/* bound (1 + 1/m)^n, m > 0, n >= 0 */
void
mag_binpow_uiui(mag_t b, ulong m, ulong n)
{
mag_t t;
if (m == 0)
{
mag_inf(b);
return;
}
mag_init(t);
/* bound by exp(n/m) <= 1 + (n/m) + (n/m)^2 */
if (m > n)
{
mag_set_ui(t, n); /* x = n/m */
mag_div_ui(t, t, m);
mag_mul(b, t, t); /* x^2 */
mag_add(b, b, t); /* x */
mag_one(t);
mag_add(b, b, t); /* 1 */
}
else
{
mag_one(b);
mag_div_ui(b, b, m);
mag_one(t);
mag_add(t, t, b);
mag_pow_ui(b, t, n);
}
mag_clear(t);
}
static void
acb_rising_get_mag2_right(mag_t bound, const arb_t a, const arb_t b, ulong n)
{
mag_t t, u;
ulong k;
mag_init(t);
mag_init(u);
arb_get_mag(t, a);
arb_get_mag(u, b);
mag_mul(bound, t, t);
mag_addmul(bound, u, u);
mag_set(u, bound);
mag_mul_2exp_si(t, t, 1);
for (k = 1; k < n; k++)
{
mag_add_ui_2exp_si(u, u, 2 * k - 1, 0);
mag_add(u, u, t);
mag_mul(bound, bound, u);
}
mag_clear(t);
mag_clear(u);
}
static void
acb_hypgeom_mag_Cn(mag_t Cn, int R, const mag_t nu, const mag_t sigma, ulong n)
{
if (R == 1)
{
mag_one(Cn);
}
else
{
acb_hypgeom_mag_chi(Cn, n);
if (R == 3)
{
mag_t tmp;
mag_init(tmp);
mag_mul(tmp, nu, nu);
mag_mul(tmp, tmp, sigma);
if (n != 1)
mag_mul_ui(tmp, tmp, n);
mag_add(Cn, Cn, tmp);
mag_pow_ui(tmp, nu, n);
mag_mul(Cn, Cn, tmp);
mag_clear(tmp);
}
}
}
void
arb_mat_bound_inf_norm(mag_t b, const arb_mat_t A)
{
slong i, j, r, c;
mag_t s, t;
r = arb_mat_nrows(A);
c = arb_mat_ncols(A);
mag_zero(b);
if (r == 0 || c == 0)
return;
mag_init(s);
mag_init(t);
for (i = 0; i < r; i++)
{
mag_zero(s);
for (j = 0; j < c; j++)
{
arb_get_mag(t, arb_mat_entry(A, i, j));
mag_add(s, s, t);
}
mag_max(b, b, s);
}
mag_clear(s);
mag_clear(t);
}
/* Differential equation for F(a,b,c,y+z):
(y+z)(y-1+z) F''(z) + ((y+z)(a+b+1) - c) F'(z) + a b F(z) = 0
Coefficients in the Taylor series are bounded by
A * binomial(N+k, k) * nu^k
using the Cauchy-Kovalevskaya majorant method.
See J. van der Hoeven, "Fast evaluation of holonomic functions near
and in regular singularities"
*/
static void
bound(mag_t A, mag_t nu, mag_t N,
const acb_t a, const acb_t b, const acb_t c, const acb_t y,
const acb_t f0, const acb_t f1)
{
mag_t M0, M1, t, u;
acb_t d;
acb_init(d);
mag_init(M0);
mag_init(M1);
mag_init(t);
mag_init(u);
/* nu = max(1/|y-1|, 1/|y|) = 1/min(|y-1|, |y|) */
acb_get_mag_lower(t, y);
acb_sub_ui(d, y, 1, MAG_BITS);
acb_get_mag_lower(u, d);
mag_min(t, t, u);
mag_one(u);
mag_div(nu, u, t);
/* M0 = 2 nu |ab| */
acb_get_mag(t, a);
acb_get_mag(u, b);
mag_mul(M0, t, u);
mag_mul(M0, M0, nu);
mag_mul_2exp_si(M0, M0, 1);
/* M1 = 2 nu |(a+b+1)y-c| + 2|a+b+1| */
acb_add(d, a, b, MAG_BITS);
acb_add_ui(d, d, 1, MAG_BITS);
acb_get_mag(t, d);
acb_mul(d, d, y, MAG_BITS);
acb_sub(d, d, c, MAG_BITS);
acb_get_mag(u, d);
mag_mul(u, u, nu);
mag_add(M1, t, u);
mag_mul_2exp_si(M1, M1, 1);
/* N = max(sqrt(2 M0), 2 M1) / nu */
mag_mul_2exp_si(M0, M0, 1);
mag_sqrt(M0, M0);
mag_mul_2exp_si(M1, M1, 1);
mag_max(N, M0, M1);
mag_div(N, N, nu);
/* A = max(|f0|, |f1| / (nu (N+1)) */
acb_get_mag(t, f0);
acb_get_mag(u, f1);
mag_div(u, u, nu);
mag_div(u, u, N); /* upper bound for dividing by N+1 */
mag_max(A, t, u);
acb_clear(d);
mag_clear(M0);
mag_clear(M1);
mag_clear(t);
mag_clear(u);
}
void
mag_add_ui(mag_t y, const mag_t x, ulong k)
{
mag_t t;
