int main()
{
slong iter;
flint_rand_t state;
flint_printf("set_interval_mpfr....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 100000 * arb_test_multiplier(); iter++)
{
arb_t x;
arf_t a, b;
mpfr_t aa, bb;
arb_init(x);
arf_init(a);
arf_init(b);
mpfr_init2(aa, 200);
mpfr_init2(bb, 200);
arf_randtest_special(a, state, 200, 10);
arf_randtest_special(b, state, 200, 10);
if (arf_cmp(a, b) > 0)
arf_swap(a, b);
arf_get_mpfr(aa, a, MPFR_RNDD);
arf_get_mpfr(bb, b, MPFR_RNDU);
arb_set_interval_mpfr(x, aa, bb, 2 + n_randint(state, 200));
if (!arb_contains_arf(x, a) || !arb_contains_arf(x, b))
{
flint_printf("FAIL:\n\n");
flint_printf("x = "); arb_print(x); flint_printf("\n\n");
flint_printf("a = "); arf_print(a); flint_printf("\n\n");
flint_printf("b = "); arf_print(b); flint_printf("\n\n");
flint_abort();
}
arb_clear(x);
arf_clear(a);
arf_clear(b);
mpfr_clear(aa);
mpfr_clear(bb);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
static __inline__ void
zeta_bsplit_init(zeta_bsplit_t S)
{
arb_init(S->A);
arb_init(S->B);
arb_init(S->C);
arb_init(S->D);
arb_init(S->Q1);
arb_init(S->Q2);
arb_init(S->Q3);
}
static void
bound_rfac(arb_ptr F, const acb_t s, ulong n, slong len, slong wp)
{
if (len == 1)
{
acb_rising_ui_get_mag(arb_radref(F), s, n);
arf_set_mag(arb_midref(F), arb_radref(F));
mag_zero(arb_radref(F + 0));
}
else
{
arb_struct sx[2];
arb_init(sx + 0);
arb_init(sx + 1);
acb_abs(sx + 0, s, wp);
arb_one(sx + 1);
_arb_vec_zero(F, len);
_arb_poly_rising_ui_series(F, sx, 2, n, len, wp);
arb_clear(sx + 0);
arb_clear(sx + 1);
}
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("const_e....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 250 * arb_test_multiplier(); iter++)
{
arb_t r;
mpfr_t s;
slong accuracy, prec;
prec = 2 + n_randint(state, 1 << n_randint(state, 16));
arb_init(r);
mpfr_init2(s, prec + 1000);
arb_const_e(r, prec);
mpfr_set_ui(s, 1, MPFR_RNDN);
mpfr_exp(s, s, MPFR_RNDN);
if (!arb_contains_mpfr(r, s))
{
flint_printf("FAIL: containment\n\n");
flint_printf("prec = %wd\n", prec);
flint_printf("r = "); arb_printd(r, prec / 3.33); flint_printf("\n\n");
flint_abort();
}
accuracy = arb_rel_accuracy_bits(r);
if (accuracy < prec - 4)
{
flint_printf("FAIL: poor accuracy\n\n");
flint_printf("prec = %wd\n", prec);
flint_printf("r = "); arb_printd(r, prec / 3.33); flint_printf("\n\n");
flint_abort();
}
arb_clear(r);
mpfr_clear(s);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
void
_arb_poly_evaluate_horner(arb_t y, arb_srcptr f, slong len,
const arb_t x, slong prec)
{
if (len == 0)
{
arb_zero(y);
}
else if (len == 1 || arb_is_zero(x))
{
arb_set_round(y, f, prec);
}
else if (len == 2)
{
arb_mul(y, x, f + 1, prec);
arb_add(y, y, f + 0, prec);
}
else
{
slong i = len - 1;
arb_t t, u;
arb_init(t);
arb_init(u);
arb_set(u, f + i);
for (i = len - 2; i >= 0; i--)
{
arb_mul(t, u, x, prec);
