/4, the error on v is
+             bounded by ulp(v)/2 + 2^(err_s+1-w). */
+	  if (err_s + 2 > w)
+	    {
+	      w += err_s + 2;
+	      ok = 0;
+	    }
+	  else
+	    {
+	      err_s += 1 - MPFR_EXP(v);
+	      err_s = (err_s >= 0) ? err_s + 1 : 0;
+	      /* the error on v is bounded by 2^err_s ulps */
+	      err_u += MPFR_EXP(u); /* absolute error on u */
+	      err_s += MPFR_EXP(v); /* absolute error on v */
+	      mpfr_sub (s, v, u, GMP_RNDN);
+	      /* the total error on s is bounded by ulp(s)/2 + 2^(err_u-w)
+		 + 2^(err_s-w) <= ulp(s)/2 + 2^(max(err_u,err_s)+1-w) */
+	      err_s = (err_s >= err_u) ? err_s : err_u;
+	      err_s += 1 - MPFR_EXP(s); /* error is 2^err_s ulp(s) */
+	      err_s = (err_s >= 0) ? err_s + 1 : 0;
+	      ok = mpfr_can_round (s, w - err_s, GMP_RNDN, GMP_RNDZ, precy
+				   + (rnd == GMP_RNDN));
+	    }
+        }
+      while (ok == 0);
+      goto end;
+    }
+
+  /* now z0 > 1 */
+
+  /* since k is O(w), the value of log(z0*...*(z0+k-1)) is about w*log(w),
+     so there is a cancellation of ~log(w) in the argument reconstruction */
+  w = precy + __gmpfr_ceil_log2 ((double) precy);
+
+  do
+    {
+      w += __gmpfr_ceil_log2 ((double) w) + 13;
+      MPFR_ASSERTD (w >= 3);
+
+      mpfr_set_prec (s, 53);
+      /* we need z >= w*log(2)/(2*Pi) to get an absolute error less than 2^(-w)
+         but the optimal value is about 0.155665*w */
+      mpfr_set_d (s, mpfr_gamma_alpha (precy) * (double) w, GMP_RNDU);
+      if (mpfr_cmp (z0, s) < 0)
+        {
+          mpfr_sub (s, s, z0, GMP_RNDU);
+          k = mpfr_get_ui (s, GMP_RNDU);
+          if (k < 3)
+            k = 3;
+        }
+      else
+        k = 3;
+
+  mpfr_set_prec (s, w);
+  mpfr_set_prec (t, w);
+  mpfr_set_prec (u, w);
+  mpfr_set_prec (v, w);
+  mpfr_set_prec (z, w);
+
+  mpfr_add_ui (z, z0, k, GMP_RNDN); /* z = (z0+k)*(1+t1) with |t1| <= 2^(-w) */
+
+  /* z >= 4 ensures the relative error on log(z) is small,
+     and also (z-1/2)*log(z)-z >= 0 */
+  MPFR_ASSERTD (mpfr_cmp_ui (z, 4) >= 0);
+
+  mpfr_log (s, z, GMP_RNDN); /* log(z) */
+  /* we have s = log((z0+k)*(1+t1))*(1+t2) with |t1|, |t2| <= 2^(-w).
