int
arf_sub_special(arf_t z, const arf_t x, const arf_t y, slong prec, arf_rnd_t rnd)
{
if (arf_is_zero(x))
{
if (arf_is_zero(y))
{
arf_zero(z);
return 0;
}
else
return arf_neg_round(z, y, prec, rnd);
}
else if (arf_is_zero(y))
{
return arf_set_round(z, x, prec, rnd);
}
else if (arf_is_nan(x) || arf_is_nan(y)
|| (arf_is_pos_inf(x) && arf_is_pos_inf(y))
|| (arf_is_neg_inf(x) && arf_is_neg_inf(y)))
{
arf_nan(z);
return 0;
}
else if (arf_is_special(x))
{
arf_set(z, x);
return 0;
}
else
{
arf_neg(z, y);
return 0;
}
}
void arf_twobytwo_diag(arf_t u1, arf_t u2, const arf_t a, const arf_t b, const arf_t d, slong prec) {
// Compute the orthogonal matrix that diagonalizes
//
// A = [a b]
// [b d]
//
// This matrix will have the form
//
// U = [cos x , -sin x]
// [sin x, cos x]
//
// where the diagonal matrix is U^t A U.
// We set u1 = cos x, u2 = -sin x.
if(arf_is_zero(b)) {
arf_set_ui(u1, 1);
arf_set_ui(u2, 0);
return;
}
arf_t x; arf_init(x);
arf_mul(u1, b, b, prec, ARF_RND_NEAR); // u1 = b^2
arf_sub(u2, a, d, prec, ARF_RND_NEAR); // u2 = a - d
arf_mul_2exp_si(u2, u2, -1); // u2 = (a - d)/2
arf_mul(u2, u2, u2, prec, ARF_RND_NEAR); // u2 = ( (a - d)/2 )^2
arf_add(u1, u1, u2, prec, ARF_RND_NEAR); // u1 = b^2 + ( (a-d)/2 )^2
arf_sqrt(u1, u1, prec, ARF_RND_NEAR); // u1 = sqrt(above)
arf_mul_2exp_si(u1, u1, 1); // u1 = 2 (sqrt (above) )
arf_add(u1, u1, d, prec, ARF_RND_NEAR); // u1 += d
arf_sub(u1, u1, a, prec, ARF_RND_NEAR); // u1 -= a
arf_mul_2exp_si(u1, u1, -1); // u1 = (d - a)/2 + sqrt(b^2 + ( (a-d)/2 )^2)
arf_mul(x, u1, u1, prec, ARF_RND_NEAR);
arf_addmul(x, b, b, prec, ARF_RND_NEAR); // x = u1^2 + b^2
arf_sqrt(x, x, prec, ARF_RND_NEAR); // x = sqrt(u1^2 + b^2)
arf_div(u2, u1, x, prec, ARF_RND_NEAR);
arf_div(u1, b, x, prec, ARF_RND_NEAR);
arf_neg(u1, u1);
arf_clear(x);
}
void
_acb_poly_reciprocal_majorant(arb_ptr res, acb_srcptr vec, long len, long prec)
{
long i;
for (i = 0; i < len; i++)
{
if (i == 0)
{
acb_get_abs_lbound_arf(arb_midref(res + i), vec + i, prec);
mag_zero(arb_radref(res + i));
}
else
{
acb_get_abs_ubound_arf(arb_midref(res + i), vec + i, prec);
arf_neg(arb_midref(res + i), arb_midref(res + i));
mag_zero(arb_radref(res + i));
}
}
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("fundamental_domain_approx....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 10000 * arb_test_multiplier(); iter++)
{
fmpq_t x, y;
psl2z_t g;
arf_t one_minus_eps, tol;
acb_t z, w, w2;
arb_t t;
slong prec;
fmpq_init(x);
fmpq_init(y);
psl2z_init(g);
acb_init(z);
acb_init(w);
acb_init(w2);
arf_init(one_minus_eps);
arf_init(tol);
arb_init(t);
/* pick an exact point in the upper half plane */
fmpq_randtest(x, state, 1 + n_randint(state, 500));
do {
fmpq_randtest(y, state, 1 + n_randint(state, 500));
} while (fmpz_sgn(fmpq_numref(y)) <= 0);
/* pick a tolerance */
arf_set_ui_2exp_si(tol, 1, -(slong) n_randint(state, 500));