arb_add(u, f + i, t, prec);
}
arb_swap(y, u);
arb_clear(t);
arb_clear(u);
}
}
renf_elem_class::operator double() const noexcept
{
if (nf == nullptr)
{
arb_t s;
arb_init(s);
arb_set_fmpq(s, b, 128);
double ans = arf_get_d(arb_midref(s), ARF_RND_NEAR);
arb_clear(s);
return ans;
}
else
return renf_elem_get_d(a, nf->renf_t(), ARF_RND_NEAR);
}
int main()
{
int iter;
FLINT_TEST_INIT(state);
test_field1(state);
test_field2(state);
for (iter = 0; iter < 100; iter++)
{
renf_t nf;
renf_elem_t a;
fmpz_t f;
arb_t e;
fmpz_init(f);
arb_init(e);
renf_randtest(nf, state,
2 + n_randint(state, 20), /* length */
8 + n_randint(state, 2408), /* prec */
10 + n_randint(state, 10) /* bits */
);
renf_elem_init(a, nf);
renf_elem_randtest(a, state, 30 + n_randint(state, 10), nf);
renf_elem_ceil(f, a, nf);
arb_sub_fmpz(e, a->emb, f, 1024);
if (arb_is_positive(e))
{
printf("FAIL:\n");
abort();
}
arb_add_ui(e, e, 1, 1024);
if (arb_is_negative(e))
{
printf("FAIL:\n");
abort();
}
renf_elem_clear(a, nf);
renf_clear(nf);
fmpz_clear(f);
arb_clear(e);
}
FLINT_TEST_CLEANUP(state);
return 0;
}
/* TODO: BSGS can reduce to nv mul */
void
acb_dirichlet_arb_quadratic_powers(arb_ptr v, slong nv, const arb_t x, slong prec)
{
slong i;
arb_t dx, x2;
arb_init(dx);
arb_init(x2);
arb_set(dx, x);
arb_mul(x2, x, x, prec);
for (i = 0; i < nv; i++)
{
if (i == 0)
arb_one(v + i);
else if (i == 1)
arb_set_round(v + i, x, prec);
else
{
arb_mul(dx, dx, x2, prec);
arb_mul(v + i, v + i - 1, dx, prec);
}
}
arb_clear(dx);
arb_clear(x2);
}
std::string renf_elem_class::to_string(int flags) const noexcept
{
std::string s;
assert(!((flags & EANTIC_STR_D) && (flags & EANTIC_STR_ARB)));
// call to renf_elem_get_str_pretty
if (nf == nullptr)
{
if (flags & EANTIC_STR_ALG)
{
char * u = fmpq_get_str(nullptr, 10, b);
s += u;
flint_free(u);
if (flags & (EANTIC_STR_D | EANTIC_STR_ARB)) s += " ~ ";
}
if (flags & EANTIC_STR_D)
{
char * u = (char *) flint_malloc(20 * sizeof(char));
sprintf(u, "%lf", fmpq_get_d(b));
s += u;
flint_free(u);
}
if (flags & EANTIC_STR_ARB)
{
char * u;
arb_t x;
arb_init(x);
arb_set_fmpq(x, b, 128);
u = arb_get_str(x, 64, 0);
s += u;
arb_clear(x);
flint_free(u);
}
}
else
{
char * u = renf_elem_get_str_pretty(renf_elem_t(), parent().gen_name().c_str(), parent().renf_t(), 10, flags);
s += u;
flint_free(u);
}
if (flags != EANTIC_STR_ALG && flags != EANTIC_STR_D && flags != EANTIC_STR_ARB)
return "(" + s + ")";
else
return s;
}
void
_arb_poly_tan_series(arb_ptr g,
arb_srcptr h, slong hlen, slong len, slong prec)
{
hlen = FLINT_MIN(hlen, len);
if (hlen == 1)
{
arb_tan(g, h, prec);
_arb_vec_zero(g + 1, len - 1);
}
else if (len == 2)
{
arb_t t;
arb_init(t);
arb_tan(g, h, prec);
arb_mul(t, g, g, prec);
arb_add_ui(t, t, 1, prec);
arb_mul(g + 1, t, h + 1, prec); /* safe since hlen >= 2 */
arb_clear(t);
}
else
{
arb_ptr t, u;
