started review of rootofunity.c

1 files changed, 110 insertions, 59 deletions

diff --git a/src/rootofunity.c b/src/rootofunity.c
index e4f533e..45d7cab 100644
--- a/src/rootofunity.c
+++ b/src/rootofunity.c
@@ -21,8 +21,8 @@ along with this program. If not, see http://www.gnu.org/licenses/ .
 #include "mpc-impl.h"
 #include <assert.h>
 
-static unsigned long int
-gcd (unsigned long int a, unsigned long int b)
+static unsigned long
+gcd (unsigned long a, unsigned long b)
 {
    if (b == 0)
       return a;
@@ -31,9 +31,9 @@ gcd (unsigned long int a, unsigned long int b)
 
 /* put in rop the value of exp(2*i*pi*k/n) rounded according to rnd */
 int
-mpc_rootofunity (mpc_ptr rop, unsigned long int n, unsigned long int k, mpc_rnd_t rnd)
+mpc_rootofunity (mpc_ptr rop, unsigned long n, unsigned long k, mpc_rnd_t rnd)
 {
-   unsigned long int g;
+   unsigned long g;
    mpq_t kn;
    mpfr_t t, s, c;
    mpfr_prec_t prec;
@@ -53,68 +53,106 @@ mpc_rootofunity (mpc_ptr rop, unsigned long int n, unsigned long int k, mpc_rnd_
    k /= g;
    n /= g;
 