/* now increase the precision until convergence */
for (prec = 32; ; prec *= 2)
{
if (prec > 16384)
{
flint_printf("FAIL (no convergence)\n");
flint_printf("x = "); fmpq_print(x); flint_printf("\n\n");
flint_printf("y = "); fmpq_print(y); flint_printf("\n\n");
flint_printf("z = "); acb_printd(z, 50); flint_printf("\n\n");
flint_printf("w = "); acb_printd(w, 50); flint_printf("\n\n");
flint_printf("w2 = "); acb_printd(w2, 50); flint_printf("\n\n");
flint_printf("g = "); psl2z_print(g); flint_printf("\n\n");
abort();
}
arb_set_fmpq(acb_realref(z), x, prec);
arb_set_fmpq(acb_imagref(z), y, prec);
arf_set_ui_2exp_si(one_minus_eps, 1, -prec / 4);
arf_sub_ui(one_minus_eps, one_minus_eps, 1, prec, ARF_RND_DOWN);
arf_neg(one_minus_eps, one_minus_eps);
acb_modular_fundamental_domain_approx(w, g, z, one_minus_eps, prec);
acb_modular_transform(w2, g, z, prec);
if (!psl2z_is_correct(g) || !acb_overlaps(w, w2))
{
flint_printf("FAIL (incorrect transformation)\n");
flint_printf("x = "); fmpq_print(x); flint_printf("\n\n");
flint_printf("y = "); fmpq_print(y); flint_printf("\n\n");
flint_printf("z = "); acb_printd(z, 50); flint_printf("\n\n");
flint_printf("w = "); acb_printd(w, 50); flint_printf("\n\n");
flint_printf("w2 = "); acb_printd(w2, 50); flint_printf("\n\n");
flint_printf("g = "); psl2z_print(g); flint_printf("\n\n");
abort();
}
/* success */
if (acb_modular_is_in_fundamental_domain(w, tol, prec))
break;
}
/* check that g^(-1) * w contains x+yi */
psl2z_inv(g, g);
acb_modular_transform(w2, g, w, 2 + n_randint(state, 1000));
if (!arb_contains_fmpq(acb_realref(w2), x) ||
!arb_contains_fmpq(acb_imagref(w2), y))
{
flint_printf("FAIL (inverse containment)\n");
flint_printf("x = "); fmpq_print(x); flint_printf("\n\n");
flint_printf("y = "); fmpq_print(y); flint_printf("\n\n");
flint_printf("z = "); acb_printd(z, 50); flint_printf("\n\n");
flint_printf("w = "); acb_printd(w, 50); flint_printf("\n\n");
flint_printf("w2 = "); acb_printd(w2, 50); flint_printf("\n\n");
flint_printf("g = "); psl2z_print(g); flint_printf("\n\n");
abort();
}
fmpq_clear(x);
fmpq_clear(y);
psl2z_clear(g);
acb_clear(z);
acb_clear(w);
acb_clear(w2);
arf_clear(one_minus_eps);
arf_clear(tol);
arb_clear(t);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
int arb_mat_jacobi(arb_mat_t D, arb_mat_t P, const arb_mat_t A, slong prec) {
//
// Given a d x d real symmetric matrix A, compute an orthogonal matrix
// P and a diagonal D such that A = P D P^t = P D P^(-1).
//
// D should have already been initialized as a d x 1 matrix, and Pp
// should have already been initialized as a d x d matrix.
//
// If the eigenvalues can be certified as unique, then a nonzero int is
// returned, and the eigenvectors should have reasonable error bounds. If
// the eigenvalues cannot be certified as unique, then some of the
// eigenvectors will have infinite error radius.
