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try to speed up nzeros

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This commit is contained in:
p15-git-acc 2019-02-20 13:30:32 -06:00
parent 1faec8c283
commit eb9aa858ab
1 changed files with 160 additions and 66 deletions
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226
acb_dirichlet/isolate_hardy_z_zero.c
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@ -1166,50 +1166,67 @@ _acb_set_arf(acb_t res, const arf_t t)
arb_set_arf(acb_realref(res), t);
}
void
_acb_dirichlet_exact_zeta_nzeros(fmpz_t res, const arf_t t)
/*
* Find the index of the largest Gram point less than t.
* Requires t > 10.
*/
static void
gram_index(fmpz_t res, const arf_t t)
{
if (arf_cmp_si(t, 10) < 0)
{
flint_printf("gram_index requires t > 10\n");
flint_abort();
}
else
{
acb_t z;
slong prec = arf_abs_bound_lt_2exp_si(t) + 8;
acb_init(z);
while (1)
{
_acb_set_arf(z, t);
acb_dirichlet_hardy_theta(z, z, NULL, NULL, 1, prec);
arb_const_pi(acb_imagref(z), prec);
arb_div(acb_realref(z), acb_realref(z), acb_imagref(z), prec);
arb_floor(acb_realref(z), acb_realref(z), prec);
if (arb_get_unique_fmpz(res, acb_realref(z)))
{
break;
}
prec *= 2;
}
}
}
/*
* Compute nzeros(t) for up to len values of t
* and return the number computed. p must be in increasing order,
* and all entries must be greater than 10.
*/
static slong
_exact_zeta_multi_nzeros(fmpz *res, arf_srcptr points, slong len)
{
zz_node_ptr U, V, u, v, p;
arb_t pi, x;
acb_t z;
fmpz_t n;
int s; /* sign of Z(t) */
slong prec = arf_bits(t) + 8;
arb_t x;
fmpz_t n, N;
slong i;
arf_srcptr t;
int s;
slong prec;
if (arf_cmp_si(t, 14) < 0)
{
fmpz_zero(res);
return;
}
fmpz_init(n);
arb_init(pi);
arb_init(x);
acb_init(z);
fmpz_init(n);
fmpz_init(N);
/*
* t is located between two Gram points. Find the unique n such that the
* nth zero is also predicted to be located between these Gram points.
* The first point is located between two Gram points. Find the unique n
* such that the nth zero is also predicted to be located between these
* Gram points.
*/
while (1)
{
arb_const_pi(pi, prec);
_acb_set_arf(z, t);
acb_dirichlet_hardy_theta(z, z, NULL, NULL, 1, prec);
acb_get_real(x, z);
arb_div(x, x, pi, prec);
arb_floor(x, x, prec);
if (arb_get_unique_fmpz(n, x))
{
break;
}
prec *= 2;
}
gram_index(n, points);
fmpz_add_ui(n, n, 2);
/* Compute the sign of Z(t). */
s = _acb_dirichlet_definite_hardy_z(x, t, &prec);
/* Get a list of points that fully separate zeros of Z. */
if (fmpz_cmp_si(n, GRAMS_LAW_MAX) <= 0)
{
@ -1238,32 +1255,63 @@ _acb_dirichlet_exact_zeta_nzeros(fmpz_t res, const arf_t t)
flint_printf("U and V must be good Gram points\n");
flint_abort();
}
if (U == V)
{
flint_printf("the list must include at least one interval\n");
flint_abort();
}
/* Find the position of t relative to points in the list. */
p = U;
fmpz_add_ui(res, U->gram, 1);
while (p != V)
fmpz_add_ui(N, U->gram, 1);
/*
* It's possible that one or more points are located between
* the first Gram point and the arf_t representative of that Gram point.