mag_init(t); /* no need to free */
mag_set_ui(t, k);
mag_add(y, x, t);
}
void
arb_sub(arb_t z, const arb_t x, const arb_t y, slong prec)
{
int inexact;
inexact = arf_sub(arb_midref(z), arb_midref(x), arb_midref(y), prec, ARB_RND);
mag_add(arb_radref(z), arb_radref(x), arb_radref(y));
if (inexact)
arf_mag_add_ulp(arb_radref(z), arb_radref(z), arb_midref(z), prec);
}
void
arb_sinc(arb_t z, const arb_t x, slong prec)
{
mag_t c, r;
mag_init(c);
mag_init(r);
mag_set_ui_2exp_si(c, 5, -1);
arb_get_mag_lower(r, x);
if (mag_cmp(c, r) < 0)
{
/* x is not near the origin */
_arb_sinc_direct(z, x, prec);
}
else if (mag_cmp_2exp_si(arb_radref(x), 1) < 0)
{
/* determine error magnitude using the derivative bound */
if (arb_is_exact(x))
{
mag_zero(c);
}
else
{
_arb_sinc_derivative_bound(r, x);
mag_mul(c, arb_radref(x), r);
}
/* evaluate sinc at the midpoint of x */
if (arf_is_zero(arb_midref(x)))
{
arb_one(z);
}
else
{
arb_get_mid_arb(z, x);
_arb_sinc_direct(z, z, prec);
}
/* add the error */
mag_add(arb_radref(z), arb_radref(z), c);
}
else
{
/* x has a large radius and includes points near the origin */
arf_zero(arb_midref(z));
mag_one(arb_radref(z));
}
mag_clear(c);
mag_clear(r);
}
void
arb_root_ui_algebraic(arb_t res, const arb_t x, ulong k, slong prec)
{
mag_t r, msubr, m1k, t;
if (arb_is_exact(x))
{
arb_root_arf(res, arb_midref(x), k, prec);
return;
}
if (!arb_is_nonnegative(x))
{
arb_indeterminate(res);
return;
}
mag_init(r);
mag_init(msubr);
mag_init(m1k);
mag_init(t);
/* x = [m-r, m+r] */
mag_set(r, arb_radref(x));
/* m - r */
arb_get_mag_lower(msubr, x);
/* m^(1/k) */
arb_root_arf(res, arb_midref(x), k, prec);
/* bound for m^(1/k) */
arb_get_mag(m1k, res);
/* C = min(1, log(1+r/(m-r))/k) */
mag_div(t, r, msubr);
mag_log1p(t, t);
mag_div_ui(t, t, k);
if (mag_cmp_2exp_si(t, 0) > 0)
mag_one(t);
/* C m^(1/k) */
mag_mul(t, m1k, t);
mag_add(arb_radref(res), arb_radref(res), t);
mag_clear(r);
mag_clear(msubr);
mag_clear(m1k);
mag_clear(t);
}
int arb_calc_newton_step(arb_t xnew, arb_calc_func_t func,
void * param, const arb_t x, const arb_t conv_region,
const arf_t conv_factor, slong prec)
{
mag_t err, v;
arb_t t;
arb_struct u[2];
int result;
mag_init(err);
mag_init(v);
arb_init(t);
arb_init(u + 0);
arb_init(u + 1);
mag_mul(err, arb_radref(x), arb_radref(x));
arf_get_mag(v, conv_factor);
mag_mul(err, err, v);
arf_set(arb_midref(t), arb_midref(x));
mag_zero(arb_radref(t));
func(u, t, param, 2, prec);
arb_div(u, u, u + 1, prec);
arb_sub(u, t, u, prec);
mag_add(arb_radref(u), arb_radref(u), err);
if (arb_contains(conv_region, u) &&
(mag_cmp(arb_radref(u), arb_radref(x)) < 0))
{
arb_swap(xnew, u);
result = ARB_CALC_SUCCESS;
}
else
{
arb_set(xnew, x);
result = ARB_CALC_NO_CONVERGENCE;
}
arb_clear(t);
arb_clear(u);
arb_clear(u + 1);
mag_clear(err);
mag_clear(v);
return result;
}
void
arb_zeta_ui_borwein_bsplit(arb_t x, ulong s, slong prec)
{
zeta_bsplit_t sum;
mag_t err;
slong wp, n;
/* zeta(0) = -1/2 */
if (s == 0)
{
arb_set_si(x, -1);
arb_mul_2exp_si(x, x, -1);
return;
}
if (s == 1)
{
flint_printf("zeta_ui_borwein_bsplit: zeta(1)");
abort();
}
n = prec / ERROR_B + 2;
wp = prec + 30;
zeta_bsplit_init(sum);
zeta_bsplit(sum, 0, n + 1, n, s, 0, wp);
/* A/Q3 - B/Q3 / (C/Q1) = (A*C - B*Q1) / (Q3*C) */
