text: /*
      Copyright (C) 2014 Fredrik Johansson
  
      This file is part of Arb.
  
      Arb is free software: you can redistribute it and/or modify it under
      the terms of the GNU Lesser General Public License (LGPL) as published
      by the Free Software Foundation; either version 2.1 of the License, or
      (at your option) any later version.  See <http://www.gnu.org/licenses/>.
  */
  
  #include "flint/fmpz_poly.h"
  #include "acb.h"
  
  void rising_difference_polynomial(fmpz * s, fmpz * c, ulong m);
  
  void
  acb_rising_ui_rs(acb_t y, const acb_t x, ulong n, ulong m, slong prec)
  {
      acb_ptr xs;
      acb_t t, u, v;
      ulong i, k, rem;
      fmpz_t c, h;
      fmpz *s, *d;
      slong wp;
  
      if (n == 0)
      {
          acb_one(y);
          return;
      }
  
      if (n == 1)
      {
          acb_set_round(y, x, prec);
          return;
      }
  
      wp = ARF_PREC_ADD(prec, FLINT_BIT_COUNT(n));
  
      acb_init(t);
      acb_init(u);
      acb_init(v);
      fmpz_init(c);
      fmpz_init(h);
  
      if (m == 0)
      {
          ulong m1, m2;
          m1 = 0.2 * pow(2.0 * wp, 0.4);
          m2 = n_sqrt(n);
          m = FLINT_MIN(m1, m2);
      }
  
      m = FLINT_MIN(m, n);
      m = FLINT_MAX(m, 1);
  
      xs = _acb_vec_init(m + 1);
      d = _fmpz_vec_init(m * m);
      s = _fmpz_vec_init(m + 1);
  
      _acb_vec_set_powers(xs, x, m + 1, wp);
  
      rising_difference_polynomial(s, d, m);
  
      /* tail */
      rem = m;
      while (rem + m <= n)
          rem += m;
      acb_one(y);
      for (k = rem; k < n; k++)
      {
          acb_add_ui(t, xs + 1, k, wp);
          acb_mul(y, y, t, wp);
      }
  
      /* initial rising factorial */
      acb_zero(t);
      for (i = 1; i <= m; i++)
          acb_addmul_fmpz(t, xs + i, s + i, wp);
  
      acb_mul(y, y, t, wp);
  
      /* the leading coefficient is always the same */
      acb_mul_fmpz(xs + m - 1, xs + m - 1, d + m - 1 + 0, wp);
  
      for (k = 0; k + 2 * m <= n; k += m)
      {
          for (i = 0; i < m - 1; i++)
          {
              fmpz_set_ui(h, k);
              _fmpz_poly_evaluate_horner_fmpz(c, d + i * m, m - i, h);
  
              if (i == 0)
                  acb_add_fmpz(t, t, c, wp);
              else
                  acb_addmul_fmpz(t, xs + i, c, wp);
          }
  
          acb_add(t, t, xs + m - 1, wp);
          acb_mul(y, y, t, wp);
      }
  
      acb_set_round(y, y, prec);
  
      acb_clear(t);
      acb_clear(u);
      acb_clear(v);
      _acb_vec_clear(xs, m + 1);
      _fmpz_vec_clear(d, m * m);
      _fmpz_vec_clear(s, m + 1);
      fmpz_clear(c);
      fmpz_clear(h);
  }
  
  