void
_arb_poly_sinh_cosh_series_exponential(arb_ptr s, arb_ptr c,
const arb_srcptr h, slong hlen, slong len, slong prec)
{
arb_ptr t, u, v;
arb_t s0, c0;
hlen = FLINT_MIN(hlen, len);
if (hlen == 1)
{
arb_sinh_cosh(s, c, h, prec);
_arb_vec_zero(s + 1, len - 1);
_arb_vec_zero(c + 1, len - 1);
return;
}
arb_init(s0);
arb_init(c0);
t = _arb_vec_init(3 * len);
u = t + len;
v = u + len;
arb_sinh_cosh(s0, c0, h, prec);
_arb_vec_set(t + 1, h + 1, hlen - 1);
_arb_poly_exp_series(t, t, len, len, prec);
/* todo: part of the inverse could be avoided since exp computes
it internally to half the length */
_arb_poly_inv_series(u, t, len, len, prec);
/* hyperbolic sine */
_arb_vec_sub(s, t, u, len, prec);
_arb_vec_scalar_mul_2exp_si(s, s, len, -1);
/* hyperbolic cosine */
_arb_vec_add(c, t, u, len, prec);
_arb_vec_scalar_mul_2exp_si(c, c, len, -1);
/* sinh(h0 + h1) = cosh(h0) sinh(h1) + sinh(h0) cosh(h1)
cosh(h0 + h1) = cosh(h0) cosh(h1) + sinh(h0) sinh(h1) */
if (!arb_is_zero(s0))
{
_arb_vec_scalar_mul(t, s, len, c0, prec);
_arb_vec_scalar_mul(u, c, len, s0, prec);
_arb_vec_scalar_mul(v, s, len, s0, prec);
_arb_vec_add(s, t, u, len, prec);
_arb_vec_scalar_mul(t, c, len, c0, prec);
_arb_vec_add(c, t, v, len, prec);
}
_arb_vec_clear(t, 3 * len);
arb_clear(s0);
arb_clear(c0);
}
/* with inverse=1 simultaneously computes g = exp(-x) to length n
with inverse=0 uses g as scratch space, computing
g = exp(-x) only to length (n+1)/2 */
static void
_arb_poly_exp_series_newton(arb_ptr f, arb_ptr g,
arb_srcptr h, slong len, slong prec, int inverse, slong cutoff)
{
slong alloc;
arb_ptr T, U, hprime;
alloc = 3 * len;
T = _arb_vec_init(alloc);
U = T + len;
hprime = U + len;
_arb_poly_derivative(hprime, h, len, prec);
arb_zero(hprime + len - 1);
NEWTON_INIT(cutoff, len)
/* f := exp(h) + O(x^m), g := exp(-h) + O(x^m2) */
NEWTON_BASECASE(n)
_arb_poly_exp_series_basecase(f, h, n, n, prec);
_arb_poly_inv_series(g, f, (n + 1) / 2, (n + 1) / 2, prec);
NEWTON_END_BASECASE
/* extend from length m to length n */
NEWTON_LOOP(m, n)
slong m2 = (m + 1) / 2;
slong l = m - 1; /* shifted for derivative */
/* g := exp(-h) + O(x^m) */
_arb_poly_mullow(T, f, m, g, m2, m, prec);
_arb_poly_mullow(g + m2, g, m2, T + m2, m - m2, m - m2, prec);
_arb_vec_neg(g + m2, g + m2, m - m2);
/* U := h' + g (f' - f h') + O(x^(n-1))
Note: should replace h' by h' mod x^(m-1) */
_arb_vec_zero(f + m, n - m);
_arb_poly_mullow(T, f, n, hprime, n, n, prec); /* should be mulmid */
_arb_poly_derivative(U, f, n, prec);
arb_zero(U + n - 1); /* should skip low terms */
_arb_vec_sub(U + l, U + l, T + l, n - l, prec);
_arb_poly_mullow(T + l, g, n - m, U + l, n - m, n - m, prec);
_arb_vec_add(U + l, hprime + l, T + l, n - m, prec);
/* f := f + f * (h - int U) + O(x^n) = exp(h) + O(x^n) */
_arb_poly_integral(U, U, n, prec); /* should skip low terms */
_arb_vec_sub(U + m, h + m, U + m, n - m, prec);
_arb_poly_mullow(f + m, f, n - m, U + m, n - m, n - m, prec);
/* g := exp(-h) + O(x^n) */
/* not needed if we only want exp(x) */
if (n == len && inverse)
{
_arb_poly_mullow(T, f, n, g, m, n, prec);
