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https://github.com/vale981/arb
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document and remove chi_vec_sieve function
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Pascal
parent
fe9116073f
commit
c37fc67884
4 changed files with 119 additions and 103 deletions
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@ -190,7 +190,6 @@ ulong acb_dirichlet_ui_chi(const acb_dirichlet_group_t G, const acb_dirichlet_ch
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void acb_dirichlet_ui_vec_set_null(ulong *v, const acb_dirichlet_group_t G, slong nv);
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void acb_dirichlet_ui_chi_vec_loop(ulong *v, const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi, slong nv);
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void acb_dirichlet_ui_chi_vec_primeloop(ulong *v, const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi, slong nv);
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void acb_dirichlet_ui_chi_vec_sieve(ulong *v, const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi, slong nv);
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void acb_dirichlet_ui_chi_vec(ulong *v, const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi, slong nv);
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/* precompute powers of a root of unity */
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@ -27,13 +27,13 @@
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#include "profiler.h"
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typedef void (*dir_f) (ulong *v, const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi, slong nv);
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void
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static void
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dir_empty(ulong *v, const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi, slong nv)
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{
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return;
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}
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void
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static void
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vecloop(dir_f dir, ulong minq, ulong maxq, ulong * rand, ulong nr, ulong * v, ulong nv)
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{
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ulong q;
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@ -62,6 +62,38 @@ vecloop(dir_f dir, ulong minq, ulong maxq, ulong * rand, ulong nr, ulong * v, ul
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flint_printf("\n");
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}
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static void
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acb_dirichlet_ui_chi_vec_sieve(ulong *v, const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi, slong nv)
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{
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slong k, p, pmax;
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n_primes_t iter;
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n_primes_init(iter);
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pmax = (nv < G->q) ? nv : G->q;
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v[1] = 0;
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while ((p = n_primes_next(iter)) < pmax)
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{
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if (G->q % p == 0)
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{
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for (k = p; k < nv; k += p)
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v[k] = ACB_DIRICHLET_CHI_NULL;
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}
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else
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{
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long chip;
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chip = acb_dirichlet_ui_chi(G, chi, p);
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for (k = p; k < nv; k += p)
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if (v[k] != -1)
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v[k] = nmod_add(v[k], chip, chi->order);
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}
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}
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n_primes_clear(iter);
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}
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int main()
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{
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slong iter, k, nv, nref, r, nr;
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@ -107,11 +139,9 @@ int main()
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fflush(stdout);
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vecloop(acb_dirichlet_ui_chi_vec_sieve, minq, maxq, rand, nr, v, nv);
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/*
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flint_printf("generic........ ");
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flint_printf("generic................. ");
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fflush(stdout);
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vecloop(acb_dirichlet_chi_vec, minq, maxq, rand, nr, v, nv);
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*/
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vecloop(acb_dirichlet_ui_chi_vec, minq, maxq, rand, nr, v, nv);
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}
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flint_free(v);
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@ -1,63 +0,0 @@
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/*=============================================================================
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This file is part of ARB.
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ARB is free software; you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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ARB is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with ARB; if not, write to the Free Software
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Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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=============================================================================*/
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/******************************************************************************
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Copyright (C) 2016 Pascal Molin
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******************************************************************************/
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#include "acb_dirichlet.h"
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/* sieve on primes */
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/* TODO: see if really more efficient than primeloop using dlog
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* this one is cheaper propagating the values, but not sure this
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* is noticeable ...
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*/
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void
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acb_dirichlet_ui_chi_vec_sieve(ulong *v, const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi, slong nv)
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{
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slong k, p, pmax;
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n_primes_t iter;
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n_primes_init(iter);
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pmax = (nv < G->q) ? nv : G->q;
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v[1] = 0;
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while ((p = n_primes_next(iter)) < pmax)
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{
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if (G->q % p == 0)
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{
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for (k = p; k < nv; k += p)
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v[k] = ACB_DIRICHLET_CHI_NULL;
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}
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else
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{
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long chip;
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chip = acb_dirichlet_ui_chi(G, chi, p);
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for (k = p; k < nv; k += p)
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if (v[k] != -1)
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v[k] = nmod_add(v[k], chip, chi->order);
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}
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}
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n_primes_clear(iter);
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}
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@ -69,6 +69,11 @@ logarithms.
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exceed `10^{12}` (an abort will be raised). This restriction could
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be removed in the future.
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.. function:: void acb_dirichlet_subgroup_init(acb_dirichlet_group_t H, const acb_dirichlet_group_t G, ulong h)
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Given an already computed group *G* mod `q`, initialize its subgroup *H*
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defined mod `h\mid q`. Precomputed discrete logs tables are kept.
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.. function:: void acb_dirichlet_group_clear(acb_dirichlet_group_t G)
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Clears *G*.
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@ -80,8 +85,8 @@ logarithms.
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If *num* gets very large, the entire group may be indexed.
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Conrey index
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...............................................................................
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Conrey elements
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-------------------------------------------------------------------------------
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.. type:: acb_dirichlet_conrey_struct
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@ -134,34 +139,6 @@ the group *G* is isomorphic to its dual.
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The returned value is the numerator of the actual value exponent mod the group exponent *G->expo*.
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Character properties
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...............................................................................
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As a consequence of the Conrey numbering, properties of
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characters such that the order, the parity or the conductor are available
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at the level of their number, whithout any discrete log computation,
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or at the Conrey index level.
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.. function:: ulong acb_dirichlet_ui_order(const acb_dirichlet_group_t G, ulong a)
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.. function:: ulong acb_dirichlet_conrey_order(const acb_dirichlet_group_t G, const acb_dirichlet_conrey_t x)
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Compute the order of a mod q.
