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
zeta_ui_borwein_bsplit(fmprb_t x, ulong s, long prec)
{
zeta_bsplit_t sum;
fmpr_t err;
long wp, n;
/* zeta(0) = -1/2 */
if (s == 0)
{
fmpr_set_si_2exp_si(fmprb_midref(x), -1, -1);
fmpr_zero(fmprb_radref(x));
return;
}
if (s == 1)
{
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) */
fmprb_mul(sum->A, sum->A, sum->C, wp);
fmprb_mul(sum->B, sum->B, sum->Q1, wp);
fmprb_sub(sum->A, sum->A, sum->B, wp);
fmprb_mul(sum->Q3, sum->Q3, sum->C, wp);
fmprb_div(sum->C, sum->A, sum->Q3, wp);
fmpr_init(err);
borwein_error(err, n);
fmprb_add_error_fmpr(sum->C, err);
fmpr_clear(err);
/* convert from eta(s) to zeta(s) */
fmprb_div_2expm1_ui(x, sum->C, s - 1, wp);
fmprb_mul_2exp_si(x, x, s - 1);
zeta_bsplit_clear(sum);
}
static void
zeta_bsplit(zeta_bsplit_t L, slong a, slong b,
slong n, slong s, int cont, slong bits)
{
if (a + 1 == b)
{
zeta_coeff_k(L, a, n, s);
}
else
{
zeta_bsplit_t R;
slong m = (a + b) / 2;
zeta_bsplit(L, m, b, n, s, 1, bits);
zeta_bsplit_init(R);
zeta_bsplit(R, a, m, n, s, 1, bits);
arb_mul(L->B, L->B, R->D, bits);
arb_addmul(L->B, L->A, R->C, bits);
arb_mul(L->B, L->B, R->Q2, bits);
arb_addmul(L->B, R->B, L->Q3, bits);
arb_mul(L->A, L->A, R->Q3, bits);
arb_addmul(L->A, R->A, L->Q3, bits);
arb_mul(L->C, L->C, R->D, bits);
arb_addmul(L->C, R->C, L->Q1, bits);
if (cont)
{
arb_mul(L->D, L->D, R->D, bits);
arb_mul(L->Q2, L->Q2, R->Q2, bits);
}
arb_mul(L->Q1, L->Q1, R->Q1, bits);
arb_mul(L->Q3, L->Q3, R->Q3, bits);
zeta_bsplit_clear(R);
}
}