_arb_poly_mullow(g + m, g, m, T + m, n - m, n - m, prec);
_arb_vec_neg(g + m, g + m, n - m);
}
NEWTON_END_LOOP
NEWTON_END
_arb_vec_clear(T, alloc);
}
void
keiper_li_series(arb_ptr z, slong len, slong prec)
{
arb_ptr t, u, v;
t = _arb_vec_init(len);
u = _arb_vec_init(len);
v = _arb_vec_init(len);
/* -zeta(s) */
flint_printf("zeta: ");
TIMEIT_ONCE_START
arb_zero(t + 0);
arb_one(t + 1);
arb_one(u);
_arb_poly_zeta_series(v, t, 2, u, 0, len, prec);
_arb_vec_neg(v, v, len);
TIMEIT_ONCE_STOP
SHOW_MEMORY_USAGE
/* logarithm */
flint_printf("log: ");
TIMEIT_ONCE_START
_arb_poly_log_series(t, v, len, len, prec);
TIMEIT_ONCE_STOP
/* add log(gamma(1+s/2)) */
flint_printf("gamma: ");
TIMEIT_ONCE_START
arb_one(u);
arb_one(u + 1);
arb_mul_2exp_si(u + 1, u + 1, -1);
_arb_poly_lgamma_series(v, u, 2, len, prec);
_arb_vec_add(t, t, v, len, prec);
TIMEIT_ONCE_STOP
/* subtract 0.5 s log(pi) */
arb_const_pi(u, prec);
arb_log(u, u, prec);
arb_mul_2exp_si(u, u, -1);
arb_sub(t + 1, t + 1, u, prec);
/* add log(1-s) */
arb_one(u);
arb_set_si(u + 1, -1);
_arb_poly_log_series(v, u, 2, len, prec);
_arb_vec_add(t, t, v, len, prec);
/* binomial transform */
flint_printf("binomial transform: ");
TIMEIT_ONCE_START
arb_set(z, t);
_arb_vec_neg(t + 1, t + 1, len - 1);
_arb_poly_binomial_transform(z + 1, t + 1, len - 1, len - 1, prec);
TIMEIT_ONCE_STOP
_arb_vec_clear(t, len);
_arb_vec_clear(u, len);
_arb_vec_clear(v, len);
}
void
_acb_poly_mullow_transpose_gauss(acb_ptr res,
acb_srcptr poly1, slong len1,
acb_srcptr poly2, slong len2, slong n, slong prec)
{
arb_ptr a, b, c, d, e, f, w;
arb_ptr t, u, v;
slong i;
len1 = FLINT_MIN(len1, n);
len2 = FLINT_MIN(len2, n);
w = flint_malloc(sizeof(arb_struct) * (2 * (len1 + len2 + n)));
a = w;
b = a + len1;
c = b + len1;
d = c + len2;
e = d + len2;
f = e + n;
t = _arb_vec_init(n);
u = _arb_vec_init(n);
v = _arb_vec_init(n);
for (i = 0; i < len1; i++)
{
a[i] = *acb_realref(poly1 + i);
b[i] = *acb_imagref(poly1 + i);
}
for (i = 0; i < len2; i++)
{
c[i] = *acb_realref(poly2 + i);
d[i] = *acb_imagref(poly2 + i);
}
for (i = 0; i < n; i++)
{
e[i] = *acb_realref(res + i);
f[i] = *acb_imagref(res + i);
}
_arb_vec_add(t, a, b, len1, prec);
_arb_vec_add(u, c, d, len2, prec);
_arb_poly_mullow(v, t, len1, u, len2, n, prec);
_arb_poly_mullow(t, a, len1, c, len2, n, prec);
_arb_poly_mullow(u, b, len1, d, len2, n, prec);
_arb_vec_sub(e, t, u, n, prec);
_arb_vec_sub(f, v, t, n, prec);
_arb_vec_sub(f, f, u, n, prec);
for (i = 0; i < n; i++)
{
*acb_realref(res + i) = e[i];
*acb_imagref(res + i) = f[i];
}
_arb_vec_clear(t, n);
_arb_vec_clear(u, n);
_arb_vec_clear(v, n);
flint_free(w);
}
void
_arb_poly_zeta_series(arb_ptr res, arb_srcptr h, long hlen, const arb_t a, int deflate, long len, long prec)
{
long i;
acb_t cs, ca;
acb_ptr z;
arb_ptr t, u;
if (arb_contains_nonpositive(a))
{
_arb_vec_indeterminate(res, len);
return;
}
hlen = FLINT_MIN(hlen, len);
z = _acb_vec_init(len);
t = _arb_vec_init(len);
u = _arb_vec_init(len);
acb_init(cs);
acb_init(ca);
/* use reflection formula */
if (arf_sgn(arb_midref(h)) < 0 && arb_is_one(a))