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.. function:: ulong acb_dirichlet_ui_conductor(const acb_dirichlet_group_t G, ulong a)
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.. function:: ulong acb_dirichlet_conrey_conductor(const acb_dirichlet_group_t G, const acb_dirichlet_conrey_t x)
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Compute the conductor of a mod q, that is the smallest r dividing q such
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that a corresponds to an element defined modulo r.
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.. function:: ulong acb_dirichlet_ui_parity(const acb_dirichlet_group_t G, ulong a)
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.. function:: int acb_dirichlet_conrey_parity(const acb_dirichlet_group_t G, const acb_dirichlet_conrey_t x)
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Compute the parity of a mod q, which is the parity of the corresponding
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Dirichlet character.
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Character type
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-------------------------------------------------------------------------------
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@ -191,13 +168,44 @@ Character type
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Sets *chi* to the Dirichlet character of Conrey index *x*.
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Character properties
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...............................................................................
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-------------------------------------------------------------------------------
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.. function:: ulong acb_dirichlet_char_order(const acb_dirichlet_char_t chi)
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As a consequence of the Conrey numbering, all these numbers are available at the
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level off *number* and *index*, and for *char*.
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No discrete log computation is performed.
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.. function:: ulong acb_dirichlet_number_primitive(const acb_dirichlet_group_t G)
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return the number of primitive elements in *G*.
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.. function:: ulong acb_dirichlet_ui_conductor(const acb_dirichlet_group_t G, ulong a)
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.. function:: ulong acb_dirichlet_conrey_conductor(const acb_dirichlet_group_t G, const acb_dirichlet_conrey_t x)
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.. function:: ulong acb_dirichlet_char_conductor(const acb_dirichlet_char_t chi)
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return the *conductor* of `\chi_q(a,\cdot)`, that is the smallest `r` dividing `q`
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such `\chi_q(a,\cdot)` can be obtained as a character mod `r`.
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This number is precomputed for the *char* type.
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.. function:: int acb_dirichlet_ui_parity(const acb_dirichlet_group_t G, ulong a)
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.. function:: int acb_dirichlet_conrey_parity(const acb_dirichlet_group_t G, const acb_dirichlet_conrey_t x)
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.. function:: int acb_dirichlet_char_parity(const acb_dirichlet_char_t chi)
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.. function:: ulong acb_dirichlet_char_conductor(const acb_dirichlet_char_t chi)
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return the *parity* `\lambda` in `\{0,1\}` of `\chi_q(a,\cdot)`, such that
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`\chi_q(a,-1)=(-1)^\lambda`.
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This number is precomputed for the *char* type.
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.. function:: ulong acb_dirichlet_ui_order(const acb_dirichlet_group_t G, ulong a)
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.. function:: int acb_dirichlet_conrey_order(const acb_dirichlet_group_t G, const acb_dirichlet_conrey_t x)
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.. function:: ulong acb_dirichlet_char_order(const acb_dirichlet_char_t chi)
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return the order of `\chi_q(a,\cdot)` which is the order of `a\nmod q`.
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This number is precomputed for the *char* type.
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Character evaluation
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-------------------------------------------------------------------------------
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@ -206,6 +214,12 @@ The image of a Dirichlet character is a finite cyclic group. Dirichlet
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character evaluations are either exponents in this group, or an *acb_t* root of
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unity.
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.. function:: ulong acb_dirichlet_ui_chi_conrey(const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi, const acb_dirichlet_conrey_t x)
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.. function:: ulong acb_dirichlet_ui_chi(const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi, ulong n)
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compute that value `\chi(a)` as the exponent mod the order of `\chi`.
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.. function:: void acb_dirichlet_chi(acb_t res, const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi, ulong n, slong prec)
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Sets *res* to `\chi(n)`, the value of the Dirichlet character *chi*
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@ -213,13 +227,49 @@ unity.
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There are no restrictions on *n*.
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Vector evaluation
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-------------------------------------------------------------------------------
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.. function:: void acb_dirichlet_ui_chi_vec(ulong * v, const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi, slong nv)
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compute the list of exponent values *v[k]* for `0\leq k < nv`
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.. function:: void acb_dirichlet_chi_vec(acb_ptr v, const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi, slong nv, slong prec)
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compute the *nv* first Dirichlet values
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Operations
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-------------------------------------------------------------------------------
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.. function:: void acb_dirichlet_conrey_mul(acb_dirichlet_conrey_t c, const acb_dirichlet_group_t G, const acb_dirichlet_conrey_t a, const acb_dirichlet_conrey_t b)
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.. function:: void acb_dirichlet_char_mul(acb_dirichlet_char_t chi12, const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi1, const acb_dirichlet_char_t chi2)
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multiply two characters in the same group
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.. function:: void acb_dirichlet_conrey_pow(acb_dirichlet_conrey_t c, const acb_dirichlet_group_t G, const acb_dirichlet_conrey_t a, ulong n)
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take the power of some character
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Gauss and Jacobi sums
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-------------------------------------------------------------------------------
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.. function:: void acb_dirichlet_gauss_sum(acb_t res, const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi, slong prec)
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compute the Gauss sum
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.. math::
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G_q(a) = \sum_{x \mod q} \chi_q(a, x)e^{\frac{2i\pi x}q}
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.. function:: void acb_dirichlet_jacobi_sum(acb_t res, const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi1, const acb_dirichlet_char_t chi2, slong prec)
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compute the Jacobi sum
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.. math::
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J_q(a,b) = \sum_{x \mod q} \chi_q(a, x)\chi_q(b, 1-x)
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Euler products
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-------------------------------------------------------------------------------
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