void fmpq_poly_resultant(fmpq_t r, const fmpq_poly_t f, const fmpq_poly_t g)
{
const long len1 = f->length;
const long len2 = g->length;
if (len1 == 0 || len2 == 0)
{
fmpq_zero(r);
}
else
{
if (len1 >= len2)
{
_fmpq_poly_resultant(fmpq_numref(r), fmpq_denref(r),
f->coeffs, f->den, len1,
g->coeffs, g->den, len2);
}
else
{
_fmpq_poly_resultant(fmpq_numref(r), fmpq_denref(r),
g->coeffs, g->den, len2,
f->coeffs, f->den, len1);
if (((len1 | len2) & 1L) == 0L)
fmpq_neg(r, r);
}
}
}
renf_elem_class renf_elem_class::operator-() const noexcept
{
renf_elem_class ans(*this);
if (nf == nullptr)
fmpq_neg(ans.b, ans.b);
else
renf_elem_neg(ans.a, ans.a, ans.nf->renf_t());
return ans;
}
void _dgsl_rot_mp_sqrt_sigma_2(fmpq_poly_t rop, const fmpz_poly_t g, const mpfr_t sigma,
const int r, const long n, const mpfr_prec_t prec, const oz_flag_t flags) {
fmpq_poly_zero(rop);
fmpq_t r_q2;
fmpq_init(r_q2);
fmpq_set_si(r_q2, r, 1);
fmpq_mul(r_q2, r_q2, r_q2);
fmpq_neg(r_q2, r_q2);
fmpq_poly_set_coeff_fmpq(rop, 0, r_q2);
fmpq_clear(r_q2);
fmpq_poly_t g_q; fmpq_poly_init(g_q);
fmpq_poly_set_fmpz_poly(g_q, g);
fmpq_poly_t ng; fmpq_poly_init(ng);
fmpq_poly_oz_invert_approx(ng, g_q, n, prec, flags);
fmpq_poly_t ngt;
fmpq_poly_init(ngt);
fmpq_poly_oz_conjugate(ngt, ng, n);
fmpq_poly_t nggt;
fmpq_poly_init(nggt);
fmpq_poly_oz_mul(nggt, ng, ngt, n);
/**
We compute sqrt(g^-T · g^-1) to use it as the starting point for
convergence on sqrt(σ^2 · g^-T · g^-1 - r^2) below. We can compute the
former with less precision than the latter
*/
mpfr_t norm;
mpfr_init2(norm, prec);
fmpz_poly_eucl_norm_mpfr(norm, g, MPFR_RNDN);
double p = mpfr_get_d(norm, MPFR_RNDN);
/**
|g^-1| ~= 1/|g|
|g^-T| ~= |g^-1|
|g^-1·g^-T| ~= sqrt(n)·|g^-T|·|g^-1|
*/
fmpq_poly_t sqrt_start; fmpq_poly_init(sqrt_start);
p = log2(n) + 4*log2(p);
int fail = -1;
while (fail) {
p = 2*p;
if (fail<0)
fail = fmpq_poly_oz_sqrt_approx_db(sqrt_start, nggt, n, p, prec/2, flags, NULL);
else
fail = fmpq_poly_oz_sqrt_approx_db(sqrt_start, nggt, n, p, prec/2, flags, sqrt_start);
if(fail)
fprintf(stderr, "FAILED for precision %7.1f with code (%d), doubling precision.\n", p, fail);
}
fmpq_t sigma2;
fmpq_init(sigma2);
fmpq_set_mpfr(sigma2, sigma, MPFR_RNDN);
fmpq_poly_scalar_mul_fmpq(sqrt_start, sqrt_start, sigma2);
fmpq_mul(sigma2, sigma2, sigma2);
fmpq_poly_scalar_mul_fmpq(nggt, nggt, sigma2);
fmpq_clear(sigma2);
fmpq_poly_add(rop, rop, nggt);
p = p + 2*log2(mpfr_get_d(sigma, MPFR_RNDN));
fmpq_poly_oz_sqrt_approx_babylonian(rop, rop, n, p, prec, flags, sqrt_start);
mpfr_clear(norm);
fmpq_poly_clear(g_q);
fmpq_poly_clear(ng);
fmpq_poly_clear(ngt);
fmpq_poly_clear(nggt);
fmpq_poly_clear(sqrt_start);
}
void test_field2(flint_rand_t state)
{
/* test in QQ[3^(1/4)] */
renf_t nf;
renf_elem_t a;
fmpq_t d, k;
fmpq_poly_t p;
fmpq_init(d);
fmpq_poly_init(p);
fmpq_set_si(d, 3, 1);
renf_init_nth_root_fmpq(nf, d, 4, 10 + n_randint(state, 10));
fmpq_clear(d);
