/* 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);
}
static void
acb_log1p_tiny(acb_t r, const acb_t z, slong prec)
{
mag_t b, c;
acb_t t;
int real;
mag_init(b);
mag_init(c);
acb_init(t);
real = acb_is_real(z);
/* if |z| < 1, then |log(1+z) - [z - z^2/2]| <= |z|^3/(1-|z|) */
acb_get_mag(b, z);
mag_one(c);
mag_sub_lower(c, c, b);
mag_pow_ui(b, b, 3);
mag_div(b, b, c);
acb_mul(t, z, z, prec);
acb_mul_2exp_si(t, t, -1);
acb_sub(r, z, t, prec);
if (real && mag_is_finite(b))
arb_add_error_mag(acb_realref(r), b);
else
acb_add_error_mag(r, b);
mag_clear(b);
mag_clear(c);
acb_clear(t);
}
void
mag_geom_series(mag_t res, const mag_t x, ulong n)
{
if (mag_is_zero(x))
{
if (n == 0)
mag_one(res);
else
mag_zero(res);
}
else if (mag_is_inf(x))
{
mag_inf(res);
}
else
{
mag_t t;
mag_init(t);
mag_one(t);
mag_sub_lower(t, t, x);
if (mag_is_zero(t))
{
mag_inf(res);
}
else
{
mag_pow_ui(res, x, n);
mag_div(res, res, t);
}
mag_clear(t);
}
}
static void
arb_sqrt1pm1_tiny(arb_t r, const arb_t z, slong prec)
{
mag_t b, c;
arb_t t;
mag_init(b);
mag_init(c);
arb_init(t);
/* if |z| < 1, then |(sqrt(1+z)-1) - (z/2-z^2/8)| <= |z|^3/(1-|z|)/16 */
arb_get_mag(b, z);
mag_one(c);
mag_sub_lower(c, c, b);
mag_pow_ui(b, b, 3);
mag_div(b, b, c);
mag_mul_2exp_si(b, b, -4);
arb_mul(t, z, z, prec);
arb_mul_2exp_si(t, t, -2);
arb_sub(r, z, t, prec);
arb_mul_2exp_si(r, r, -1);
if (mag_is_finite(b))
arb_add_error_mag(r, b);
else
arb_indeterminate(r);
mag_clear(b);
mag_clear(c);
arb_clear(t);
}
void
acb_hypgeom_mag_chi(mag_t chi, ulong n)
{
mag_t p, q;
ulong k;
mag_init(p);
mag_init(q);
if (n % 2 == 0)
{
mag_one(p);
mag_one(q);
}
else
{
/* upper bound for pi/2 */
mag_set_ui_2exp_si(p, 843314857, -28);
mag_one(q);
}
for (k = n; k >= 2; k -= 2)
{
mag_mul_ui(p, p, k);
mag_mul_ui_lower(q, q, k - 1);
}
mag_div(chi, p, q);
mag_clear(p);
mag_clear(q);
}
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);
}
void
arb_div(arb_t z, const arb_t x, const arb_t y, long prec)
{
mag_t zr, xm, ym, yl, yw;
int inexact;
if (arb_is_exact(y))
{
arb_div_arf(z, x, arb_midref(y), prec);
}
else if (mag_is_inf(arb_radref(x)) || mag_is_inf(arb_radref(y)))
{
arf_div(arb_midref(z), arb_midref(x), arb_midref(y), prec, ARB_RND);
mag_inf(arb_radref(z));
}
else
{
mag_init_set_arf(xm, arb_midref(x));
mag_init_set_arf(ym, arb_midref(y));
mag_init(zr);
mag_init(yl);
mag_init(yw);
/* (|x|*yrad + |y|*xrad)/(y*(|y|-yrad)) */
mag_mul(zr, xm, arb_radref(y));
mag_addmul(zr, ym, arb_radref(x));
arb_get_mag_lower(yw, y);
arf_get_mag_lower(yl, arb_midref(y));
mag_mul_lower(yl, yl, yw);
mag_div(zr, zr, yl);