#define B(i,j) arb_mat_entry(B, i, j)
#define D(i) arb_mat_entry(D, i, 0)
#define P(i,j) arb_mat_entry(P, i, j)
int dim = arb_mat_nrows(A);
if(dim == 1) {
arb_mat_set(D, A);
arb_mat_one(P);
return 0;
}
arb_mat_t B;
arb_mat_init(B, dim, dim);
arf_t * B1 = (arf_t*)malloc(dim * sizeof(arf_t));
arf_t * B2 = (arf_t*)malloc(dim * sizeof(arf_t));
arf_t * row_max = (arf_t*)malloc((dim - 1) * sizeof(arf_t));
int * row_max_indices = (int*)malloc((dim - 1) * sizeof(int));
for(int k = 0; k < dim; k++) {
arf_init(B1[k]);
arf_init(B2[k]);
}
for(int k = 0; k < dim - 1; k++) {
arf_init(row_max[k]);
}
arf_t x1, x2;
arf_init(x1);
arf_init(x2);
arf_t Gii, Gij, Gji, Gjj;
arf_init(Gii);
arf_init(Gij);
arf_init(Gji);
arf_init(Gjj);
arb_mat_set(B, A);
arb_mat_one(P);
for(int i = 0; i < dim - 1; i++) {
for(int j = i + 1; j < dim; j++) {
arf_abs(x1, arb_midref(B(i,j)));
if(arf_cmp(row_max[i], x1) < 0) {
arf_set(row_max[i], x1);
row_max_indices[i] = j;
}
}
}
int finished = 0;
while(!finished) {
arf_zero(x1);
int i = 0;
int j = 0;
for(int k = 0; k < dim - 1; k++) {
if(arf_cmp(x1, row_max[k]) < 0) {
arf_set(x1, row_max[k]);
i = k;
}
}
j = row_max_indices[i];
slong bound = arf_abs_bound_lt_2exp_si(x1);
if(bound < -prec * .9) {
finished = 1;
break;
}
else {
//printf("%ld\n", arf_abs_bound_lt_2exp_si(x1));
//arb_mat_printd(B, 10);
//printf("\n");
}
arf_twobytwo_diag(Gii, Gij, arb_midref(B(i,i)), arb_midref(B(i,j)), arb_midref(B(j,j)), 2*prec);
arf_neg(Gji, Gij);
arf_set(Gjj, Gii);
//printf("%d %d\n", i, j);
//arf_printd(Gii, 100);
//printf(" ");
//arf_printd(Gij, 100);
//printf("\n");
if(arf_is_zero(Gij)) { // If this happens, we're
finished = 1; // not going to do any better
break; // without increasing the precision.
}
for(int k = 0; k < dim; k++) {
arf_mul(B1[k], Gii, arb_midref(B(i,k)), prec, ARF_RND_NEAR);
arf_addmul(B1[k], Gji, arb_midref(B(j,k)), prec, ARF_RND_NEAR);
arf_mul(B2[k], Gij, arb_midref(B(i,k)), prec, ARF_RND_NEAR);
arf_addmul(B2[k], Gjj, arb_midref(B(j,k)), prec, ARF_RND_NEAR);
}
for(int k = 0; k < dim; k++) {
arf_set(arb_midref(B(i,k)), B1[k]);
arf_set(arb_midref(B(j,k)), B2[k]);
}
for(int k = 0; k < dim; k++) {
arf_mul(B1[k], Gii, arb_midref(B(k,i)), prec, ARF_RND_NEAR);
arf_addmul(B1[k], Gji, arb_midref(B(k,j)), prec, ARF_RND_NEAR);
arf_mul(B2[k], Gij, arb_midref(B(k,i)), prec, ARF_RND_NEAR);
arf_addmul(B2[k], Gjj, arb_midref(B(k,j)), prec, ARF_RND_NEAR);
}
for(int k = 0; k < dim; k++) {
arf_set(arb_midref(B(k,i)), B1[k]);
arf_set(arb_midref(B(k,j)), B2[k]);
}
for(int k = 0; k < dim; k++) {
arf_mul(B1[k], Gii, arb_midref(P(k,i)), prec, ARF_RND_NEAR);
arf_addmul(B1[k], Gji, arb_midref(P(k,j)), prec, ARF_RND_NEAR);
arf_mul(B2[k], Gij, arb_midref(P(k,i)), prec, ARF_RND_NEAR);
arf_addmul(B2[k], Gjj, arb_midref(P(k,j)), prec, ARF_RND_NEAR);
}
for(int k = 0; k < dim; k++) {
arf_set(arb_midref(P(k,i)), B1[k]);
arf_set(arb_midref(P(k,j)), B2[k]);
}
if(i < dim - 1)
arf_set_ui(row_max[i], 0);
if(j < dim - 1)
arf_set_ui(row_max[j], 0);
// Update the max in any row where the maximum
// was in a column that changed.