arb_mul(sum->A, sum->A, sum->C, wp);
arb_mul(sum->B, sum->B, sum->Q1, wp);
arb_sub(sum->A, sum->A, sum->B, wp);
arb_mul(sum->Q3, sum->Q3, sum->C, wp);
arb_div(sum->C, sum->A, sum->Q3, wp);
mag_init(err);
mag_borwein_error(err, n);
mag_add(arb_radref(sum->C), arb_radref(sum->C), err);
mag_clear(err);
/* convert from eta(s) to zeta(s) */
arb_div_2expm1_ui(x, sum->C, s - 1, wp);
arb_mul_2exp_si(x, x, s - 1);
zeta_bsplit_clear(sum);
}
void
arb_hypgeom_infsum(arb_t P, arb_t Q, hypgeom_t hyp, long target_prec, long prec)
{
mag_t err, z;
long n;
mag_init(err);
mag_init(z);
mag_set_fmpz(z, hyp->P->coeffs + hyp->P->length - 1);
mag_div_fmpz(z, z, hyp->Q->coeffs + hyp->Q->length - 1);
if (!hyp->have_precomputed)
{
hypgeom_precompute(hyp);
hyp->have_precomputed = 1;
}
n = hypgeom_bound(err, hyp->r, hyp->boundC, hyp->boundD,
hyp->boundK, hyp->MK, z, target_prec);
arb_hypgeom_sum(P, Q, hyp, n, prec);
if (arf_sgn(arb_midref(Q)) < 0)
{
arb_neg(P, P);
arb_neg(Q, Q);
}
/* We have p/q = s + err i.e. (p + q*err)/q = s */
{
mag_t u;
mag_init(u);
arb_get_mag(u, Q);
mag_mul(u, u, err);
mag_add(arb_radref(P), arb_radref(P), u);
mag_clear(u);
}
mag_clear(z);
mag_clear(err);
}
void
arb_atan(arb_t z, const arb_t x, slong prec)
{
if (arb_is_exact(x))
{
arb_atan_arf(z, arb_midref(x), prec);
}
else
{
mag_t t, u;
mag_init(t);
mag_init(u);
arb_get_mag_lower(t, x);
if (mag_is_zero(t))
{
mag_set(t, arb_radref(x));
}
else
{
mag_mul_lower(t, t, t);
mag_one(u);
mag_add_lower(t, t, u);
mag_div(t, arb_radref(x), t);
}
if (mag_cmp_2exp_si(t, 0) > 0)
{
mag_const_pi(u);
mag_min(t, t, u);
}
arb_atan_arf(z, arb_midref(x), prec);
mag_add(arb_radref(z), arb_radref(z), t);
mag_clear(t);
mag_clear(u);
}
}
void
arb_log(arb_t y, const arb_t x, slong prec)
{
if (arb_is_exact(x))
{
arb_log_arf(y, arb_midref(x), prec);
}
else
{
/*
Let the input be [a-b, a+b]. We require a > b >= 0 (otherwise the
interval contains zero or a negative number and the logarithm is not
defined). The error is largest at a-b, and we have
log(a) - log(a-b) = log(1 + b/(a-b)).
*/
mag_t err;
mag_init(err);
arb_get_mag_lower_nonnegative(err, x);
if (mag_is_zero(err))
{
mag_inf(err);
}
else
{
mag_div(err, arb_radref(x), err);
mag_log1p(err, err);
}
arb_log_arf(y, arb_midref(x), prec);
mag_add(arb_radref(y), arb_radref(y), err);
mag_clear(err);
}
}
void
_arb_bell_sum_taylor(arb_t res, const fmpz_t n,
const fmpz_t a, const fmpz_t b, const fmpz_t mmag, long tol)
{
fmpz_t m, r, R, tmp;
mag_t B, C, D, bound;
arb_t t, u;
long wp, k, N;
if (_fmpz_sub_small(b, a) < 5)
{
arb_bell_sum_bsplit(res, n, a, b, mmag, tol);
return;
}
fmpz_init(m);
fmpz_init(r);
fmpz_init(R);
fmpz_init(tmp);
/* r = max(m - a, b - m) */
/* m = a + (b - a) / 2 */
fmpz_sub(r, b, a);
fmpz_cdiv_q_2exp(r, r, 1);
fmpz_add(m, a, r);
fmpz_mul_2exp(R, r, RADIUS_BITS);
mag_init(B);
mag_init(C);
mag_init(D);
mag_init(bound);
arb_init(t);
arb_init(u);
if (fmpz_cmp(R, m) >= 0)
{
mag_inf(C);
mag_inf(D);
}
else
{
/* C = exp(R * |F'(m)| + (1/2) R^2 * (n/(m-R)^2 + 1/(m-R))) */
/* C = exp(R * (|F'(m)| + (1/2) R * (n/(m-R) + 1)/(m-R))) */
/* D = (1/2) R * (n/(m-R) + 1)/(m-R) */
fmpz_sub(tmp, m, R);
mag_set_fmpz(D, n);
mag_div_fmpz(D, D, tmp);
mag_one(C);
mag_add(D, D, C);
mag_div_fmpz(D, D, tmp);
mag_mul_fmpz(D, D, R);