t = _arb_vec_init(2 * len);
u = t + len;
NEWTON_INIT(TAN_NEWTON_CUTOFF, len)
NEWTON_BASECASE(n)
_arb_poly_sin_cos_series_basecase(t, u, h, hlen, n, prec, 0);
_arb_poly_div_series(g, t, n, u, n, n, prec);
NEWTON_END_BASECASE
NEWTON_LOOP(m, n)
_arb_poly_mullow(u, g, m, g, m, n, prec);
arb_add_ui(u, u, 1, prec);
_arb_poly_atan_series(t, g, m, n, prec);
_arb_poly_sub(t + m, h + m, FLINT_MAX(0, hlen - m), t + m, n - m, prec);
_arb_poly_mullow(g + m, u, n, t + m, n - m, n - m, prec);
NEWTON_END_LOOP
NEWTON_END
_arb_vec_clear(t, 2 * len);
}
}
void
nd_axis_init(nd_axis_t axis,
const char *name, int total_count, column_reduction_t r,
int component_axis_count, struct nd_component_axis *component_axes,
slong prec)
{
int i, idx;
/* set compound axis hack data */
axis->component_axis_count = component_axis_count;
axis->component_axes = component_axes;
/* set the name */
axis->name = malloc(strlen(name) + 1);
strcpy(axis->name, name);
/* set the counts */
axis->n = total_count;
axis->k = r->selection_len;
/* allocate some bookkeeping vectors */
axis->selection = calloc(axis->k, sizeof(int));
axis->is_selected = calloc(axis->n, sizeof(int));
axis->request_update = calloc(axis->n, sizeof(int));
/* initialize the bookkeeping vectors */
for (i = 0; i < axis->k; i++)
{
idx = r->selection[i];
axis->selection[i] = idx;
axis->is_selected[idx] = 1;
axis->request_update[idx] = 1;
}
/* initialize the arbitrary precision weight arrays */
if (r->agg_mode == AGG_NONE)
{
axis->agg_weights = NULL;
}
else
{
axis->agg_weights = _arb_vec_init(axis->n);
arb_init(axis->agg_weight_divisor);
}
/* initialize weight values */
nd_axis_update_precision(axis, r, prec);
}
/* The floor+vec method *requires* n <= 1498 for floor(p(n)/2^64)
to be equal to floor(T/2^64). It is faster up to n ~= 1200.
With doubles, it is faster up to n ~= 500. */
void
_partitions_fmpz_ui(fmpz_t res, ulong n, int use_doubles)
{
if (n < NUMBER_OF_SMALL_PARTITIONS)
{
fmpz_set_ui(res, partitions_lookup[n]);
}
else if (FLINT_BITS == 64 && (n < 500 || (!use_doubles && n < 1200)))
{
mp_ptr tmp = flint_malloc((n + 1) * sizeof(mp_limb_t));
if (n < 417) /* p(n) < 2^64 */
{
partitions_vec(tmp, n + 1);
fmpz_set_ui(res, tmp[n]);
}
else
{
arb_t x;
arb_init(x);
fmpz_set_ui(res, n);
partitions_leading_fmpz(x, res, 4 * sqrt(n) - 50);
arb_mul_2exp_si(x, x, -64);
arb_floor(x, x, 4 * sqrt(n) - 50);
if (arb_get_unique_fmpz(res, x))
{
fmpz_mul_2exp(res, res, 64);
partitions_vec(tmp, n + 1);
fmpz_add_ui(res, res, tmp[n]);
}
else
{
flint_printf("warning: failed at %wu\n", n);
fmpz_set_ui(res, n);
partitions_fmpz_fmpz_hrr(res, res, use_doubles);
}
arb_clear(x);
}
flint_free(tmp);
}
else
{
fmpz_set_ui(res, n);
partitions_fmpz_fmpz_hrr(res, res, use_doubles);
}
}
/* compose by poly2 = a*x^n + c, no aliasing; n >= 1 */
void
_arb_poly_compose_axnc(arb_ptr res, arb_srcptr poly1, slong len1,
