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/*
Copyright (C) 2012 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 "arb_poly.h"
/* allow changing this from the test code */
ARB_DLL slong arb_poly_newton_exp_cutoff = 0;
/* 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
_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
{
slong cutoff;
if (arb_poly_newton_exp_cutoff != 0)
cutoff = arb_poly_newton_exp_cutoff;
else if (prec <= 256)
cutoff = 750;
else
cutoff = 1e5 / pow(log(prec), 3);
if (hlen <= 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, 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);
}
}
}
void
arb_poly_exp_series(arb_poly_t f, const arb_poly_t h, slong n, slong prec)
{
slong hlen = h->length;
if (n == 0)
{
arb_poly_zero(f);
return;
}
if (hlen == 0)
{
arb_poly_one(f);
return;
}
if (hlen == 1)
n = 1;
arb_poly_fit_length(f, n);
_arb_poly_exp_series(f->coeffs, h->coeffs, hlen, n, prec);
_arb_poly_set_length(f, n);
_arb_poly_normalise(f);
}
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