+   /* Now 0 <= k < n and gcd(k,n)=1. */
+
    /* We assume that only n=1, 2, 3, 4, 6 and 12 may yield exact results
       and treat them separately; n=8 is also treated here for efficiency
       reasons. */
    if (n == 1)
-      return mpc_set_ui_ui (rop, 1, 0, rnd);
+     {
+       /* necessarily k=0 thus we want exp(0)=1 */
+       MPC_ASSERT (k == 0);
+       return mpc_set_ui_ui (rop, 1, 0, rnd);
+     }
    else if (n == 2)
-      return mpc_set_si_si (rop, -1, 0, rnd);
+     {
+       /* since gcd(k,n)=1, necessarily k=1, thus we want exp(i*pi)=-1 */
+       MPC_ASSERT (k == 1);
+       return mpc_set_si_si (rop, -1, 0, rnd);
+     }
    else if (n == 4)
-      if (k == 1)
+     {
+       /* since gcd(k,n)=1, necessarily k=1 or k=3, thus we want
+          exp(2*i*pi/4)=i or exp(2*i*pi*3/4)=-i */
+       MPC_ASSERT (k == 1 || k == 3);
+       if (k == 1)
          return mpc_set_ui_ui (rop, 0, 1, rnd);
-      else
+       else
          return mpc_set_si_si (rop, 0, -1, rnd);
-   else if (n == 3 || n == 6) {
-      inex_re = mpfr_set_si (mpc_realref (rop), (n == 3 ? -1 : 1),
-                             MPC_RND_RE (rnd));
-      /* inverse the rounding mode for -sqrt(3)/2 for zeta_3^2 and zeta_6^5 */
-      rnd_im = MPC_RND_IM (rnd);
-      if (k != 1)
+     }
+   else if (n == 3 || n == 6)
+     {
+       MPC_ASSERT ((n == 3 && (k == 1 || k == 2)) ||
+                   (n == 6 && (k == 1 || k == 5)));
+       /* for n=3, necessarily k=1 or k=2: -1/2+/-1/2*sqrt(3)*i;
+          for n=6, necessarily k=1 or k=5: 1/2+/-1/2*sqrt(3)*i */
+       inex_re = mpfr_set_si (mpc_realref (rop), (n == 3 ? -1 : 1),
+                              MPC_RND_RE (rnd));
+       /* inverse the rounding mode for -sqrt(3)/2 for zeta_3^2 and zeta_6^5 */
+       rnd_im = MPC_RND_IM (rnd);
+       if (k != 1)
          rnd_im = INV_RND (rnd_im);
-      inex_im = mpfr_sqrt_ui (mpc_imagref (rop), 3, rnd_im);
-      mpc_div_2ui (rop, rop, 1, MPC_RNDNN);
-      if (k != 1) {
-         mpfr_neg (mpc_imagref (rop), mpc_imagref (rop), MPFR_RNDN);
-         inex_im = -inex_im;
-      }
-      return MPC_INEX (inex_re, inex_im);
-   }
-   else if (n == 12) {
-      /* inverse the rounding mode for -sqrt(3)/2 for zeta_12^5 and zeta_12^7 */
-      rnd_re = MPC_RND_RE (rnd);
-      if (k == 5 || k == 7)
+       inex_im = mpfr_sqrt_ui (mpc_imagref (rop), 3, rnd_im);
+       mpc_div_2ui (rop, rop, 1, MPC_RNDNN);
+       if (k != 1)
+         {
+           mpfr_neg (mpc_imagref (rop), mpc_imagref (rop), MPFR_RNDN);
+           inex_im = -inex_im;
+         }
+       return MPC_INEX (inex_re, inex_im);
+     }
+   else if (n == 12)
+     {
+       /* necessarily k=1, 5, 7, 11:
+          k=1: 1/2*sqrt(3) + 1/2*I
+          k=5: -1/2*sqrt(3) + 1/2*I
+          k=7: -1/2*sqrt(3) - 1/2*I
+          k=11: 1/2*sqrt(3) - 1/2*I */
+       MPC_ASSERT (k == 1 || k == 5 || k == 7 || k == 11);
+       /* inverse the rounding mode for -sqrt(3)/2 for zeta_12^5 and zeta_12^7 */
+       rnd_re = MPC_RND_RE (rnd);
+       if (k == 5 || k == 7)
          rnd_re = INV_RND (rnd_re);
-      inex_re = mpfr_sqrt_ui (mpc_realref (rop), 3, rnd_re);
-      inex_im = mpfr_set_si (mpc_imagref (rop), (k == 1 || k == 5 ? 1 : -1),
-         MPC_RND_IM (rnd));
-      mpc_div_2ui (rop, rop, 1u, MPC_RNDNN);
-      if (k == 5 || k == 7) {
-         mpfr_neg (mpc_realref (rop), mpc_realref (rop), MPFR_RNDN);
-         inex_re = -inex_re;
-      }
-      return MPC_INEX (inex_re, inex_im);
-   }
-   else if (n == 8) {
-      rnd_re = MPC_RND_RE (rnd);
-      if (k == 3 || k == 5)
+       inex_re = mpfr_sqrt_ui (mpc_realref (rop), 3, rnd_re);
+       inex_im = mpfr_set_si (mpc_imagref (rop), k < 6 ? 1 : -1,
+                              MPC_RND_IM (rnd));
+       mpc_div_2ui (rop, rop, 1, MPC_RNDNN);
+       if (k == 5 || k == 7)
+         {
+           mpfr_neg (mpc_realref (rop), mpc_realref (rop), MPFR_RNDN);
+           inex_re = -inex_re;
+         }
+       return MPC_INEX (inex_re, inex_im);
+     }
+   else if (n == 8)
+     {
+       /* k=1, 3, 5 or 7:
+          k=1: (1/2*I + 1/2)*sqrt(2)
+          k=3: (1/2*I - 1/2)*sqrt(2)
+          k=5: -(1/2*I + 1/2)*sqrt(2)
+          k=7: -(1/2*I - 1/2)*sqrt(2) */
+       MPC_ASSERT (k == 1 || k == 3 || k == 5 || k == 7);
+       rnd_re = MPC_RND_RE (rnd);
+       if (k == 3 || k == 5)
          rnd_re = INV_RND (rnd_re);
-      rnd_im = MPC_RND_IM (rnd);
-      if (k == 5 || k == 7)
+       rnd_im = MPC_RND_IM (rnd);
+       if (k > 4)
          rnd_im = INV_RND (rnd_im);
-      inex_re = mpfr_sqrt_ui (mpc_realref (rop), 2, rnd_re);
-      inex_im = mpfr_sqrt_ui (mpc_imagref (rop), 2, rnd_im);
-      mpc_div_2ui (rop, rop, 1u, MPC_RNDNN);
-      if (k == 3 || k == 5) {
-         mpfr_neg (mpc_realref (rop), mpc_realref (rop), MPFR_RNDN);
-         inex_re = -inex_re;
-      }
-      if (k == 5 || k == 7) {
-         mpfr_neg (mpc_imagref (rop), mpc_imagref (rop), MPFR_RNDN);
-         inex_im = -inex_im;
-      }
-      return MPC_INEX (inex_re, inex_im);
-   }
+       inex_re = mpfr_sqrt_ui (mpc_realref (rop), 2, rnd_re);
+       inex_im = mpfr_sqrt_ui (mpc_imagref (rop), 2, rnd_im);
+       mpc_div_2ui (rop, rop, 1, MPC_RNDNN);
+       if (k == 3 || k == 5)
+         {
+           mpfr_neg (mpc_realref (rop), mpc_realref (rop), MPFR_RNDN);
+           inex_re = -inex_re;
+         }
+       if (k > 4)
+         {
+           mpfr_neg (mpc_imagref (rop), mpc_imagref (rop), MPFR_RNDN);
+           inex_im = -inex_im;
+         }
+       return MPC_INEX (inex_re, inex_im);
+     }
 