for(int k = 0; k < dim - 1; k++) {
if(row_max_indices[k] == j || row_max_indices[k] == i) {
arf_abs(row_max[k], arb_midref(B(k,k+1)));
row_max_indices[k] = k+1;
for(int l = k+2; l < dim; l++) {
arf_abs(x1, arb_midref(B(k,l)));
if(arf_cmp(row_max[k], x1) < 0) {
arf_set(row_max[k], x1);
row_max_indices[k] = l;
}
}
}
}
// Update the max in the ith row.
for(int k = i + 1; k < dim; k++) {
arf_abs(x1, arb_midref(B(i, k)));
if(arf_cmp(row_max[i], x1) < 0) {
arf_set(row_max[i], x1);
row_max_indices[i] = k;
}
}
// Update the max in the jth row.
for(int k = j + 1; k < dim; k++) {
arf_abs(x1, arb_midref(B(j, k)));
if(arf_cmp(row_max[j], x1) < 0) {
arf_set(row_max[j], x1);
row_max_indices[j] = k;
}
}
// Go through column i to see if any of
// the new entries are larger than the
// max of their row.
for(int k = 0; k < i; k++) {
if(k == dim) continue;
arf_abs(x1, arb_midref(B(k, i)));
if(arf_cmp(row_max[k], x1) < 0) {
arf_set(row_max[k], x1);
row_max_indices[k] = i;
}
}
// And then column j.
for(int k = 0; k < j; k++) {
if(k == dim) continue;
arf_abs(x1, arb_midref(B(k, j)));
if(arf_cmp(row_max[k], x1) < 0) {
arf_set(row_max[k], x1);
row_max_indices[k] = j;
}
}
}
for(int k = 0; k < dim; k++) {
arb_set(D(k), B(k,k));
arb_set_exact(D(k));
}
// At this point we've done that diagonalization and all that remains is
// to certify the correctness and compute error bounds.