inexact = arf_div(arb_midref(z), arb_midref(x), arb_midref(y), prec, ARB_RND);
if (inexact)
arf_mag_add_ulp(arb_radref(z), zr, arb_midref(z), prec);
else
mag_swap(arb_radref(z), zr);
mag_clear(xm);
mag_clear(ym);
mag_clear(zr);
mag_clear(yl);
mag_clear(yw);
}
}
void
arb_div_arf(arb_t z, const arb_t x, const arf_t y, long prec)
{
mag_t zr, ym;
int inexact;
if (arf_is_zero(y))
{
arb_zero_pm_inf(z);
}
else if (arb_is_exact(x))
{
inexact = arf_div(arb_midref(z), arb_midref(x), y, prec, ARB_RND);
if (inexact)
arf_mag_set_ulp(arb_radref(z), arb_midref(z), prec);
else
mag_zero(arb_radref(z));
}
else if (mag_is_inf(arb_radref(x)))
{
arf_div(arb_midref(z), arb_midref(x), y, prec, ARB_RND);
mag_inf(arb_radref(z));
}
else
{
mag_init(ym);
mag_init(zr);
arf_get_mag_lower(ym, y);
mag_div(zr, arb_radref(x), ym);
inexact = arf_div(arb_midref(z), arb_midref(x), y, prec, ARB_RND);
if (inexact)
arf_mag_add_ulp(arb_radref(z), zr, arb_midref(z), prec);
else
mag_swap(arb_radref(z), zr);
mag_clear(ym);
mag_clear(zr);
}
}
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
mag_polylog_tail(mag_t u, const mag_t z, long sigma, ulong d, ulong N)
{
mag_t TN, UN, t;
if (N < 2)
{
mag_inf(u);
return;
}
mag_init(TN);
mag_init(UN);
mag_init(t);
if (mag_cmp_2exp_si(z, 0) >= 0)
{
mag_inf(u);
}
else
{
/* Bound T(N) */
mag_pow_ui(TN, z, N);
/* multiply by log(N)^d */
if (d > 0)
{
mag_log_ui(t, N);
mag_pow_ui(t, t, d);
mag_mul(TN, TN, t);
}
/* multiply by 1/k^s */
if (sigma > 0)
{
mag_set_ui_lower(t, N);
mag_pow_ui_lower(t, t, sigma);
mag_div(TN, TN, t);
}
else if (sigma < 0)
{
mag_set_ui(t, N);
mag_pow_ui(t, t, -sigma);
mag_mul(TN, TN, t);
}
/* Bound U(N) */
mag_set(UN, z);
/* multiply by (1 + 1/N)**S */
if (sigma < 0)
{
mag_binpow_uiui(t, N, -sigma);
mag_mul(UN, UN, t);
}
/* multiply by (1 + 1/(N log(N)))^d */
if (d > 0)
{
ulong nl;
/* rounds down */
nl = mag_d_log_lower_bound(N) * N * (1 - 1e-13);
mag_binpow_uiui(t, nl, d);
mag_mul(UN, UN, t);
}
/* T(N) / (1 - U(N)) */
if (mag_cmp_2exp_si(UN, 0) >= 0)
{
mag_inf(u);
}
else
{
mag_one(t);
mag_sub_lower(t, t, UN);
mag_div(u, TN, t);
}
}
mag_clear(TN);
mag_clear(UN);
mag_clear(t);
}
slong
hypgeom_bound(mag_t error, int r,
slong A, slong B, slong K, const mag_t TK, const mag_t z, slong tol_2exp)
{
mag_t Tn, t, u, one, tol, num, den;
slong n, m;
mag_init(Tn);
mag_init(t);
mag_init(u);
mag_init(one);
mag_init(tol);
mag_init(num);
mag_init(den);
mag_one(one);
mag_set_ui_2exp_si(tol, UWORD(1), -tol_2exp);
/* approximate number of needed terms */
n = hypgeom_estimate_terms(z, r, tol_2exp);