    prec = MPC_MAX_PREC(rop);
 
@@ -123,7 +161,7 @@ mpc_rootofunity (mpc_ptr rop, unsigned long int n, unsigned long int k, mpc_rnd_
    mpfr_init2 (c, 67);
    mpq_init (kn);
    mpq_set_ui (kn, k, n);
-   mpq_mul_2exp (kn, kn, 1);
+   mpq_mul_2exp (kn, kn, 1); /* kn=2*k/n < 2 */
 
    do {
       prec += mpc_ceil_log2 (prec) + 5;
@@ -132,18 +170,31 @@ mpc_rootofunity (mpc_ptr rop, unsigned long int n, unsigned long int k, mpc_rnd_
       mpfr_set_prec (s, prec);
       mpfr_set_prec (c, prec);
 
-      mpfr_const_pi (t, MPFR_RNDN); /* error 0.5 ulp */
-      mpfr_mul_q (t, t, kn, MPFR_RNDN); /* error 2*1.5+0.5=3.5 ulp */
+      mpfr_const_pi (t, MPFR_RNDN); /* error <= 0.5 ulp but since
+                                       ulp(t)=2^(2-prec), the absolute error
+                                       is bounded by 2^(1-prec), and the
+                                       relative error is bounded by
+                                       2^(1-prec)/pi <= 0.64*2^(-prec) */
+      mpfr_mul_q (t, t, kn, MPFR_RNDN); /* error <= 1.15 ulp(t) */
+      /* Indeed, the error is bounded by 0.64*2^(-prec)*pi*kn + 0.5 ulp [1].
+         Applying 2^(-prec)*|x| <= ulp(x) to x=pi*kn, we get a bound of:
+         0.64*ulp(pi*kn)+0.5 ulp.
+         Now since ulp(pi*kn) <= 2*ulp(t), we get: |t-pi*kn| <= 1.78*ulp(t).
+         If we plug this into [1] we get:
+         0.64*2^(-prec)*t + 0.64*2^(-prec)*1.78*ulp(t) + 0.5 ulp(t)
+         <= 0.64*ulp(t) + 0.64*2^(-prec)*1.78*ulp(t) + 0.5 ulp(t)
+         <= 1.15*ulp(t) for prec >= 7. */
+
       mpfr_sin_cos (s, c, t, MPFR_RNDN);
          /* error (1.5*2^{Exp (t) - Exp (s resp. c)} + 0.5) ulp
             We have 0<t<2*pi, so Exp (t) <= 3.
             Unfortunately s or c can be close to 0, but with n<2^64,
             we lose at most about 64 bits:
             Where the minimum of s and c over all primitive n-th roots of
-            unity is reached depends on n mod 4. 
+            unity is reached depends on n mod 4.
             To simplify the argument, we will consider the 4*n-th roots of
             unity, which contain the n-th roots of unity and which are
-            symmmetrically distributed with respect to the real and imaginary
+            symmetrically distributed with respect to the real and imaginary
             axes, so that it becomes enough to consider only s for k=1.
             With n<2^64, the absolute values of all s or c are at least
             sin (2 * pi * 2^(-64) / 4) > 2^(-64) of exponent at least -63.