const arb_t c, const arb_t a, slong n, slong prec)
{
slong i;
_arb_vec_set_round(res, poly1, len1, prec);
/* shift by c (c = 0 case will be fast) */
_arb_poly_taylor_shift(res, c, len1, prec);
/* multiply by powers of a */
if (!arb_is_one(a))
{
if (arb_equal_si(a, -1))
{
for (i = 1; i < len1; i += 2)
arb_neg(res + i, res + i);
}
else if (len1 == 2)
{
arb_mul(res + 1, res + 1, a, prec);
}
else
{
arb_t t;
arb_init(t);
arb_set(t, a);
for (i = 1; i < len1; i++)
{
arb_mul(res + i, res + i, t, prec);
if (i + 1 < len1)
arb_mul(t, t, a, prec);
}
arb_clear(t);
}
}
/* stretch */
for (i = len1 - 1; i >= 1 && n > 1; i--)
{
arb_swap(res + i * n, res + i);
_arb_vec_zero(res + (i - 1) * n + 1, n - 1);
}
}
static void
_acb_hypgeom_li_offset(acb_t res, const acb_t z, long prec)
{
if (acb_is_int(z) && arf_cmp_2exp_si(arb_midref(acb_realref(z)), 1) == 0)
{
acb_zero(res);
}
else
{
arb_t t;
arb_init(t);
_acb_hypgeom_const_li2(t, prec);
_acb_hypgeom_li(res, z, prec);
arb_sub(acb_realref(res), acb_realref(res), t, prec);
arb_clear(t);
}
}
void test_field1(flint_rand_t state)
{
/* tests in QQ[sqrt(5)] */
int iter;
fmpq_t k;
fmpq_poly_t p;
arb_t emb;
renf_t nf;
renf_elem_t a;
fmpq_poly_init(p);
fmpq_poly_set_coeff_si(p, 2, 1);
fmpq_poly_set_coeff_si(p, 1, -1);
fmpq_poly_set_coeff_si(p, 0, -1);
arb_init(emb);
arb_set_d(emb, 1.61803398874989);
arb_add_error_2exp_si(emb, -20);
renf_init(nf, p, emb, 20 + n_randint(state, 20));
arb_clear(emb);
renf_elem_init(a, nf);
fmpq_init(k);
/* (1+sqrt(5))/2 vs Fibonacci */
fmpq_poly_zero(p);
fmpq_poly_set_coeff_si(p, 1, -1);
for (iter = 1; iter < 50; iter++)
{
fprintf(stderr, "start iter = %d\n", iter);
fflush(stderr);
fmpz_fib_ui(fmpq_numref(k), iter+1);
fmpz_fib_ui(fmpq_denref(k), iter);
fmpq_poly_set_coeff_fmpq(p, 0, k);
renf_elem_set_fmpq_poly(a, p, nf);
check_ceil(a, nf, 1 - iter % 2, "sqrt(5)");
fprintf(stderr, "end\n");
fflush(stderr);
}
renf_elem_clear(a, nf);
renf_clear(nf);
fmpq_clear(k);
fmpq_poly_clear(p);
}
void
arb_bernoulli_fmpz(arb_t res, const fmpz_t n, slong prec)
{
if (fmpz_cmp_ui(n, UWORD_MAX) <= 0)
{
if (fmpz_sgn(n) >= 0)
arb_bernoulli_ui(res, fmpz_get_ui(n), prec);
else
arb_zero(res);
}
else if (fmpz_is_odd(n))
{
arb_zero(res);
}
else
{
arb_t t;
slong wp;
arb_init(t);
wp = prec + 2 * fmpz_bits(n);
/* zeta(n) ~= 1 */
arf_one(arb_midref(res));
mag_one(arb_radref(res));
mag_mul_2exp_si(arb_radref(res), arb_radref(res), WORD_MIN);
/* |B_n| = 2 * n! / (2*pi)^n * zeta(n) */
arb_gamma_fmpz(t, n, wp);
arb_mul_fmpz(t, t, n, wp);
arb_mul(res, res, t, wp);
arb_const_pi(t, wp);
arb_mul_2exp_si(t, t, 1);
arb_pow_fmpz(t, t, n, wp);
arb_div(res, res, t, prec);
arb_mul_2exp_si(res, res, 1);
if (fmpz_fdiv_ui(n, 4) == 0)
arb_neg(res, res);
arb_clear(t);
}
}
slong renf_set_embeddings_fmpz_poly(renf * nf, fmpz_poly_t pol, slong lim, slong prec)
{
slong i, n, n_exact, n_interval;