mag_mul_2exp_si(D, D, -1);
/* C = |F'(m)| */
wp = 20 + 1.05 * fmpz_bits(n);
arb_set_fmpz(t, n);
arb_div_fmpz(t, t, m, wp);
fmpz_add_ui(tmp, m, 1);
arb_set_fmpz(u, tmp);
arb_digamma(u, u, wp);
arb_sub(t, t, u, wp);
arb_get_mag(C, t);
/* C = exp(R * (C + D)) */
mag_add(C, C, D);
mag_mul_fmpz(C, C, R);
mag_exp(C, C);
}
if (mag_cmp_2exp_si(C, tol / 4 + 2) > 0)
{
_arb_bell_sum_taylor(res, n, a, m, mmag, tol);
_arb_bell_sum_taylor(t, n, m, b, mmag, tol);
arb_add(res, res, t, 2 * tol);
}
else
{
arb_ptr mx, ser1, ser2, ser3;
/* D = T(m) */
wp = 20 + 1.05 * fmpz_bits(n);
arb_set_fmpz(t, m);
arb_pow_fmpz(t, t, n, wp);
fmpz_add_ui(tmp, m, 1);
arb_gamma_fmpz(u, tmp, wp);
arb_div(t, t, u, wp);
arb_get_mag(D, t);
/* error bound: (b-a) * C * D * B^N / (1 - B), B = r/R */
/* ((b-a) * C * D * 2) * 2^(-N*RADIUS_BITS) */
/* ((b-a) * C * D * 2) */
mag_mul(bound, C, D);
mag_mul_2exp_si(bound, bound, 1);
fmpz_sub(tmp, b, a);
mag_mul_fmpz(bound, bound, tmp);
/* N = (tol + log2((b-a)*C*D*2) - mmag) / RADIUS_BITS */
if (mmag == NULL)
{
/* estimate D ~= 2^mmag */
fmpz_add_ui(tmp, MAG_EXPREF(C), tol);
fmpz_cdiv_q_ui(tmp, tmp, RADIUS_BITS);
}
else
{
fmpz_sub(tmp, MAG_EXPREF(bound), mmag);
fmpz_add_ui(tmp, tmp, tol);
fmpz_cdiv_q_ui(tmp, tmp, RADIUS_BITS);
}
if (fmpz_cmp_ui(tmp, 5 * tol / 4) > 0)
N = 5 * tol / 4;
else if (fmpz_cmp_ui(tmp, 2) < 0)
N = 2;
else
N = fmpz_get_ui(tmp);
/* multiply by 2^(-N*RADIUS_BITS) */
mag_mul_2exp_si(bound, bound, -N * RADIUS_BITS);
mx = _arb_vec_init(2);
ser1 = _arb_vec_init(N);
ser2 = _arb_vec_init(N);
ser3 = _arb_vec_init(N);
/* estimate (this should work for moderate n and tol) */
wp = 1.1 * tol + 1.05 * fmpz_bits(n) + 5;
/* increase precision until convergence */
while (1)
{
/* (m+x)^n / gamma(m+1+x) */
arb_set_fmpz(mx, m);
arb_one(mx + 1);
_arb_poly_log_series(ser1, mx, 2, N, wp);
for (k = 0; k < N; k++)
arb_mul_fmpz(ser1 + k, ser1 + k, n, wp);
arb_add_ui(mx, mx, 1, wp);
_arb_poly_lgamma_series(ser2, mx, 2, N, wp);
_arb_vec_sub(ser1, ser1, ser2, N, wp);
_arb_poly_exp_series(ser3, ser1, N, N, wp);
/* t = a - m, u = b - m */
arb_set_fmpz(t, a);
arb_sub_fmpz(t, t, m, wp);
arb_set_fmpz(u, b);
arb_sub_fmpz(u, u, m, wp);
arb_power_sum_vec(ser1, t, u, N, wp);
arb_zero(res);
for (k = 0; k < N; k++)
arb_addmul(res, ser3 + k, ser1 + k, wp);
if (mmag != NULL)
{
if (_fmpz_sub_small(MAG_EXPREF(arb_radref(res)), mmag) <= -tol)
break;
}
else
{
if (arb_rel_accuracy_bits(res) >= tol)
break;
}
wp = 2 * wp;
}
/* add the series truncation bound */
arb_add_error_mag(res, bound);
_arb_vec_clear(mx, 2);
_arb_vec_clear(ser1, N);
_arb_vec_clear(ser2, N);
_arb_vec_clear(ser3, N);
}
mag_clear(B);
mag_clear(C);
mag_clear(D);
mag_clear(bound);
arb_clear(t);
arb_clear(u);
fmpz_clear(m);
fmpz_clear(r);
fmpz_clear(R);
fmpz_clear(tmp);
}
void
acb_gamma_stirling_eval(acb_t s, const acb_t z, long nterms, int digamma, long prec)
{
acb_t t, logz, zinv, zinv2;
arb_t b;
mag_t err;
long k, term_prec;
double z_mag, term_mag;
acb_init(t);
acb_init(logz);
acb_init(zinv);
acb_init(zinv2);
arb_init(b);
acb_log(logz, z, prec);
acb_inv(zinv, z, prec);
nterms = FLINT_MAX(nterms, 1);
acb_zero(s);
if (nterms > 1)
{
acb_mul(zinv2, zinv, zinv, prec);
z_mag = arf_get_d(arb_midref(acb_realref(logz)), ARF_RND_UP) * 1.44269504088896;