fmpq_init(k);
renf_elem_init(a, nf);
/* test rationals */
/* --> 3^(1/4) */
fmpq_poly_set_coeff_si(p, 1, 1);
renf_elem_set_fmpq_poly(a, p, nf);
check_ceil(a, nf, 2, "3^(1/4)");
/* --> 3^(1/4) - p_34 / q_34 */
/* ceil = 1 */
fmpz_set_str(fmpq_numref(k), "3871793620206447926", 10);
fmpz_set_str(fmpq_denref(k), "2941926960111028069", 10);
fmpq_neg(k, k);
fmpq_poly_set_coeff_fmpq(p, 0, k);
renf_elem_set_fmpq_poly(a, p, nf);
check_ceil(a, nf, 1, "3^(1/4)");
/* --> 3^(1/4) - p_35 / q_35 */
/* ceil = 0 */
fmpz_set_str(fmpq_numref(k), "4393442218385055959", 10);
fmpz_set_str(fmpq_denref(k), "3338294180377262795", 10);
fmpq_neg(k, k);
fmpq_poly_set_coeff_fmpq(p, 0, k);
renf_elem_set_fmpq_poly(a, p, nf);
check_ceil(a, nf, 0, "3^(1/4)");
/* --> 3^(1/4) - p_200 / q_200 */
fmpz_set_str(fmpq_numref(k), "51566086581654990699052199424489069476470199719930170996263916596162993841059250500042162091", 10);
fmpz_set_str(fmpq_denref(k), "39181752754141206003124111890355840072199542360218864430892618765033598468868752146602163065", 10);
fmpq_neg(k, k);
fmpq_poly_set_coeff_fmpq(p, 0, k);
renf_elem_set_fmpq_poly(a, p, nf);
check_ceil(a, nf, 1, "3^(1/4)");
/* --> 3^(1/4) - p_201 / q_201 */
fmpz_set_str(fmpq_numref(k), "80796322887694335717970676356641716096406222234122724217891106756946083353628876437327250032", 10);
fmpz_set_str(fmpq_denref(k), "61391929399498685496270115285641595325756438975454257165479021482386018841773493669624721869", 10);
fmpq_neg(k, k);
fmpq_poly_set_coeff_fmpq(p, 0, k);
renf_elem_set_fmpq_poly(a, p, nf);
check_ceil(a, nf, 0, "3^(1/4)");
/* */
fmpz_set_str(fmpq_numref(k), "13231942875843754343234", 10);
fmpz_set_str(fmpq_denref(k), "14321431341231112121", 10);
fmpq_poly_set_coeff_fmpq(p, 3, k);
fmpz_set_str(fmpq_numref(k), "148589873455543948591", 10);
fmpz_set_str(fmpq_denref(k), "12332111221111", 10);
fmpq_neg(k, k);
fmpq_poly_set_coeff_fmpq(p, 2, k);
fmpz_set_str(fmpq_numref(k), "1233321469998711012392391", 10);
fmpz_set_str(fmpq_denref(k), "11814121556810191", 10);
fmpq_poly_set_coeff_fmpq(p, 1, k);
fmpz_set_str(fmpq_numref(k), "1249152314425433983202991363672458443993964487436329478959287771807457205881969983777233465754608376177969464841", 10);
fmpz_set_str(fmpq_denref(k), "10720278662399817731713810382544982753044312944075797382817281426908463944866446042500978893159281330135", 10);
fmpq_neg(k, k);
fmpq_poly_set_coeff_fmpq(p, 0, k);
renf_elem_set_fmpq_poly(a, p, nf);
check_ceil(a, nf, 231, "3^(1/4)");
renf_elem_clear(a, nf);
renf_clear(nf);
fmpq_clear(k);
fmpq_poly_clear(p);
}
int fmpq_mat_inv(fmpq_mat_t B, const fmpq_mat_t A)
{
long n = A->r;
if (n == 0)
{
return 1;
}
else if (n == 1)
{
if (fmpq_is_zero(fmpq_mat_entry(A, 0, 0)))
return 0;
fmpq_inv(fmpq_mat_entry(B, 0, 0), fmpq_mat_entry(A, 0, 0));
return 1;
}
else if (n == 2)
{
fmpq_t d;
int success;
fmpq_init(d);
fmpq_mul(d, fmpq_mat_entry(A, 0, 0), fmpq_mat_entry(A, 1, 1));