fmpq_poly_t p2;
arb_t a;
fmpz * c;
slong * k;
n = fmpz_poly_num_real_roots_upper_bound(pol);
c = _fmpz_vec_init(n);
k = (slong *) flint_malloc(n * sizeof(slong));
fmpz_poly_isolate_real_roots(NULL, &n_exact, c, k, &n_interval, pol);
if (n_exact)
{
fprintf(stderr, "ERROR (fmpz_poly_real_embeddings): rational roots\n");
abort();
}
arb_init(a);
fmpq_poly_init(p2);
fmpz_one(fmpq_poly_denref(p2));
fmpq_poly_fit_length(p2, pol->length);
_fmpz_vec_set(p2->coeffs, pol->coeffs, pol->length);
p2->length = pol->length;
for (i = 0; i < FLINT_MIN(lim, n_interval); i++)
{
arb_set_fmpz(a, c + i);
arb_mul_2exp_si(a, a, 1);
arb_add_si(a, a, 1, prec);
mag_one(arb_radref(a));
arb_mul_2exp_si(a, a, k[i] - 1);
renf_init(nf + i, p2, a, prec);
}
arb_clear(a);
fmpq_poly_clear(p2);
_fmpz_vec_clear(c, n);
flint_free(k);
return n_interval;
}
void
arb_fac2_ui(arb_t res, ulong n, long prec)
{
if (n % 2 == 0)
{
arb_fac_ui(res, n / 2, prec);
arb_mul_2exp_si(res, res, n / 2);
}
else
{
arb_t t;
arb_init(t);
arb_fac2_ui(t, n - 1, prec + 5);
arb_fac_ui(res, n, prec + 5);
arb_div(res, res, t, prec);
arb_clear(t);
}
}
void
_arb_pow_exp(arb_t z, const arb_t x, int negx, const arb_t y, long prec)
{
arb_t t;
arb_init(t);
if (negx)
{
arb_neg(t, x);
arb_log(t, t, prec);
}
else
arb_log(t, x, prec);
arb_mul(t, t, y, prec);
arb_exp(z, t, prec);
arb_clear(t);
}
void check_init(fmpq_poly_t p, double demb, double rad1, double rad2, slong iter, flint_rand_t state)
{
renf_t nf;
arb_t emb;
slong prec, i;
arb_init(emb);
for (i = 0; i < iter; i++)
{
prec = 8 + n_randint(state, 2048);
randomized_embedding(emb, demb, rad1, rad2);
renf_init(nf, p, emb, prec);
check_renf(nf);
renf_clear(nf);
}
arb_clear(emb);
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("zeta_ui_asymp....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 10000; iter++)
{
arb_t r;
ulong n;
mpfr_t s;
slong prec;
prec = 2 + n_randint(state, 1 << n_randint(state, 10));
arb_init(r);
mpfr_init2(s, prec + 100);
n = 2 + n_randint(state, 1 << n_randint(state, 10));
arb_zeta_ui_asymp(r, n, prec);
mpfr_zeta_ui(s, (unsigned long) FLINT_MIN(n, ULONG_MAX), MPFR_RNDN);
if (!arb_contains_mpfr(r, s))
{
flint_printf("FAIL: containment\n\n");
flint_printf("n = %wu\n\n", n);
flint_printf("r = "); arb_printd(r, prec / 3.33); flint_printf("\n\n");
flint_printf("s = "); mpfr_printf("%.275Rf\n", s); flint_printf("\n\n");
abort();
}
arb_clear(r);
mpfr_clear(s);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
int
main(void)
{
int i, result;
flint_rand_t state;
printf("get_coeff_ptr....");
fflush(stdout);
flint_randinit(state);
for (i = 0; i < 1000; i++)
{
arb_poly_t A;
arb_t a;
long n = n_randint(state, 100);
arb_poly_init(A);
arb_poly_randtest(A, state, n_randint(state, 100), 100, 10);
arb_init(a);
arb_poly_get_coeff_arb(a, A, n);
result = n < arb_poly_length(A) ?