for (k = nterms - 1; k >= 1; k--)
{
term_mag = bernoulli_bound_2exp_si(2 * k);
term_mag -= (2 * k - 1) * z_mag;
term_prec = prec + term_mag;
term_prec = FLINT_MIN(term_prec, prec);
term_prec = FLINT_MAX(term_prec, 10);
arb_gamma_stirling_coeff(b, k, digamma, term_prec);
if (prec > 2000)
{
acb_set_round(t, zinv2, term_prec);
acb_mul(s, s, t, term_prec);
}
else
acb_mul(s, s, zinv2, term_prec);
arb_add(acb_realref(s), acb_realref(s), b, term_prec);
}
if (digamma)
acb_mul(s, s, zinv2, prec);
else
acb_mul(s, s, zinv, prec);
}
/* remainder bound */
mag_init(err);
acb_gamma_stirling_bound(err, z, digamma ? 1 : 0, 1, nterms);
mag_add(arb_radref(acb_realref(s)), arb_radref(acb_realref(s)), err);
mag_add(arb_radref(acb_imagref(s)), arb_radref(acb_imagref(s)), err);
mag_clear(err);
if (digamma)
{
acb_neg(s, s);
acb_mul_2exp_si(zinv, zinv, -1);
acb_sub(s, s, zinv, prec);
acb_add(s, s, logz, prec);
}
else
{
/* (z-0.5)*log(z) - z + log(2*pi)/2 */
arb_one(b);
arb_mul_2exp_si(b, b, -1);
arb_set(acb_imagref(t), acb_imagref(z));
arb_sub(acb_realref(t), acb_realref(z), b, prec);
acb_mul(t, logz, t, prec);
acb_add(s, s, t, prec);
acb_sub(s, s, z, prec);
arb_const_log_sqrt2pi(b, prec);
arb_add(acb_realref(s), acb_realref(s), b, prec);
}
acb_clear(t);
acb_clear(logz);
acb_clear(zinv);
acb_clear(zinv2);
arb_clear(b);
}
void
mag_log1p(mag_t z, const mag_t x)
{
if (mag_is_special(x))
{
if (mag_is_zero(x))
mag_zero(z);
else
mag_inf(z);
}
else
{
fmpz exp = MAG_EXP(x);
if (!COEFF_IS_MPZ(exp))
{
/* Quick bound by x */
if (exp < -10)
{
mag_set(z, x);
return;
}
else if (exp < 1000)
{
double t;
t = ldexp(MAG_MAN(x), exp - MAG_BITS);
t = (1.0 + t) * (1 + 1e-14);
t = mag_d_log_upper_bound(t);
mag_set_d(z, t);
return;
}
}
else if (fmpz_sgn(MAG_EXPREF(x)) < 0)
{
/* Quick bound by x */
mag_set(z, x);
return;
}
/* Now we must have x >= 2^1000 */
/* Use log(2^(exp-1) * (2*v)) = exp*log(2) + log(2*v) */
{
double t;
fmpz_t b;
mag_t u;
mag_init(u);
fmpz_init(b);
/* incrementing the mantissa gives an upper bound for x+1 */
t = ldexp(MAG_MAN(x) + 1, 1 - MAG_BITS);
t = mag_d_log_upper_bound(t);
mag_set_d(u, t);
/* log(2) < 744261118/2^30 */
_fmpz_add_fast(b, MAG_EXPREF(x), -1);
fmpz_mul_ui(b, b, 744261118);
mag_set_fmpz(z, b);
_fmpz_add_fast(MAG_EXPREF(z), MAG_EXPREF(z), -30);
mag_add(z, z, u);
mag_clear(u);
fmpz_clear(b);
}
}
}
void
_arb_sin_cos_generic(arb_t s, arb_t c, const arf_t x, const mag_t xrad, slong prec)
{
int want_sin, want_cos;
slong maglim;
want_sin = (s != NULL);
want_cos = (c != NULL);
if (arf_is_zero(x) && mag_is_zero(xrad))
{
if (want_sin) arb_zero(s);
if (want_cos) arb_one(c);
return;
}
if (!arf_is_finite(x) || !mag_is_finite(xrad))
{
if (arf_is_nan(x))
{
if (want_sin) arb_indeterminate(s);
if (want_cos) arb_indeterminate(c);
}
else
{
if (want_sin) arb_zero_pm_one(s);
if (want_cos) arb_zero_pm_one(c);
}
return;
}
maglim = FLINT_MAX(65536, 4 * prec);
if (mag_cmp_2exp_si(xrad, -16) > 0 || arf_cmpabs_2exp_si(x, maglim) > 0)
{
_arb_sin_cos_wide(s, c, x, xrad, prec);
return;
}
if (arf_cmpabs_2exp_si(x, -(prec/2) - 2) <= 0)
{
mag_t t, u, v;
mag_init(t);
mag_init(u);
mag_init(v);
arf_get_mag(t, x);
mag_add(t, t, xrad);
mag_mul(u, t, t);
/* |sin(z)-z| <= z^3/6 */
if (want_sin)
