void _arb_poly_log1p_series(arb_ptr res, arb_srcptr f, slong flen, slong n, slong prec) { arb_t a; flen = FLINT_MIN(flen, n); arb_init(a); arb_log1p(a, f, prec); if (flen == 1) { _arb_vec_zero(res + 1, n - 1); } else if (n == 2) { arb_add_ui(res, f + 0, 1, prec); arb_div(res + 1, f + 1, res + 0, prec); } else if (_arb_vec_is_zero(f + 1, flen - 2)) /* f = a + bx^d */ { slong i, j, d = flen - 1; arb_add_ui(res, f + 0, 1, prec); for (i = 1, j = d; j < n; j += d, i++) { if (i == 1) arb_div(res + j, f + d, res, prec); else arb_mul(res + j, res + j - d, res + d, prec); _arb_vec_zero(res + j - d + 1, flen - 2); } _arb_vec_zero(res + j - d + 1, n - (j - d + 1)); for (i = 2, j = 2 * d; j < n; j += d, i++) arb_div_si(res + j, res + j, i % 2 ? i : -i, prec); } else { arb_ptr f_diff, f_inv; slong alloc; alloc = n + flen; f_inv = _arb_vec_init(alloc); f_diff = f_inv + n; arb_add_ui(f_diff, f, 1, prec); _arb_vec_set(f_diff + 1, f + 1, flen - 1); _arb_poly_inv_series(f_inv, f_diff, flen, n, prec); _arb_poly_derivative(f_diff, f, flen, prec); _arb_poly_mullow(res, f_inv, n - 1, f_diff, flen - 1, n - 1, prec); _arb_poly_integral(res, res, n, prec); _arb_vec_clear(f_inv, alloc); } arb_swap(res, a); arb_clear(a); }
void _arb_poly_compose(arb_ptr res, arb_srcptr poly1, slong len1, arb_srcptr poly2, slong len2, slong prec) { if (len1 == 1) { arb_set_round(res, poly1, prec); } else if (len2 == 1) { _arb_poly_evaluate(res, poly1, len1, poly2, prec); } else if (_arb_vec_is_zero(poly2 + 1, len2 - 2)) { _arb_poly_compose_axnc(res, poly1, len1, poly2, poly2 + len2 - 1, len2 - 1, prec); } else if (len1 <= 7) { _arb_poly_compose_horner(res, poly1, len1, poly2, len2, prec); } else { _arb_poly_compose_divconquer(res, poly1, len1, poly2, len2, prec); } }
void _arb_poly_compose_series(arb_ptr res, arb_srcptr poly1, slong len1, arb_srcptr poly2, slong len2, slong n, slong prec) { if (len2 == 1) { arb_set_round(res, poly1, prec); _arb_vec_zero(res + 1, n - 1); } else if (_arb_vec_is_zero(poly2 + 1, len2 - 2)) /* poly2 is a monomial */ { slong i, j; arb_t t; arb_init(t); arb_set(t, poly2 + len2 - 1); arb_set_round(res, poly1, prec); for (i = 1, j = len2 - 1; i < len1 && j < n; i++, j += len2 - 1) { arb_mul(res + j, poly1 + i, t, prec); if (i + 1 < len1 && j + len2 - 1 < n) arb_mul(t, t, poly2 + len2 - 1, prec); } if (len2 != 2) for (i = 1; i < n; i++) if (i % (len2 - 1) != 0) arb_zero(res + i); arb_clear(t); } else if (len1 < 6 || n < 6) { _arb_poly_compose_series_horner(res, poly1, len1, poly2, len2, n, prec); } else { _arb_poly_compose_series_brent_kung(res, poly1, len1, poly2, len2, n, prec); } }
void _arb_poly_sqrt_series(arb_ptr g, arb_srcptr h, slong hlen, slong len, slong prec) { hlen = FLINT_MIN(hlen, len); while (hlen > 0 && arb_is_zero(h + hlen - 1)) hlen--; if (hlen <= 1) { arb_sqrt(g, h, prec); _arb_vec_zero(g + 1, len - 1); } else if (len == 2) { arb_sqrt(g, h, prec); arb_div(g + 1, h + 1, h, prec); arb_mul(g + 1, g + 1, g, prec); arb_mul_2exp_si(g + 1, g + 1, -1); } else if (_arb_vec_is_zero(h + 1, hlen - 2)) { arb_t t; arb_init(t); arf_set_si_2exp_si(arb_midref(t), 1, -1); _arb_poly_binomial_pow_arb_series(g, h, hlen, t, len, prec); arb_clear(t); } else { arb_ptr t; t = _arb_vec_init(len); _arb_poly_rsqrt_series(t, h, hlen, len, prec); _arb_poly_mullow(g, t, len, h, hlen, len, prec); _arb_vec_clear(t, len); } }
void _arb_poly_exp_series(arb_ptr f, arb_srcptr h, slong hlen, slong n, slong prec) { hlen = FLINT_MIN(hlen, n); if (hlen == 1) { arb_exp(f, h, prec); _arb_vec_zero(f + 1, n - 1); } else if (n == 2) { arb_exp(f, h, prec); arb_mul(f + 1, f, h + 1, prec); /* safe since hlen >= 2 */ } else if (_arb_vec_is_zero(h + 1, hlen - 2)) /* h = a + bx^d */ { slong i, j, d = hlen - 1; arb_t t; arb_init(t); arb_set(t, h + d); arb_exp(f, h, prec); for (i = 1, j = d; j < n; j += d, i++) { arb_mul(f + j, f + j - d, t, prec); arb_div_ui(f + j, f + j, i, prec); _arb_vec_zero(f + j - d + 1, hlen - 2); } _arb_vec_zero(f + j - d + 1, n - (j - d + 1)); arb_clear(t); } else if (hlen <= arb_poly_newton_exp_cutoff) { _arb_poly_exp_series_basecase(f, h, hlen, n, prec); } else { arb_ptr g, t; arb_t u; int fix; g = _arb_vec_init((n + 1) / 2); fix = (hlen < n || h == f || !arb_is_zero(h)); if (fix) { t = _arb_vec_init(n); _arb_vec_set(t + 1, h + 1, hlen - 1); } else t = (arb_ptr) h; arb_init(u); arb_exp(u, h, prec); _arb_poly_exp_series_newton(f, g, t, n, prec, 0, arb_poly_newton_exp_cutoff); if (!arb_is_one(u)) _arb_vec_scalar_mul(f, f, n, u, prec); _arb_vec_clear(g, (n + 1) / 2); if (fix) _arb_vec_clear(t, n); arb_clear(u); } }