arb_equal(a, arb_poly_get_coeff_ptr(A, n)) :
arb_poly_get_coeff_ptr(A, n) == NULL;
if (!result)
{
printf("FAIL:\n");
printf("A = "), arb_poly_printd(A, 10), printf("\n\n");
printf("a = "), arb_print(a), printf("\n\n");
printf("n = %ld\n\n", n);
abort();
}
arb_poly_clear(A);
arb_clear(a);
}
flint_randclear(state);
flint_cleanup();
printf("PASS\n");
return 0;
}
int
f_aj(arb_t m, const arb_t t, params_t * p, slong prec)
{
slong k;
acb_t z, zu;
arb_t abs;
arb_init(abs);
acb_init(z);
acb_init(zu);
arb_const_pi(abs, prec);
arb_mul_2exp_si(abs, abs, -2); /* Pi/4 */
arb_set(acb_realref(z), t);
arb_set(acb_imagref(z), abs);
acb_sinh(z, z, prec);
arb_mul_2exp_si(abs, abs, 1); /* Pi/2 */
acb_mul_arb(z, z, abs, prec);
acb_tanh(z, z, prec);
arb_one(m);
for (k = 0; k < p->len; k++)
{
acb_sub(zu, z, p->z + k, prec);
if (acb_contains_zero(zu))
{
arb_clear(abs);
acb_clear(zu);
acb_clear(z);
return 0;
}
acb_abs(abs, zu, prec);
arb_mul(m, m, abs, prec);
}
arb_inv(m, m, prec);
arb_clear(abs);
acb_clear(zu);
acb_clear(z);
return 1;
}
void
arb_poly_fit_length(arb_poly_t poly, slong len)
{
slong i;
if (len > poly->alloc)
{
if (len < 2 * poly->alloc)
len = 2 * poly->alloc;
poly->coeffs = flint_realloc(poly->coeffs,
len * sizeof(arb_struct));
for (i = poly->alloc; i < len; i++)
arb_init(poly->coeffs + i);
poly->alloc = len;
}
}
/* rate matrix and edge rates have already been updated */
static void
_update_transition_matrices(cross_site_ws_t w, slong prec)
{
slong i, j;
arb_t s;
arb_init(s);
for (i = 0; i < w->rate_category_count; i++)
{
for (j = 0; j < w->edge_count; j++)
{
arb_mat_struct *tmat;
tmat = cross_site_ws_transition_matrix(w, i, j);
arb_mul(s, w->rate_mix_rates + i, w->edge_rates + j, prec);
arb_mat_scalar_mul_arb(tmat, w->rate_matrix, s, prec);
arb_mat_exp(tmat, tmat, prec);
}
}
arb_clear(s);
}
/*
* Update the frechet matrix for each rate category and edge.
* At this point the rate matrix has been normalized
* to have zero row sums, but it has not been scaled
* by the edge rate coefficients.
* The frechet matrices must already have been initialized.