+     Since w >= 2 and z0+k >= 4, we can write log((z0+k)*(1+t1))
+     = log(z0+k) * (1+t3) with |t3| <= 2^(-w), thus we have
+     s = log(z0+k) * (1+t4)^2 with |t4| <= 2^(-w) */
+  mpfr_mul_2exp (t, z, 1, GMP_RNDN); /* t = 2z * (1+t5) */
+  mpfr_sub_ui (t, t, 1, GMP_RNDN); /* t = 2z-1 * (1+t6)^3
+     since we can write 2z*(1+t5) = (2z-1)*(1+t5') with
+     t5' = 2z/(2z-1) * t5, thus |t5'| <= 8/7 * t5 */
+  mpfr_mul (s, s, t, GMP_RNDN); /* (2z-1)*log(z) * (1+t7)^6 */
+  mpfr_div_2exp (s, s, 1, GMP_RNDN); /* (z-1/2)*log(z) * (1+t7)^6 */
+  mpfr_sub (s, s, z, GMP_RNDN); /* (z-1/2)*log(z)-z */
+  /* s = [(z-1/2)*log(z)-z]*(1+u)^14, s >= 1/2 */
+
+  mpfr_ui_div (u, 1, z, GMP_RNDN); /* 1/z * (1+u), u <= 1/4 since z >= 4 */
+
+  /* the first term is B[2]/2/z = 1/12/z: t=1/12/z, C[2]=1 */
+  mpfr_div_ui (t, u, 12, GMP_RNDN); /* 1/(12z) * (1+u)^2, t <= 3/128 */
+  mpfr_set (v, t, GMP_RNDN);        /* (1+u)^2, v < 2^(-5) */
+  mpfr_add (s, s, v, GMP_RNDN);     /* (1+u)^15 */
+
+  mpfr_mul (u, u, u, GMP_RNDN); /* 1/z^2 * (1+u)^3 */
+
+  if (Bm == 0)
+    {
+      B = bernoulli (NULL, 0);
+      B = bernoulli (B, 1);
+      Bm = 2;
+    }
+
+  /* m <= maxm ensures that 2*m*(2*m+1) <= ULONG_MAX */
+  maxm = 1UL << (BITS_PER_MP_LIMB / 2 - 1);
+
+  /* s:(1+u)^15, t:(1+u)^2, t <= 3/128 */
+  
+  for (m = 2; MPFR_EXP(v) + (mp_exp_t) w >= MPFR_EXP(s); m++)
+    {
+      mpfr_mul (t, t, u, GMP_RNDN); /* (1+u)^(10m-14) */
+      if (m <= maxm)
+        {
+          mpfr_mul_ui (t, t, 2*(m-1)*(2*m-3), GMP_RNDN);
+          mpfr_div_ui (t, t, 2*m*(2*m-1), GMP_RNDN);
+          mpfr_div_ui (t, t, 2*m*(2*m+1), GMP_RNDN);
+        }
+      else
+        {
+          mpfr_mul_ui (t, t, 2*(m-1), GMP_RNDN);
+          mpfr_mul_ui (t, t, 2*m-3, GMP_RNDN);
+          mpfr_div_ui (t, t, 2*m, GMP_RNDN);
+          mpfr_div_ui (t, t, 2*m-1, GMP_RNDN);
+          mpfr_div_ui (t, t, 2*m, GMP_RNDN);
+          mpfr_div_ui (t, t, 2*m+1, GMP_RNDN);
+        }
+      /* (1+u)^(10m-8) */
+      /* invariant: t=1/(2m)/(2m-1)/z^(2m-1)/(2m+1)! */
+      if (Bm <= m)
+	{
+	  B = bernoulli (B, m); /* B[2m]*(2m+1)!, exact */
+	  Bm ++;
+	}
+      mpfr_mul_z (v, t, B[m], GMP_RNDN); /* (1+u)^(10m-7) */
+      MPFR_ASSERTN(MPFR_EXP(v) <= - (2 * m + 3));
+      mpfr_add (s, s, v, GMP_RNDN);
+    }
+  /* m <= 1/2*Pi*e*z ensures that |v[m]| < 1/2^(2m+3) */
+  MPFR_ASSERTN ((double) m <= 4.26 * mpfr_get_d (z, GMP_RNDZ));
+
+  /* We have sum([(1+u)^(10m-7)-1]*1/2^(2m+3), m=2..infinity)
+     <= 1.46*u for u <= 2^(-3).
+     We have 0 < lngamma(z) - [(z - 1/2) ln(z) - z + 1/2 ln(2 Pi)] < 0.021
+     for z >= 4, thus since the initial s >= 0.85, the different values of
+     s differ by at most one binade, and the total rounding error on s 
+     in the for-loop is bounded by 2*(m-1)*ulp(final_s).
+     The error coming from the v's is bounded by 1.46*2^(-w) <= 2*ulp(final_s).
+     Thus the total error so far is bounded by [(1+u)^15-1]*s+2m*ulp(s)
+     <= (2m+47)*ulp(s).
+     Taking into account the truncation error (which is bounded by the last
+     term v[] according to 6.1.42 in A&S), the bound is (2m+48)*ulp(s).
