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/*
Copyright (C) 2021 Fredrik Johansson
This file is part of FLINT.
FLINT is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 3 of the License, or
(at your option) any later version. See <https://www.gnu.org/licenses/>.
*/
#include "qqbar/impl.h"
#include "ca.h"
#include "ca/impl.h"
#include "ca_vec.h"
/* todo: Re, Im, Abs, Sgn ... */
static void
ca_rewrite_ext_complex_normal_form(ca_t res, ca_ext_ptr ext, int deep, ca_ctx_t ctx)
{
switch (CA_EXT_HEAD(ext))
{
case CA_QQBar:
if (qqbar_is_i(CA_EXT_QQBAR(ext)))
{
ca_set_ext(res, ext, ctx);
}
else if (qqbar_is_root_of_unity(NULL, NULL, CA_EXT_QQBAR(ext)))
{
ca_set_ext(res, ext, ctx);
}
else
{
fmpq_poly_t poly;
qqbar_t zeta;
ulong q;
qqbar_init(zeta);
fmpq_poly_init(poly);
q = qqbar_try_as_cyclotomic(zeta, poly, CA_EXT_QQBAR(ext));
if (q != 0)
{
ca_set_qqbar(res, zeta, ctx);
ca_fmpq_poly_evaluate(res, poly, res, ctx);
}
else
{
ca_set_ext(res, ext, ctx);
}
qqbar_clear(zeta);
fmpq_poly_clear(poly);
}
break;
case CA_Re:
case CA_Im:
case CA_Conjugate:
case CA_Abs:
case CA_Sign:
case CA_Arg:
{
ca_t t, u;
truth_t is_zero;
ca_init(t, ctx);
ca_init(u, ctx);
if (deep)
ca_rewrite_complex_normal_form(t, CA_EXT_FUNC_ARGS(ext), deep, ctx);
else
ca_set(t, CA_EXT_FUNC_ARGS(ext), ctx);
switch (CA_EXT_HEAD(ext))
{
case CA_Re:
ca_conj_deep(u, t, ctx);
ca_add(res, t, u, ctx);
ca_div_ui(res, res, 2, ctx);
break;
case CA_Im:
ca_conj_deep(u, t, ctx);
ca_sub(res, t, u, ctx);
ca_div_ui(res, res, 2, ctx);
ca_neg_i(t, ctx);
ca_mul(res, res, t, ctx);
break;
case CA_Conjugate:
ca_conj_deep(res, t, ctx);
break;
case CA_Abs:
ca_conj_deep(u, t, ctx);
ca_mul(t, t, u, ctx);
ca_sqrt(res, t, ctx);
break;
case CA_Sign:
is_zero = ca_check_is_zero(t, ctx);
if (is_zero == T_TRUE)
{
ca_zero(res, ctx);
}
else if (is_zero == T_FALSE)
{
ca_conj_deep(u, t, ctx);
ca_mul(u, t, u, ctx);
ca_sqrt(u, u, ctx);
/* todo: division by zero is impossible; optimizations here */
ca_div(res, t, u, ctx);
}
else
{
ca_sgn(res, t, ctx);
}
break;
case CA_Arg:
is_zero = ca_check_is_zero(t, ctx);
if (is_zero == T_TRUE)
{
ca_zero(res, ctx);
}
else if (is_zero == T_FALSE)
{
/* todo: better to create two logs? */
ca_conj_deep(u, t, ctx);
ca_mul(u, t, u, ctx);
ca_sqrt(u, u, ctx);
ca_div(u, t, u, ctx);
ca_log(u, u, ctx);
ca_neg_i(t, ctx);
ca_mul(res, t, u, ctx);
}
else
{
ca_arg(res, t, ctx);
}
break;
default:
flint_throw(FLINT_ERROR, "(%s)\n", __func__);
}
ca_clear(t, ctx);
ca_clear(u, ctx);
}
break;
/* sin(x) = (exp(ix) - 1/exp(ix))/(2i) */
/* cos(x) = (exp(ix) + 1/exp(ix))/2 */
/* tan(x) = 2i/(exp(2ix)+1) */
case CA_Sin:
case CA_Cos:
case CA_Tan:
{
ca_t t, u;
ca_init(t, ctx);
ca_init(u, ctx);
if (deep)
ca_rewrite_complex_normal_form(t, CA_EXT_FUNC_ARGS(ext), deep, ctx);
else
ca_set(t, CA_EXT_FUNC_ARGS(ext), ctx);
if (CA_EXT_HEAD(ext) == CA_Sin)
ca_sin_cos_exponential(res, NULL, t, ctx);
else if (CA_EXT_HEAD(ext) == CA_Cos)
ca_sin_cos_exponential(NULL, res, t, ctx);
else
{
ca_sin_cos_exponential(t, u, t, ctx);
ca_div(res, t, u, ctx);
}
ca_clear(t, ctx);