*/
static void
_update_state_pair_frechet_matrices(
cross_site_ws_t csw, model_and_data_t m,
nd_axis_struct *edge_axis,
slong first_state, slong second_state, slong prec)
{
arb_mat_t P, L, Q;
slong cat;
arb_t rate;
slong state_count = model_and_data_state_count(m);
slong edge_count = model_and_data_edge_count(m);
slong rate_category_count = model_and_data_rate_category_count(m);
arb_init(rate);
arb_mat_init(P, state_count, state_count);
arb_mat_init(L, state_count, state_count);
arb_mat_init(Q, state_count, state_count);
arb_set(arb_mat_entry(L, first_state, second_state),
arb_mat_entry(csw->rate_matrix, first_state, second_state));
for (cat = 0; cat < rate_category_count; cat++)
{
slong edge;
const arb_struct * cat_rate = csw->rate_mix_rates + cat;
for (edge = 0; edge < edge_count; edge++)
{
slong idx = m->edge_map->order[edge];
const arb_struct * edge_rate = csw->edge_rates + idx;
arb_mat_struct *fmat;
if (!edge_axis->request_update[edge]) continue;
fmat = cross_site_ws_trans_frechet_matrix(csw, cat, idx);
arb_mul(rate, edge_rate, cat_rate, prec);
arb_mat_scalar_mul_arb(Q, csw->rate_matrix, rate, prec);
_arb_mat_exp_frechet(P, fmat, Q, L, prec);
}
}
arb_clear(rate);
arb_mat_clear(P);
arb_mat_clear(L);
arb_mat_clear(Q);
}
void arb_twobytwo_diag(arb_t u1, arb_t u2, const arb_t a, const arb_t b, const arb_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(arb_contains_zero(b)) {
// this is not quite right (doesn't set error intervals)
arb_set_ui(u1, 1);
arb_set_ui(u2, 0);
return;
}
arb_t x; arb_init(x);
arb_mul(u1, b, b, prec); // u1 = b^2
arb_sub(u2, a, d, prec); // u2 = a - d
arb_mul_2exp_si(u2, u2, -1); // u2 = (a - d)/2
arb_mul(u2, u2, u2, prec); // u2 = ( (a - d)/2 )^2
arb_add(u1, u1, u2, prec); // u1 = b^2 + ( (a-d)/2 )^2
arb_sqrt(u1, u1, prec); // u1 = sqrt(above)
arb_mul_2exp_si(u1, u1, 1); // u1 = 2 (sqrt (above) )
arb_add(u1, u1, d, prec); // u1 += d
arb_sub(u1, u1, a, prec); // u1 -= a
arb_mul_2exp_si(u1, u1, -1); // u1 = (d - a)/2 + sqrt(b^2 + ( (a-d)/2 )^2)
arb_mul(x, u1, u1, prec);
arb_addmul(x, b, b, prec); // x = u1^2 + b^2
arb_sqrt(x, x, prec); // x = sqrt(u1^2 + b^2)
arb_div(u2, u1, x, prec);
arb_div(u1, b, x, prec);
arb_neg(u1, u1);
arb_clear(x);
}
/* series of c^(d+x) */
static __inline__ void
_arb_poly_pow_cpx(arb_ptr res, const arb_t c, const arb_t d, long trunc, long prec)
{
long i;
arb_t logc;
arb_init(logc);
arb_log(logc, c, prec);
arb_mul(res + 0, logc, d, prec);
arb_exp(res + 0, res + 0, prec);
for (i = 1; i < trunc; i++)
{
arb_mul(res + i, res + i - 1, logc, prec);
arb_div_ui(res + i, res + i, i, prec);
}
arb_clear(logc);
}
/* invalid in (-1,0) */
int
_acb_hypgeom_legendre_q_single_valid(const acb_t z)
{
arb_t t;
int ok;
if (!arb_contains_zero(acb_imagref(z)))
return 1;
if (arb_is_positive(acb_imagref(z)))
return 1;
arb_init(t);
arb_one(t);
arb_neg(t, t);
ok = arb_lt(acb_realref(z), t);
arb_clear(t);
return ok;
}
void
_arb_poly_compose_series(arb_ptr res, arb_srcptr poly1, slong len1,
arb_srcptr poly2, slong len2, slong n, slong prec)
{
if (len2 == 1)
{
arb_set_round(res, poly1, prec);
_arb_vec_zero(res + 1, n - 1);
}
else if (_arb_vec_is_zero(poly2 + 1, len2 - 2)) /* poly2 is a monomial */
{
slong i, j;
arb_t t;
arb_init(t);
arb_set(t, poly2 + len2 - 1);
arb_set_round(res, poly1, prec);
for (i = 1, j = len2 - 1; i < len1 && j < n; i++, j += len2 - 1)
{
arb_mul(res + j, poly1 + i, t, prec);
if (i + 1 < len1 && j + len2 - 1 < n)
arb_mul(t, t, poly2 + len2 - 1, prec);
}
if (len2 != 2)
for (i = 1; i < n; i++)
if (i % (len2 - 1) != 0)
arb_zero(res + i);
arb_clear(t);
}
else if (len1 < 6 || n < 6)
{
_arb_poly_compose_series_horner(res, poly1, len1, poly2, len2, n, prec);
}
else
{
_arb_poly_compose_series_brent_kung(res, poly1, len1, poly2, len2, n, prec);
}
}