arb_mat_t e;
arb_t error_norms[dim];
for(int k = 0; k < dim; k++) arb_init(error_norms[k]);
arb_mat_init(e, dim, 1);
arb_t z1, z2;
arb_init(z1);
arb_init(z2);
for(int j = 0; j < dim; j++) {
arb_mat_set(B, A);
for(int k = 0; k < dim; k++) {
arb_sub(B(k, k), B(k, k), D(j), prec);
}
for(int k = 0; k < dim; k++) {
arb_set(arb_mat_entry(e, k, 0), P(k, j));
}
arb_mat_L2norm(z2, e, prec);
arb_mat_mul(e, B, e, prec);
arb_mat_L2norm(error_norms[j], e, prec);
arb_div(z2, error_norms[j], z2, prec); // and now z1 is an upper bound for the
// error in the eigenvalue
arb_add_error(D(j), z2);
}
int unique_eigenvalues = 1;
for(int j = 0; j < dim; j++) {
if(j == 0) {
arb_sub(z1, D(j), D(1), prec);
}
else {
arb_sub(z1, D(j), D(0), prec);
}
arb_get_abs_lbound_arf(x1, z1, prec);
for(int k = 1; k < dim; k++) {
if(k == j) continue;
arb_sub(z1, D(j), D(k), prec);
arb_get_abs_lbound_arf(x2, z1, prec);
if(arf_cmp(x2, x1) < 0) {
arf_set(x1, x2);
}
}
if(arf_is_zero(x1)) {
unique_eigenvalues = 0;
}
arb_div_arf(z1, error_norms[j], x1, prec);
for(int k = 0; k < dim; k++) {
arb_add_error(P(k, j), z1);
}
}
arb_mat_clear(e);
arb_clear(z1);
arb_clear(z2);
for(int k = 0; k < dim; k++) arb_clear(error_norms[k]);
arf_clear(x1);
arf_clear(x2);
arb_mat_clear(B);
for(int k = 0; k < dim; k++) {
arf_clear(B1[k]);
arf_clear(B2[k]);
}
for(int k = 0; k < dim - 1; k++) {
arf_clear(row_max[k]);
}
arf_clear(Gii);
arf_clear(Gij);
arf_clear(Gji);
arf_clear(Gjj);
free(B1);
free(B2);
free(row_max);
free(row_max_indices);
if(unique_eigenvalues) return 0;
else return 1;
#undef B
#undef D
#undef P
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("floor....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 10000; iter++)
{
arf_t x, y;
int result;
arf_init(x);
arf_init(y);
arf_randtest_special(x, state, 2000, 100);
arf_randtest_special(y, state, 2000, 100);
arf_floor(y, x);
result = 1;
if (arf_is_int(x) || !arf_is_finite(x))
{
result = arf_equal(y, x);
}
else if (!arf_is_int(y))
{
result = 0;
}
else if (arf_cmp(y, x) >= 0)
{
result = 0;
}
else
{
arf_t s, t[3];
/* check floor(x) - x + 1 > 0 */
arf_init(s);
arf_init(t[0]);
arf_init(t[1]);
arf_init(t[2]);
arf_set(t[0], y);
arf_neg(t[1], x);
arf_one(t[2]);
arf_sum(s, (arf_ptr) t, 3, 32, ARF_RND_DOWN);
result = arf_sgn(s) > 0;
arf_clear(s);
arf_clear(t[0]);
arf_clear(t[1]);
arf_clear(t[2]);
}
if (!result)
{
flint_printf("FAIL!\n");
flint_printf("x = "); arf_print(x); flint_printf("\n\n");
flint_printf("y = "); arf_print(y); flint_printf("\n\n");
abort();
}
arf_floor(x, x);
if (!arf_equal(x, y))
{
flint_printf("FAIL (aliasing)!\n");
flint_printf("x = "); arf_print(x); flint_printf("\n\n");
flint_printf("y = "); arf_print(y); flint_printf("\n\n");
abort();
}
arf_clear(x);
arf_clear(y);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
void
acb_inv(acb_t res, const acb_t z, slong prec)
{
mag_t am, bm;
slong hprec;
#define a arb_midref(acb_realref(z))
#define b arb_midref(acb_imagref(z))
#define x arb_radref(acb_realref(z))
#define y arb_radref(acb_imagref(z))