/* required for 1 + O(1/k) part to be decreasing */
n = FLINT_MAX(n, K + 1);
/* required for z^k / (k!)^r to be decreasing */
m = hypgeom_root_bound(z, r);
n = FLINT_MAX(n, m);
/* We now have |R(k)| <= G(k) where G(k) is monotonically decreasing,
and can bound the tail using a geometric series as soon
as soon as G(k) < 1. */
/* bound T(n-1) */
hypgeom_term_bound(Tn, TK, K, A, B, r, z, n-1);
while (1)
{
/* bound R(n) */
mag_mul_ui(num, z, n);
mag_mul_ui(num, num, n - B);
mag_set_ui_lower(den, n - A);
mag_mul_ui_lower(den, den, n - 2*B);
if (r != 0)
{
mag_set_ui_lower(u, n);
mag_pow_ui_lower(u, u, r);
mag_mul_lower(den, den, u);
}
mag_div(t, num, den);
/* multiply bound for T(n-1) by bound for R(n) to bound T(n) */
mag_mul(Tn, Tn, t);
/* geometric series termination check */
/* u = max(1-t, 0), rounding down [lower bound] */
mag_sub_lower(u, one, t);
if (!mag_is_zero(u))
{
mag_div(u, Tn, u);
if (mag_cmp(u, tol) < 0)
{
mag_set(error, u);
break;
}
}
/* move on to next term */
n++;
}
mag_clear(Tn);
mag_clear(t);
mag_clear(u);
mag_clear(one);
mag_clear(tol);
mag_clear(num);
mag_clear(den);
return n;
}
void
acb_hypgeom_pfq_sum_rs(acb_t res, acb_t term, acb_srcptr a, slong p,
acb_srcptr b, slong q, const acb_t z, slong n, slong prec)
{
acb_ptr zpow;
acb_t s, t, u;
slong i, j, k, m;
mag_t B, C;
if (n == 0)
{
acb_zero(res);
acb_one(term);
return;
}
if (n < 0)
abort();
m = n_sqrt(n);
m = FLINT_MIN(m, 150);
mag_init(B);
mag_init(C);
acb_init(s);
acb_init(t);
acb_init(u);
zpow = _acb_vec_init(m + 1);
_acb_vec_set_powers(zpow, z, m + 1, prec);
mag_one(B);
for (k = n; k >= 0; k--)
{
j = k % m;
if (k < n)
acb_add(s, s, zpow + j, prec);
if (k > 0)
{
if (p > 0)
{
acb_add_ui(u, a, k - 1, prec);
for (i = 1; i < p; i++)
{
acb_add_ui(t, a + i, k - 1, prec);
acb_mul(u, u, t, prec);
}
if (k < n)
acb_mul(s, s, u, prec);
acb_get_mag(C, u);
mag_mul(B, B, C);
}
if (q > 0)
{
acb_add_ui(u, b, k - 1, prec);
for (i = 1; i < q; i++)
{
acb_add_ui(t, b + i, k - 1, prec);
acb_mul(u, u, t, prec);
}
if (k < n)
acb_div(s, s, u, prec);
acb_get_mag_lower(C, u);
mag_div(B, B, C);
}
if (j == 0 && k < n)
{
acb_mul(s, s, zpow + m, prec);
}
}
}
acb_get_mag(C, z);
mag_pow_ui(C, C, n);
mag_mul(B, B, C);
acb_zero(term);
if (_acb_vec_is_real(a, p) && _acb_vec_is_real(b, q) && acb_is_real(z))
arb_add_error_mag(acb_realref(term), B);
else
acb_add_error_mag(term, B);
acb_set(res, s);
mag_clear(B);
mag_clear(C);
acb_clear(s);
acb_clear(t);
acb_clear(u);
_acb_vec_clear(zpow, m + 1);
}
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
}
/* 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);
}