{
arf_set(arb_midref(s), x);
mag_set(arb_radref(s), xrad);
arb_set_round(s, s, prec);
mag_mul(v, u, t);
mag_div_ui(v, v, 6);
arb_add_error_mag(s, v);
}
/* |cos(z)-1| <= z^2/2 */
if (want_cos)
{
arf_one(arb_midref(c));
mag_mul_2exp_si(arb_radref(c), u, -1);
}
mag_clear(t);
mag_clear(u);
mag_clear(v);
return;
}
if (mag_is_zero(xrad))
{
arb_sin_cos_arf_generic(s, c, x, prec);
}
else
{
mag_t t;
slong exp, radexp;
mag_init_set(t, xrad);
exp = arf_abs_bound_lt_2exp_si(x);
radexp = MAG_EXP(xrad);
if (radexp < MAG_MIN_LAGOM_EXP || radexp > MAG_MAX_LAGOM_EXP)
radexp = MAG_MIN_LAGOM_EXP;
if (want_cos && exp < -2)
prec = FLINT_MIN(prec, 20 - FLINT_MAX(exp, radexp) - radexp);
else
prec = FLINT_MIN(prec, 20 - radexp);
arb_sin_cos_arf_generic(s, c, x, prec);
/* todo: could use quadratic bound */
if (want_sin) mag_add(arb_radref(s), arb_radref(s), t);
if (want_cos) mag_add(arb_radref(c), arb_radref(c), t);
mag_clear(t);
}
}
static __inline__ void
_arb_poly_addmullow_rad(arb_ptr z, fmpz * zz,
const fmpz * xz, const double * xdbl, const fmpz * xexps,
const slong * xblocks, slong xlen,
const fmpz * yz, const double * ydbl, const fmpz * yexps,
const slong * yblocks, slong ylen, slong n)
{
slong i, j, k, ii, xp, yp, xl, yl, bn;
fmpz_t zexp;
mag_t t;
fmpz_init(zexp);
mag_init(t);
for (i = 0; (xp = xblocks[i]) != xlen; i++)
{
for (j = 0; (yp = yblocks[j]) != ylen; j++)
{
if (xp + yp >= n)
continue;
xl = xblocks[i + 1] - xp;
yl = yblocks[j + 1] - yp;
bn = FLINT_MIN(xl + yl - 1, n - xp - yp);
xl = FLINT_MIN(xl, bn);
yl = FLINT_MIN(yl, bn);
fmpz_add_inline(zexp, xexps + i, yexps + j);
if (xl > 1 && yl > 1 &&
(xl < DOUBLE_BLOCK_MAX_LENGTH || yl < DOUBLE_BLOCK_MAX_LENGTH))
{
fmpz_add_ui(zexp, zexp, 2 * DOUBLE_BLOCK_SHIFT);
for (k = 0; k < bn; k++)
{
/* Classical multiplication (may round down!) */
double ss = 0.0;
for (ii = FLINT_MAX(0, k - yl + 1);
ii <= FLINT_MIN(xl - 1, k); ii++)
{
ss += xdbl[xp + ii] * ydbl[yp + k - ii];
}
/* Compensate for rounding error */
ss *= DOUBLE_ROUNDING_FACTOR;
mag_set_d_2exp_fmpz(t, ss, zexp);
mag_add(arb_radref(z + xp + yp + k),
arb_radref(z + xp + yp + k), t);
}
}
else
{
if (xl >= yl)
_fmpz_poly_mullow(zz, xz + xp, xl, yz + yp, yl, bn);
else
_fmpz_poly_mullow(zz, yz + yp, yl, xz + xp, xl, bn);
for (k = 0; k < bn; k++)
{
mag_set_fmpz_2exp_fmpz(t, zz + k, zexp);
mag_add(arb_radref(z + xp + yp + k),
arb_radref(z + xp + yp + k), t);
}
}
}
}
fmpz_clear(zexp);
mag_clear(t);
}
/* computes the factors that are independent of n (all are upper bounds) */
void
acb_hypgeom_u_asymp_bound_factors(int * R, mag_t alpha,
mag_t nu, mag_t sigma, mag_t rho, mag_t zinv,
const acb_t a, const acb_t b, const acb_t z)
{
mag_t r, u, zre, zim, zlo, sigma_prime;
acb_t t;
mag_init(r);
mag_init(u);
mag_init(zre);
mag_init(zim);
mag_init(zlo);
mag_init(sigma_prime);
acb_init(t);
/* lower bounds for |re(z)|, |im(z)|, |z| */
arb_get_mag_lower(zre, acb_realref(z));
arb_get_mag_lower(zim, acb_imagref(z));
acb_get_mag_lower(zlo, z); /* todo: hypot */
/* upper bound for 1/|z| */
mag_one(u);
mag_div(zinv, u, zlo);
/* upper bound for r = |b - 2a| */
acb_mul_2exp_si(t, a, 1);
acb_sub(t, b, t, MAG_BITS);
acb_get_mag(r, t);
/* determine region */
*R = 0;
if (mag_cmp(zlo, r) >= 0)
{