{
/* zeta(s) = (2*pi)**s * sin(pi*s/2) / pi * gamma(1-s) * zeta(1-s) */
arb_t pi;
arb_ptr f, s1, s2, s3, s4;
arb_init(pi);
f = _arb_vec_init(2);
s1 = _arb_vec_init(len);
s2 = _arb_vec_init(len);
s3 = _arb_vec_init(len);
s4 = _arb_vec_init(len);
arb_const_pi(pi, prec);
/* s1 = (2*pi)**s */
arb_mul_2exp_si(pi, pi, 1);
_arb_poly_pow_cpx(s1, pi, h, len, prec);
arb_mul_2exp_si(pi, pi, -1);
/* s2 = sin(pi*s/2) / pi */
arb_set(f, h);
arb_one(f + 1);
arb_mul_2exp_si(f, f, -1);
arb_mul_2exp_si(f + 1, f + 1, -1);
_arb_poly_sin_pi_series(s2, f, 2, len, prec);
_arb_vec_scalar_div(s2, s2, len, pi, prec);
/* s3 = gamma(1-s) */
arb_sub_ui(f, h, 1, prec);
arb_neg(f, f);
arb_set_si(f + 1, -1);
_arb_poly_gamma_series(s3, f, 2, len, prec);
/* s4 = zeta(1-s) */
arb_sub_ui(f, h, 1, prec);
arb_neg(f, f);
acb_set_arb(cs, f);
acb_one(ca);
_acb_poly_zeta_cpx_series(z, cs, ca, 0, len, prec);
for (i = 0; i < len; i++)
arb_set(s4 + i, acb_realref(z + i));
for (i = 1; i < len; i += 2)
arb_neg(s4 + i, s4 + i);
_arb_poly_mullow(u, s1, len, s2, len, len, prec);
_arb_poly_mullow(s1, s3, len, s4, len, len, prec);
_arb_poly_mullow(t, u, len, s1, len, len, prec);
/* add 1/(1-(s+t)) = 1/(1-s) + t/(1-s)^2 + ... */
if (deflate)
{
arb_sub_ui(u, h, 1, prec);
arb_neg(u, u);
arb_inv(u, u, prec);
for (i = 1; i < len; i++)
arb_mul(u + i, u + i - 1, u, prec);
_arb_vec_add(t, t, u, len, prec);
}
arb_clear(pi);
_arb_vec_clear(f, 2);
_arb_vec_clear(s1, len);
_arb_vec_clear(s2, len);
_arb_vec_clear(s3, len);
_arb_vec_clear(s4, len);
}
else
{
acb_set_arb(cs, h);
acb_set_arb(ca, a);
_acb_poly_zeta_cpx_series(z, cs, ca, deflate, len, prec);
for (i = 0; i < len; i++)
arb_set(t + i, acb_realref(z + i));
}
/* compose with nonconstant part */
arb_zero(u);
_arb_vec_set(u + 1, h + 1, hlen - 1);
_arb_poly_compose_series(res, t, len, u, hlen, len, prec);
_acb_vec_clear(z, len);
_arb_vec_clear(t, len);
_arb_vec_clear(u, len);
acb_init(cs);
acb_init(ca);
}
void
_arb_poly_lgamma_series(arb_ptr res, arb_srcptr h, slong hlen, slong len, slong prec)
{
int reflect;
slong r, n, wp;
arb_t zr;
arb_ptr t, u;
if (!arb_is_positive(h))
{
_arb_vec_indeterminate(res, len);
return;
}
hlen = FLINT_MIN(hlen, len);
wp = prec + FLINT_BIT_COUNT(prec);
t = _arb_vec_init(len);
u = _arb_vec_init(len);
arb_init(zr);
/* use zeta values at small integers */
if (arb_is_int(h) && (arf_cmpabs_ui(arb_midref(h), prec / 2) < 0))
{
r = arf_get_si(arb_midref(h), ARF_RND_DOWN);
if (r <= 0)
{
_arb_vec_indeterminate(res, len);
goto cleanup;
}
else
{
_arb_poly_lgamma_series_at_one(u, len, wp);
if (r != 1)
{
arb_one(zr);
_log_rising_ui_series(t, zr, r - 1, len, wp);
_arb_vec_add(u, u, t, len, wp);
}
}
}
else if (len <= 2)
{
arb_lgamma(u, h, wp);
if (len == 2)
arb_digamma(u + 1, h, wp);
}
else
{
/* otherwise use Stirling series */
arb_gamma_stirling_choose_param(&reflect, &r, &n, h, 0, 0, wp);
arb_add_ui(zr, h, r, wp);
_arb_poly_gamma_stirling_eval(u, zr, n, len, wp);
if (r != 0)
{
_log_rising_ui_series(t, h, r, len, wp);
_arb_vec_sub(u, u, t, len, wp);
}
}
/* compose with nonconstant part */