/* choose precision for the floating-point approximation of a^2+b^2 so
that the double rounding result in less than
2 ulp error; also use at least MAG_BITS bits since the
value will be recycled for error bounds */
hprec = FLINT_MAX(prec + 3, MAG_BITS);
if (arb_is_zero(acb_imagref(z)))
{
arb_inv(acb_realref(res), acb_realref(z), prec);
arb_zero(acb_imagref(res));
return;
}
if (arb_is_zero(acb_realref(z)))
{
arb_inv(acb_imagref(res), acb_imagref(z), prec);
arb_neg(acb_imagref(res), acb_imagref(res));
arb_zero(acb_realref(res));
return;
}
if (!acb_is_finite(z))
{
acb_indeterminate(res);
return;
}
if (mag_is_zero(x) && mag_is_zero(y))
{
int inexact;
arf_t a2b2;
arf_init(a2b2);
inexact = arf_sosq(a2b2, a, b, hprec, ARF_RND_DOWN);
if (arf_is_special(a2b2))
{
acb_indeterminate(res);
}
else
{
_arb_arf_div_rounded_den(acb_realref(res), a, a2b2, inexact, prec);
_arb_arf_div_rounded_den(acb_imagref(res), b, a2b2, inexact, prec);
arf_neg(arb_midref(acb_imagref(res)), arb_midref(acb_imagref(res)));
}
arf_clear(a2b2);
return;
}
mag_init(am);
mag_init(bm);
/* first bound |a|-x, |b|-y */
arb_get_mag_lower(am, acb_realref(z));
arb_get_mag_lower(bm, acb_imagref(z));
if ((mag_is_zero(am) && mag_is_zero(bm)))
{
acb_indeterminate(res);
}
else
{
/*
The propagated error in the real part is given exactly by
(a+x')/((a+x')^2+(b+y'))^2 - a/(a^2+b^2) = P / Q,
P = [(b^2-a^2) x' - a (x'^2+y'^2 + 2y'b)]
Q = [(a^2+b^2)((a+x')^2+(b+y')^2)]
where |x'| <= x and |y'| <= y, and analogously for the imaginary part.
*/
mag_t t, u, v, w;
arf_t a2b2;
int inexact;
mag_init(t);
mag_init(u);
mag_init(v);
mag_init(w);
arf_init(a2b2);
inexact = arf_sosq(a2b2, a, b, hprec, ARF_RND_DOWN);
/* compute denominator */
/* t = (|a|-x)^2 + (|b|-x)^2 (lower bound) */
mag_mul_lower(t, am, am);
mag_mul_lower(u, bm, bm);
mag_add_lower(t, t, u);
/* u = a^2 + b^2 (lower bound) */
arf_get_mag_lower(u, a2b2);
/* t = ((|a|-x)^2 + (|b|-x)^2)(a^2 + b^2) (lower bound) */
mag_mul_lower(t, t, u);
/* compute numerator */
/* real: |a^2-b^2| x + |a| ((x^2 + y^2) + 2 |b| y)) */
/* imag: |a^2-b^2| y + |b| ((x^2 + y^2) + 2 |a| x)) */
/* am, bm = upper bounds for a, b */
arf_get_mag(am, a);
arf_get_mag(bm, b);
/* v = x^2 + y^2 */
mag_mul(v, x, x);
mag_addmul(v, y, y);
/* u = |a| ((x^2 + y^2) + 2 |b| y) */
mag_mul_2exp_si(u, bm, 1);
mag_mul(u, u, y);
mag_add(u, u, v);
mag_mul(u, u, am);
/* v = |b| ((x^2 + y^2) + 2 |a| x) */
mag_mul_2exp_si(w, am, 1);
mag_addmul(v, w, x);
mag_mul(v, v, bm);
/* w = |b^2 - a^2| (upper bound) */
if (arf_cmpabs(a, b) >= 0)
mag_mul(w, am, am);
else
mag_mul(w, bm, bm);
mag_addmul(u, w, x);
mag_addmul(v, w, y);
mag_div(arb_radref(acb_realref(res)), u, t);
mag_div(arb_radref(acb_imagref(res)), v, t);
_arb_arf_div_rounded_den_add_err(acb_realref(res), a, a2b2, inexact, prec);
_arb_arf_div_rounded_den_add_err(acb_imagref(res), b, a2b2, inexact, prec);
arf_neg(arb_midref(acb_imagref(res)), arb_midref(acb_imagref(res)));
mag_clear(t);
mag_clear(u);
mag_clear(v);
mag_clear(w);
arf_clear(a2b2);
}
mag_clear(am);
mag_clear(bm);
#undef a
#undef b
#undef x
#undef y
}