int znonneg = arb_is_nonnegative(acb_realref(z));
if (znonneg && mag_cmp(zre, r) >= 0)
{
*R = 1;
}
else if (mag_cmp(zim, r) >= 0 || znonneg)
{
*R = 2;
}
else
{
mag_mul_2exp_si(u, r, 1);
if (mag_cmp(zlo, u) >= 0)
*R = 3;
}
}
if (R == 0)
{
mag_inf(alpha);
mag_inf(nu);
mag_inf(sigma);
mag_inf(rho);
}
else
{
/* sigma = |(b-2a)/z| */
mag_mul(sigma, r, zinv);
/* nu = (1/2 + 1/2 sqrt(1-4 sigma^2))^(-1/2) <= 1 + 2 sigma^2 */
if (mag_cmp_2exp_si(sigma, -1) <= 0)
{
mag_mul(nu, sigma, sigma);
mag_mul_2exp_si(nu, nu, 1);
mag_one(u);
mag_add(nu, nu, u);
}
else
{
mag_inf(nu);
}
/* modified sigma for alpha, beta, rho when in R3 */
if (*R == 3)
mag_mul(sigma_prime, sigma, nu);
else
mag_set(sigma_prime, sigma);
/* alpha = 1/(1-sigma') */
mag_one(alpha);
mag_sub_lower(alpha, alpha, sigma_prime);
mag_one(u);
mag_div(alpha, u, alpha);
/* rho = |2a^2-2ab+b|/2 + sigma'*(1+sigma'/4)/(1-sigma')^2 */
mag_mul_2exp_si(rho, sigma_prime, -2);
mag_one(u);
mag_add(rho, rho, u);
mag_mul(rho, rho, sigma_prime);
mag_mul(rho, rho, alpha);
mag_mul(rho, rho, alpha);
acb_sub(t, a, b, MAG_BITS);
acb_mul(t, t, a, MAG_BITS);
acb_mul_2exp_si(t, t, 1);
acb_add(t, t, b, MAG_BITS);
acb_get_mag(u, t);
mag_mul_2exp_si(u, u, -1);
mag_add(rho, rho, u);
}
mag_clear(r);
mag_clear(u);
mag_clear(zre);
mag_clear(zim);
mag_clear(zlo);
mag_clear(sigma_prime);
acb_clear(t);
}
void
acb_rising_ui_get_mag(mag_t bound, const acb_t s, ulong n)
{
if (n == 0)
{
mag_one(bound);
return;
}
if (n == 1)
{
acb_get_mag(bound, s);
return;
}
if (!acb_is_finite(s))
{
mag_inf(bound);
return;
}
if (arf_sgn(arb_midref(acb_realref(s))) >= 0)
{
acb_rising_get_mag2_right(bound, acb_realref(s), acb_imagref(s), n);
}
else
{
arb_t a;
long k;
mag_t bound2, t, u;
arb_init(a);
mag_init(bound2);
mag_init(t);
mag_init(u);
arb_get_mag(u, acb_imagref(s));
mag_mul(u, u, u);
mag_one(bound);
for (k = 0; k < n; k++)
{
arb_add_ui(a, acb_realref(s), k, MAG_BITS);
if (arf_sgn(arb_midref(a)) >= 0)
{
acb_rising_get_mag2_right(bound2, a, acb_imagref(s), n - k);
mag_mul(bound, bound, bound2);
break;
}
else
{
arb_get_mag(t, a);
mag_mul(t, t, t);
mag_add(t, t, u);
mag_mul(bound, bound, t);
}
}
arb_clear(a);
mag_clear(bound2);
mag_clear(t);
mag_clear(u);
}
mag_sqrt(bound, bound);
}
void
_arb_poly_mullow_block(arb_ptr z, arb_srcptr x, slong xlen,
arb_srcptr y, slong ylen, slong n, slong prec)
{
slong xmlen, xrlen, ymlen, yrlen, i;
fmpz *xz, *yz, *zz;
fmpz *xe, *ye;
slong *xblocks, *yblocks;
int squaring;
fmpz_t scale, t;
xlen = FLINT_MIN(xlen, n);
ylen = FLINT_MIN(ylen, n);
squaring = (x == y) && (xlen == ylen);
/* Strip trailing zeros */
xmlen = xrlen = xlen;
while (xmlen > 0 && arf_is_zero(arb_midref(x + xmlen - 1))) xmlen--;
while (xrlen > 0 && mag_is_zero(arb_radref(x + xrlen - 1))) xrlen--;
if (squaring)
{
ymlen = xmlen;
yrlen = xrlen;
}
else
{
ymlen = yrlen = ylen;
while (ymlen > 0 && arf_is_zero(arb_midref(y + ymlen - 1))) ymlen--;
while (yrlen > 0 && mag_is_zero(arb_radref(y + yrlen - 1))) yrlen--;
}
/* We don't know how to deal with infinities or NaNs */
if (!_arb_vec_is_finite(x, xlen) ||
(!squaring && !_arb_vec_is_finite(y, ylen)))
{
_arb_poly_mullow_classical(z, x, xlen, y, ylen, n, prec);
return;
}
xlen = FLINT_MAX(xmlen, xrlen);
ylen = FLINT_MAX(ymlen, yrlen);