*/
for (i = 0, t = points; i < len && arf_cmp(t, &p->t) <= 0; i++, t++)
{
fmpz_set(res + i, N);
}
while (i < len && p != V)
{
if (!p->next)
{
flint_printf("prematurely reached end of list\n");
flint_printf("prematurely reached the end of the list\n");
flint_abort();
}
if (zz_node_sgn(p) != zz_node_sgn(p->next))
{
if (arf_cmp(t, &p->next->t) <= 0)
while (i < len && arf_cmp(t, &p->next->t) <= 0)
{
prec = arf_abs_bound_lt_2exp_si(t) + 8;
s = _acb_dirichlet_definite_hardy_z(x, t, &prec);
if (zz_node_sgn(p->next) == s)
{
fmpz_add_ui(res, res, 1);
fmpz_add_ui(res + i, N, 1);
}
break;
else
{
fmpz_set(res + i, N);
}
t++;
i++;
}
fmpz_add_ui(res, res, 1);
fmpz_add_ui(N, N, 1);
}
p = p->next;
}
/*
* It's possible that the first point in the array is located between
* the last Gram point and the arf_t representative of that Gram point.
*/
if (i == 0)
{
fmpz_set(res, N);
i++;
}
while (u)
{
v = u;
@ -1272,10 +1320,54 @@ _acb_dirichlet_exact_zeta_nzeros(fmpz_t res, const arf_t t)
flint_free(v);
}
fmpz_clear(n);
arb_clear(pi);
arb_clear(x);
acb_clear(z);
fmpz_clear(n);
return i;
}
/*
* Compute nzeros for len values of t. The array p must be in increasing order.
*/
static void
exact_zeta_multi_nzeros(fmpz *res, arf_srcptr p, slong len)
{
slong i, c;
arf_srcptr q;
if (len <= 0)
{
return;
}
for (i = 0, q = p; i < len - 1; i++, q++)
{
if (arf_cmp(q, p) > 0)
{
flint_printf("p must be in increasing order\n");
flint_abort();
}
}
c = 0;
while (c < len)
{
if (arf_cmp_si(p + c, 14) < 0)
{
fmpz_zero(res + c);
c++;
}
else
{
c += _exact_zeta_multi_nzeros(res + c, p + c, len - c);
}
}
}
void
_acb_dirichlet_exact_zeta_nzeros(fmpz_t res, const arf_t t)
{
exact_zeta_multi_nzeros(res, t, 1);
}
void
@ -1322,32 +1414,34 @@ _arb_set_interval_fmpz(arb_t res, const fmpz_t a, const fmpz_t b)
void
acb_dirichlet_zeta_nzeros(arb_t res, const arb_t t, slong prec)
{
fmpz_t a, b;
arf_t lb, ub;
fmpz_init(a);
fmpz_init(b);
arf_init(lb);
arf_init(ub);
if (arb_is_exact(t))
{
_acb_dirichlet_exact_zeta_nzeros(a, arb_midref(t));
arb_set_fmpz(res, a);
fmpz_t n;
fmpz_init(n);
_acb_dirichlet_exact_zeta_nzeros(n, arb_midref(t));
arb_set_fmpz(res, n);
fmpz_clear(n);
}
else
{
arb_get_lbound_arf(lb, t, prec);
arb_get_ubound_arf(ub, t, prec);
_acb_dirichlet_exact_zeta_nzeros(a, lb);
_acb_dirichlet_exact_zeta_nzeros(b, ub);
_arb_set_interval_fmpz(res, a, b);
arf_struct b[2];
fmpz n[2];
arf_init(b);
arf_init(b + 1);
fmpz_init(n);
fmpz_init(n + 1);
arb_get_lbound_arf(b, t, prec);
arb_get_ubound_arf(b + 1, t, prec);
exact_zeta_multi_nzeros(n, b, 2);
_arb_set_interval_fmpz(res, n, n + 1);
arf_clear(b);
arf_clear(b + 1);
fmpz_clear(n);
fmpz_clear(n + 1);
}
arb_set_round(res, res, prec);
fmpz_clear(a);
fmpz_clear(b);
arf_clear(lb);
arf_clear(ub);
}