ca_clear(u, ctx);
}
break;
case CA_Atan:
case CA_Asin:
case CA_Acos:
{
ca_t t;
ca_init(t, ctx);
if (deep)
ca_rewrite_complex_normal_form(t, CA_EXT_FUNC_ARGS(ext), deep, ctx);
else
ca_set(t, CA_EXT_FUNC_ARGS(ext), ctx);
if (CA_EXT_HEAD(ext) == CA_Asin)
ca_asin_logarithm(res, t, ctx);
else if (CA_EXT_HEAD(ext) == CA_Acos)
ca_acos_logarithm(res, t, ctx);
else
ca_atan_logarithm(res, t, ctx);
ca_clear(t, ctx);
}
break;
case CA_Sqrt:
case CA_Exp:
case CA_Log:
if (deep)
{
ca_rewrite_complex_normal_form(res, CA_EXT_FUNC_ARGS(ext), deep, ctx);
if (ca_equal_repr(res, CA_EXT_FUNC_ARGS(ext), ctx))
{
ca_set_ext(res, ext, ctx);
}
else
{
switch (CA_EXT_HEAD(ext))
{
case CA_Exp:
ca_exp(res, res, ctx);
break;
case CA_Log:
ca_log(res, res, ctx);
break;
case CA_Sqrt:
ca_sqrt(res, res, ctx);
break;
default:
flint_throw(FLINT_ERROR, "(%s)\n", __func__);
}
}
}
else
{
ca_set_ext(res, ext, ctx);
}
break;
default:
/* todo: deep */
ca_set_ext(res, ext, ctx);
}
}
void
ca_rewrite_complex_normal_form(ca_t res, const ca_t x, int deep, ca_ctx_t ctx)
{
if (CA_IS_SPECIAL(x))
{
if (CA_IS_SIGNED_INF(x))
{
ca_sgn(res, x, ctx);
ca_rewrite_complex_normal_form(res, res, deep, ctx);
if (!ca_is_unknown(res, ctx))
res->field |= CA_INF;
}
else
{
ca_set(res, x, ctx);
}
}
else if (CA_IS_QQ(x, ctx) || CA_IS_QQ_I(x, ctx))
{
ca_set(res, x, ctx);
}
else
{
ca_field_ptr K = CA_FIELD(x, ctx);
if (CA_FIELD_IS_NF(K))
{
if (qqbar_is_root_of_unity(NULL, NULL, CA_FIELD_NF_QQBAR(K)))
{
ca_set(res, x, ctx);
}
else
{
fmpq_poly_t poly, xpoly;
qqbar_t zeta;
ulong q;
qqbar_init(zeta);
fmpq_poly_init(poly);
q = qqbar_try_as_cyclotomic(zeta, poly, CA_FIELD_NF_QQBAR(K));
if (q != 0)
{
fmpq_poly_init(xpoly);
nf_elem_get_fmpq_poly(xpoly, CA_NF_ELEM(x), CA_FIELD_NF(K));
ca_set_qqbar(res, zeta, ctx);
ca_fmpq_poly_evaluate(res, poly, res, ctx);
ca_fmpq_poly_evaluate(res, xpoly, res, ctx);
fmpq_poly_clear(xpoly);
}
else
{
ca_set(res, x, ctx);
}
qqbar_clear(zeta);
fmpq_poly_clear(poly);
}
}
else if (0)
{
ca_set(res, x, ctx);
}
else
{
slong nvars;
int * used;
slong i;
ca_ptr cext;
nvars = CA_FIELD_LENGTH(K);
used = flint_calloc(nvars, sizeof(int));
cext = _ca_vec_init(nvars, ctx);
fmpz_mpoly_q_used_vars(used, CA_MPOLY_Q(x), CA_FIELD_MCTX(K, ctx));
for (i = 0; i < nvars; i++)
{
if (used[i])
{
ca_rewrite_ext_complex_normal_form(cext + i, CA_FIELD_EXT_ELEM(K, i), deep, ctx);
}
}
ca_fmpz_mpoly_q_evaluate_no_division_by_zero(res, CA_MPOLY_Q(x), cext, CA_FIELD_MCTX(K, ctx), ctx);
_ca_vec_clear(cext, nvars, ctx);
/* Root of unity hack: introduce artificial root
of unity to force simplifications. This should not be necessary;
cases where it helps are a sign that we are not finding
all exponential relations. */
if (0 && !CA_IS_SPECIAL(res) && !CA_FIELD_IS_QQ(CA_FIELD(res, ctx)) && !CA_FIELD_IS_NF(CA_FIELD(res, ctx)))
{
ca_t t, u;
ca_init(t, ctx);
ca_init(u, ctx);
ca_pi_i(t, ctx);
ca_div_ui(t, t, 12, ctx);
ca_exp(t, t, ctx);
ca_add(u, res, t, ctx);
ca_sub(u, u, t, ctx);
if (ca_cmp_repr(u, res, ctx) < 0)
ca_swap(res, u, ctx);
ca_clear(t, ctx);
ca_clear(u, ctx);
}
flint_free(used);
}
}
}