/* Start with the zero polynomial */
_arb_vec_zero(z, n);
/* Nothing to do */
if (xlen == 0 || ylen == 0)
return;
n = FLINT_MIN(n, xlen + ylen - 1);
fmpz_init(scale);
fmpz_init(t);
xz = _fmpz_vec_init(xlen);
yz = _fmpz_vec_init(ylen);
zz = _fmpz_vec_init(n);
xe = _fmpz_vec_init(xlen);
ye = _fmpz_vec_init(ylen);
xblocks = flint_malloc(sizeof(slong) * (xlen + 1));
yblocks = flint_malloc(sizeof(slong) * (ylen + 1));
_arb_poly_get_scale(scale, x, xlen, y, ylen);
/* Error propagation */
/* (xm + xr)*(ym + yr) = (xm*ym) + (xr*ym + xm*yr + xr*yr)
= (xm*ym) + (xm*yr + xr*(ym + yr)) */
if (xrlen != 0 || yrlen != 0)
{
mag_ptr tmp;
double *xdbl, *ydbl;
tmp = _mag_vec_init(FLINT_MAX(xlen, ylen));
xdbl = flint_malloc(sizeof(double) * xlen);
ydbl = flint_malloc(sizeof(double) * ylen);
/* (xm + xr)^2 = (xm*ym) + (xr^2 + 2 xm xr)
= (xm*ym) + xr*(2 xm + xr) */
if (squaring)
{
_mag_vec_get_fmpz_2exp_blocks(xz, xdbl, xe, xblocks, scale, x, NULL, xrlen);
for (i = 0; i < xlen; i++)
{
arf_get_mag(tmp + i, arb_midref(x + i));
mag_mul_2exp_si(tmp + i, tmp + i, 1);
mag_add(tmp + i, tmp + i, arb_radref(x + i));
}
_mag_vec_get_fmpz_2exp_blocks(yz, ydbl, ye, yblocks, scale, NULL, tmp, xlen);
_arb_poly_addmullow_rad(z, zz, xz, xdbl, xe, xblocks, xrlen, yz, ydbl, ye, yblocks, xlen, n);
}
else if (yrlen == 0)
{
/* xr * |ym| */
_mag_vec_get_fmpz_2exp_blocks(xz, xdbl, xe, xblocks, scale, x, NULL, xrlen);
for (i = 0; i < ymlen; i++)
arf_get_mag(tmp + i, arb_midref(y + i));
_mag_vec_get_fmpz_2exp_blocks(yz, ydbl, ye, yblocks, scale, NULL, tmp, ymlen);
_arb_poly_addmullow_rad(z, zz, xz, xdbl, xe, xblocks, xrlen, yz, ydbl, ye, yblocks, ymlen, n);
}
else
{
/* |xm| * yr */
for (i = 0; i < xmlen; i++)
arf_get_mag(tmp + i, arb_midref(x + i));
_mag_vec_get_fmpz_2exp_blocks(xz, xdbl, xe, xblocks, scale, NULL, tmp, xmlen);
_mag_vec_get_fmpz_2exp_blocks(yz, ydbl, ye, yblocks, scale, y, NULL, yrlen);
_arb_poly_addmullow_rad(z, zz, xz, xdbl, xe, xblocks, xmlen, yz, ydbl, ye, yblocks, yrlen, n);
/* xr*(|ym| + yr) */
if (xrlen != 0)
{
_mag_vec_get_fmpz_2exp_blocks(xz, xdbl, xe, xblocks, scale, x, NULL, xrlen);
for (i = 0; i < ylen; i++)
arb_get_mag(tmp + i, y + i);
_mag_vec_get_fmpz_2exp_blocks(yz, ydbl, ye, yblocks, scale, NULL, tmp, ylen);
_arb_poly_addmullow_rad(z, zz, xz, xdbl, xe, xblocks, xrlen, yz, ydbl, ye, yblocks, ylen, n);
}
}
_mag_vec_clear(tmp, FLINT_MAX(xlen, ylen));
flint_free(xdbl);
flint_free(ydbl);
}
/* multiply midpoints */
if (xmlen != 0 && ymlen != 0)
{
_arb_vec_get_fmpz_2exp_blocks(xz, xe, xblocks, scale, x, xmlen, prec);
if (squaring)
{
_arb_poly_addmullow_block(z, zz, xz, xe, xblocks, xmlen, xz, xe, xblocks, xmlen, n, prec, 1);
}
else
{
_arb_vec_get_fmpz_2exp_blocks(yz, ye, yblocks, scale, y, ymlen, prec);
_arb_poly_addmullow_block(z, zz, xz, xe, xblocks, xmlen, yz, ye, yblocks, ymlen, n, prec, 0);
}
}
/* Unscale. */
if (!fmpz_is_zero(scale))
{
fmpz_zero(t);
for (i = 0; i < n; i++)
{
arb_mul_2exp_fmpz(z + i, z + i, t);
fmpz_add(t, t, scale);
}
}
_fmpz_vec_clear(xz, xlen);
_fmpz_vec_clear(yz, ylen);
_fmpz_vec_clear(zz, n);
_fmpz_vec_clear(xe, xlen);
_fmpz_vec_clear(ye, ylen);
flint_free(xblocks);
flint_free(yblocks);
fmpz_clear(scale);
fmpz_clear(t);
}
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
}