+  */
+
+  /* add 1/2*log(2*Pi) and subtract log(z0*(z0+1)*...*(z0+k-1)) */
+  mpfr_const_pi (v, GMP_RNDN); /* v = Pi*(1+u) */
+  mpfr_mul_2exp (v, v, 1, GMP_RNDN); /* v = 2*Pi * (1+u) */
+  if (k)
+    {
+      unsigned long l;
+      mpfr_set (t, z0, GMP_RNDN); /* t = z0*(1+u) */
+      for (l = 1; l < k; l++)
+	{
+	  mpfr_add_ui (u, z0, l, GMP_RNDN); /* u = (z0+l)*(1+u) */
+	  mpfr_mul (t, t, u, GMP_RNDN);     /* (1+u)^(2l+1) */
+	}
+      /* now t: (1+u)^(2k-1) */
+      /* instead of computing log(sqrt(2*Pi)/t), we compute
+         1/2*log(2*Pi/t^2), which trades a square root for a square */
+      mpfr_mul (t, t, t, GMP_RNDN); /* (z0*...*(z0+k-1))^2, (1+u)^(4k-1) */
+      mpfr_div (v, v, t, GMP_RNDN); /* 2*Pi/(z0*...*(z0+k-1))^2 (1+u)^(4k+1) */
+    }
+#ifdef IS_GAMMA
+  err_s = MPFR_EXP(s);
+  mpfr_exp (s, s, GMP_RNDN);
+  /* before the exponential, we have s = s0 + h where |h| <= (2m+48)*ulp(s),
+     thus exp(s0) = exp(s) * exp(-h).
+     For |h| <= 1/4, we have |exp(h)-1| <= 1.2*|h| thus
+     |exp(s) - exp(s0)| <= 1.2 * exp(s) * (2m+48)* 2^(EXP(s)-w). */
+  d = 1.2 * (2.0 * (double) m + 48.0);
+  /* the error on s is bounded by d*2^err_s * 2^(-w) */
+  mpfr_sqrt (t, v, GMP_RNDN);
+  /* let v0 be the exact value of v. We have v = v0*(1+u)^(4k+1),
+     thus t = sqrt(v0)*(1+u)^(2k+3/2). */
+  mpfr_mul (s, s, t, GMP_RNDN);
+  /* the error on input s is bounded by (1+u)^(d*2^err_s),
+     and that on t is (1+u)^(2k+3/2), thus the
+     total error is (1+u)^(d*2^err_s+2k+5/2) */
+  err_s += __gmpfr_ceil_log2 (d);
+  err_t = __gmpfr_ceil_log2 (2.0 * (double) k + 2.5);
+  err_s = (err_s >= err_t) ? err_s + 1 : err_t + 1;
+#else
+  mpfr_log (t, v, GMP_RNDN);
+  /* let v0 be the exact value of v. We have v = v0*(1+u)^(4k+1),
+     thus log(v) = log(v0) + (4k+1)*log(1+u). Since |log(1+u)/u| <= 1.07
+     for |u| <= 2^(-3), the absolute error on log(v) is bounded by 
+     1.07*(4k+1)*u, and the rounding error by ulp(t). */
+  mpfr_div_2exp (t, t, 1, GMP_RNDN);
+  /* the error on t is now bounded by ulp(t) + 0.54*(4k+1)*2^(-w).
+     We have sqrt(2*Pi)/(z0*(z0+1)*...*(z0+k-1)) <= sqrt(2*Pi)/k! <= 0.5
+     since k>=3, thus t <= -0.5 and ulp(t) >= 2^(-w).
+     Thus the error on t is bounded by (2.16*k+1.54)*ulp(t). */
+  err_t = MPFR_EXP(t) + (mp_exp_t) __gmpfr_ceil_log2 (2.2 * (double) k + 1.6);
+  err_s = MPFR_EXP(s) + (mp_exp_t) __gmpfr_ceil_log2 (2.0 * (double) m + 48.0);
+  mpfr_add (s, s, t, GMP_RNDN); /* this is a subtraction in fact */
+  /* the final error in ulp(s) is <= 1 + 2^(err_t-EXP(s)) + 2^(err_s-EXP(s))
+     <= 2^(1+max(err_t,err_s)-EXP(s)) if err_t <> err_s
+     <= 2^(2+max(err_t,err_s)-EXP(s)) if err_t = err_s */
+  err_s = (err_t == err_s) ? 1 + err_s : ((err_t > err_s) ? err_t : err_s);
+  err_s += 1 - MPFR_EXP(s);
+#endif
+
+  ok = mpfr_can_round (s, w - err_s, GMP_RNDN, GMP_RNDZ, precy
+                       + (rnd == GMP_RNDN));
+    }
+  while (ok == 0);
+
+ end:
+  inexact = mpfr_set (y, s, rnd);
+
+  if (compared > 0)
+    {
+      while (Bm--)
+	mpz_clear (B[Bm]);
+      free (B);
+    }
+  mpfr_clear (s);
+  mpfr_clear (t);
+  mpfr_clear (u);
+  mpfr_clear (v);
+  mpfr_clear (z);
+
+  MPFR_SAVE_EXPO_FREE (expo);
+  return mpfr_check_range (y, inexact, rnd);
+}