Example #1
0
void
arb_poly_compose_series(arb_poly_t res,
                    const arb_poly_t poly1,
                    const arb_poly_t poly2, slong n, slong prec)
{
    slong len1 = poly1->length;
    slong len2 = poly2->length;
    slong lenr;

    if (len2 != 0 && !arb_is_zero(poly2->coeffs))
    {
        flint_printf("exception: compose_series: inner "
                "polynomial must have zero constant term\n");
        abort();
    }

    if (len1 == 0 || n == 0)
    {
        arb_poly_zero(res);
        return;
    }

    if (len2 == 0 || len1 == 1)
    {
        arb_poly_set_arb(res, poly1->coeffs);
        return;
    }

    lenr = FLINT_MIN((len1 - 1) * (len2 - 1) + 1, n);
    len1 = FLINT_MIN(len1, lenr);
    len2 = FLINT_MIN(len2, lenr);

    if ((res != poly1) && (res != poly2))
    {
        arb_poly_fit_length(res, lenr);
        _arb_poly_compose_series(res->coeffs, poly1->coeffs, len1,
                                        poly2->coeffs, len2, lenr, prec);
        _arb_poly_set_length(res, lenr);
        _arb_poly_normalise(res);
    }
    else
    {
        arb_poly_t t;
        arb_poly_init2(t, lenr);
        _arb_poly_compose_series(t->coeffs, poly1->coeffs, len1,
                                        poly2->coeffs, len2, lenr, prec);
        _arb_poly_set_length(t, lenr);
        _arb_poly_normalise(t);
        arb_poly_swap(res, t);
        arb_poly_clear(t);
    }
}
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);
}
Example #3
0
void
_arb_poly_rgamma_series(arb_ptr res, arb_srcptr h, long hlen, long len, long prec)
{
    int reflect;
    long i, rflen, r, n, wp;
    arb_ptr t, u, v;
    arb_struct f[2];

    hlen = FLINT_MIN(hlen, len);
    wp = prec + FLINT_BIT_COUNT(prec);

    t = _arb_vec_init(len);
    u = _arb_vec_init(len);
    v = _arb_vec_init(len);
    arb_init(f);
    arb_init(f + 1);

    /* 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);

        _arb_poly_lgamma_series_at_one(u, len, wp);

        _arb_vec_neg(u, u, len);
        _arb_poly_exp_series(t, u, len, len, wp);

        if (r == 1)
        {
            _arb_vec_swap(v, t, len);
        }
        else if (r <= 0)
        {
            arb_set(f, h);
            arb_one(f + 1);
            rflen = FLINT_MIN(len, 2 - r);
            _arb_poly_rising_ui_series(u, f, FLINT_MIN(2, len), 1 - r, rflen, wp);
            _arb_poly_mullow(v, t, len, u, rflen, len, wp);
        }
        else
        {
            arb_one(f);
            arb_one(f + 1);
            rflen = FLINT_MIN(len, r);
            _arb_poly_rising_ui_series(v, f, FLINT_MIN(2, len), r - 1, rflen, wp);

            /* TODO: use div_series? */
            _arb_poly_inv_series(u, v, rflen, len, wp);
            _arb_poly_mullow(v, t, len, u, len, len, wp);
        }
    }
    else
    {
        /* otherwise use Stirling series */
        arb_gamma_stirling_choose_param(&reflect, &r, &n, h, 1, 0, wp);

        /* rgamma(h) = (gamma(1-h+r) sin(pi h)) / (rf(1-h, r) * pi), h = h0 + t*/
        if (reflect)
        {
            /* u = gamma(r+1-h) */
            arb_sub_ui(f, h, r + 1, wp);
            arb_neg(f, f);
            _arb_poly_gamma_stirling_eval(t, f, n, len, wp);
            _arb_poly_exp_series(u, t, len, len, wp);
            for (i = 1; i < len; i += 2)
                arb_neg(u + i, u + i);

            /* v = sin(pi x) */
            arb_const_pi(f + 1, wp);
            arb_mul(f, h, f + 1, wp);
            _arb_poly_sin_series(v, f, 2, len, wp);

            _arb_poly_mullow(t, u, len, v, len, len, wp);

            /* rf(1-h,r) * pi */
            if (r == 0)
            {
                arb_const_pi(u, wp);
                _arb_vec_scalar_div(v, t, len, u, wp);
            }
            else
            {
                arb_sub_ui(f, h, 1, wp);
                arb_neg(f, f);
                arb_set_si(f + 1, -1);
                rflen = FLINT_MIN(len, r + 1);
                _arb_poly_rising_ui_series(v, f, FLINT_MIN(2, len), r, rflen, wp);
                arb_const_pi(u, wp);
                _arb_vec_scalar_mul(v, v, rflen, u, wp);

                /* divide by rising factorial */
                /* TODO: might better to use div_series, when it has a good basecase */
                _arb_poly_inv_series(u, v, rflen, len, wp);
                _arb_poly_mullow(v, t, len, u, len, len, wp);
            }
        }
        else
        {
            /* rgamma(h) = rgamma(h+r) rf(h,r) */
            if (r == 0)
            {
                arb_add_ui(f, h, r, wp);
                _arb_poly_gamma_stirling_eval(t, f, n, len, wp);
                _arb_vec_neg(t, t, len);
                _arb_poly_exp_series(v, t, len, len, wp);
            }
            else
            {
                arb_set(f, h);
                arb_one(f + 1);
                rflen = FLINT_MIN(len, r + 1);
                _arb_poly_rising_ui_series(t, f, FLINT_MIN(2, len), r, rflen, wp);

                arb_add_ui(f, h, r, wp);
                _arb_poly_gamma_stirling_eval(v, f, n, len, wp);
                _arb_vec_neg(v, v, len);
                _arb_poly_exp_series(u, v, len, len, wp);

                _arb_poly_mullow(v, u, len, t, rflen, len, wp);
            }
        }
    }

    /* compose with nonconstant part */
    arb_zero(t);
    _arb_vec_set(t + 1, h + 1, hlen - 1);
    _arb_poly_compose_series(res, v, len, t, hlen, len, prec);

    arb_clear(f);
    arb_clear(f + 1);
    _arb_vec_clear(t, len);
    _arb_vec_clear(u, len);
    _arb_vec_clear(v, len);
}
Example #4
0
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);
}