arb_zero(t);
_arb_vec_set(t + 1, h + 1, hlen - 1);
_arb_poly_compose_series(res, u, len, t, hlen, len, prec);
cleanup:
arb_clear(zr);
_arb_vec_clear(t, len);
_arb_vec_clear(u, len);
}
void
_arb_poly_sin_cos_series_tangent(arb_ptr s, arb_ptr c,
arb_srcptr h, slong hlen, slong len, slong prec, int times_pi)
{
arb_ptr t, u, v;
arb_t s0, c0;
hlen = FLINT_MIN(hlen, len);
if (hlen == 1)
{
if (times_pi)
arb_sin_cos_pi(s, c, h, prec);
else
arb_sin_cos(s, c, h, prec);
_arb_vec_zero(s + 1, len - 1);
_arb_vec_zero(c + 1, len - 1);
return;
}
/*
sin(x) = 2*tan(x/2)/(1+tan(x/2)^2)
cos(x) = (1-tan(x/2)^2)/(1+tan(x/2)^2)
*/
arb_init(s0);
arb_init(c0);
t = _arb_vec_init(3 * len);
u = t + len;
v = u + len;
/* sin, cos of h0 */
if (times_pi)
arb_sin_cos_pi(s0, c0, h, prec);
else
arb_sin_cos(s0, c0, h, prec);
/* t = tan((h-h0)/2) */
arb_zero(u);
_arb_vec_scalar_mul_2exp_si(u + 1, h + 1, hlen - 1, -1);
if (times_pi)
{
arb_const_pi(t, prec);
_arb_vec_scalar_mul(u + 1, u + 1, hlen - 1, t, prec);
}
_arb_poly_tan_series(t, u, hlen, len, prec);
/* v = 1 + t^2 */
_arb_poly_mullow(v, t, len, t, len, len, prec);
arb_add_ui(v, v, 1, prec);
/* u = 1/(1+t^2) */
_arb_poly_inv_series(u, v, len, len, prec);
/* sine */
_arb_poly_mullow(s, t, len, u, len, len, prec);
_arb_vec_scalar_mul_2exp_si(s, s, len, 1);
/* cosine */
arb_sub_ui(v, v, 2, prec);
_arb_vec_neg(v, v, len);
_arb_poly_mullow(c, v, len, u, len, len, prec);
/* sin(h0 + h1) = cos(h0) sin(h1) + sin(h0) cos(h1)
cos(h0 + h1) = cos(h0) cos(h1) - sin(h0) sin(h1) */
if (!arb_is_zero(s0))
{
_arb_vec_scalar_mul(t, s, len, c0, prec);
_arb_vec_scalar_mul(u, c, len, s0, prec);
_arb_vec_scalar_mul(v, s, len, s0, prec);
_arb_vec_add(s, t, u, len, prec);
_arb_vec_scalar_mul(t, c, len, c0, prec);
_arb_vec_sub(c, t, v, len, prec);
}
_arb_vec_clear(t, 3 * len);
arb_clear(s0);
arb_clear(c0);
}
void
_acb_poly_mullow_transpose(acb_ptr res,
acb_srcptr poly1, slong len1,
acb_srcptr poly2, slong len2, slong n, slong prec)
{
arb_ptr a, b, c, d, e, f, w;
arb_ptr t;
slong i;
len1 = FLINT_MIN(len1, n);
len2 = FLINT_MIN(len2, n);
w = flint_malloc(sizeof(arb_struct) * (2 * (len1 + len2 + n)));
a = w;
b = a + len1;
c = b + len1;
d = c + len2;
e = d + len2;
f = e + n;
/* (e+fi) = (a+bi)(c+di) = (ac - bd) + (ad + bc)i */
t = _arb_vec_init(n);
for (i = 0; i < len1; i++)
{
a[i] = *acb_realref(poly1 + i);
b[i] = *acb_imagref(poly1 + i);
}
for (i = 0; i < len2; i++)
{
c[i] = *acb_realref(poly2 + i);
d[i] = *acb_imagref(poly2 + i);
}
for (i = 0; i < n; i++)
{
e[i] = *acb_realref(res + i);
f[i] = *acb_imagref(res + i);
}
_arb_poly_mullow(e, a, len1, c, len2, n, prec);
_arb_poly_mullow(t, b, len1, d, len2, n, prec);
_arb_vec_sub(e, e, t, n, prec);
_arb_poly_mullow(f, a, len1, d, len2, n, prec);
/* squaring */
if (poly1 == poly2 && len1 == len2)
{
_arb_vec_scalar_mul_2exp_si(f, f, n, 1);
}
else
{
_arb_poly_mullow(t, b, len1, c, len2, n, prec);
_arb_vec_add(f, f, t, n, prec);
}
for (i = 0; i < n; i++)
{
*acb_realref(res + i) = e[i];
*acb_imagref(res + i) = f[i];
}
_arb_vec_clear(t, n);
flint_free(w);
}