fmpq_submul(d, fmpq_mat_entry(A, 0, 1), fmpq_mat_entry(A, 1, 0));
success = !fmpq_is_zero(d);
if (success)
{
fmpq_t t00, t01, t10, t11;
fmpq_inv(d, d);
fmpq_init(t00);
fmpq_init(t01);
fmpq_init(t10);
fmpq_init(t11);
fmpq_mul(t00, fmpq_mat_entry(A, 1, 1), d);
fmpq_mul(t01, fmpq_mat_entry(A, 0, 1), d);
fmpq_mul(t10, fmpq_mat_entry(A, 1, 0), d);
fmpq_mul(t11, fmpq_mat_entry(A, 0, 0), d);
fmpq_set(fmpq_mat_entry(B, 0, 0), t00);
fmpq_neg(fmpq_mat_entry(B, 0, 1), t01);
fmpq_neg(fmpq_mat_entry(B, 1, 0), t10);
fmpq_set(fmpq_mat_entry(B, 1, 1), t11);
fmpq_clear(t00);
fmpq_clear(t01);
fmpq_clear(t10);
fmpq_clear(t11);
}
fmpq_clear(d);
return success;
}
else
{
fmpz_mat_t Aclear, Bclear, I;
fmpz * den;
long i;
int success;
fmpz_mat_init(Aclear, n, n);
fmpz_mat_init(Bclear, n, n);
fmpz_mat_init(I, n, n);
den = _fmpz_vec_init(n);
fmpq_mat_get_fmpz_mat_rowwise(Aclear, den, A);
for (i = 0; i < n; i++)
fmpz_set(fmpz_mat_entry(I, i, i), den + i);
success = fmpz_mat_solve(Bclear, den, Aclear, I);
if (success)
fmpq_mat_set_fmpz_mat_div_fmpz(B, Bclear, den);
fmpz_mat_clear(Aclear);
fmpz_mat_clear(Bclear);
fmpz_mat_clear(I);
_fmpz_vec_clear(den, A->r);
return success;
}
}
/** @name arithmetic
@{ */
void negate(ElementType& result,const ElementType& a) const {fmpq_neg(&result,&a);}
int fmpq_poly_check_unique_real_root(const fmpq_poly_t pol, const arb_t a, slong prec)
{
if (pol->length < 2)
return 0;
else if (pol->length == 2)
{
/* linear polynomial */
fmpq_t root;
int ans;
fmpq_init(root);
fmpq_set_fmpz_frac(root, fmpq_poly_numref(pol), fmpq_poly_numref(pol) + 1);
fmpq_neg(root, root);
ans = arb_contains_fmpq(a, root);
fmpq_clear(root);
return ans;
}
else
{
arb_t b, c;
arf_t l, r;
fmpz * der;
int lsign, rsign;
fmpz_poly_t pol2;
slong n;
/* 1 - cheap test: */
/* - sign(left) * sign(right) = -1 */
/* - no zero of the derivative */
arb_init(b);
arb_init(c);
arf_init(l);
arf_init(r);
arb_get_interval_arf(l, r, a, prec);
arb_set_arf(b, l);
_fmpz_poly_evaluate_arb(c, pol->coeffs, pol->length, b, 2*prec);
lsign = arb_sgn2(c);
arb_set_arf(b, r);
_fmpz_poly_evaluate_arb(c, pol->coeffs, pol->length, b, 2*prec);
rsign = arb_sgn2(c);
arb_clear(c);
if (lsign * rsign == -1)
{
der = _fmpz_vec_init(pol->length - 1);
_fmpz_poly_derivative(der, pol->coeffs, pol->length);
_fmpz_poly_evaluate_arb(b, der, pol->length - 1, a, prec);
_fmpz_vec_clear(der, pol->length - 1);
if (!arb_contains_zero(b))
{
arf_clear(l);
arf_clear(r);
arb_clear(b);
return 1;
}
}
else
return 0;
arb_clear(b);
/* 2 - expensive testing */
fmpq_t ql, qr;
fmpq_init(ql);
fmpq_init(qr);
arf_get_fmpq(ql, l);
arf_get_fmpq(qr, r);
fmpz_poly_init(pol2);
fmpz_poly_fit_length(pol2, pol->length);
_fmpz_vec_set(pol2->coeffs, pol->coeffs, pol->length);
pol2->length = pol->length;
_fmpz_poly_scale_0_1_fmpq(pol2->coeffs, pol2->length, ql, qr);
n = fmpz_poly_num_real_roots_0_1(pol2);
fmpz_poly_clear(pol2);
fmpq_clear(ql);
fmpq_clear(qr);
return (n == 1);
}
}
