mirror of
https://github.com/vale981/arb
synced 2025-03-04 17:01:40 -05:00
factor out mag_polylog_tail
This commit is contained in:
Fredrik Johansson
parent
dd7cdca99f
commit
a22075c52e
6 changed files with 298 additions and 134 deletions
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@ -43,92 +43,6 @@ arb_get_si_lower(const arb_t x)
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return v;
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}
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void
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polylog_remainder_bound(mag_t u, const mag_t z, long sigma, ulong d, ulong N)
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{
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mag_t TN, UN, t;
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if (N < 2)
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{
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mag_inf(u);
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return;
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}
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mag_init(TN);
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mag_init(UN);
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mag_init(t);
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if (mag_cmp_2exp_si(z, 0) >= 0)
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{
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mag_inf(u);
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}
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else
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{
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/* Bound T(N) */
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mag_pow_ui(TN, z, N);
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/* multiply by log(N)^d */
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if (d > 0)
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{
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mag_log_ui(t, N);
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mag_pow_ui(t, t, d);
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mag_mul(TN, TN, t);
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}
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/* multiply by 1/k^s */
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if (sigma > 0)
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{
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mag_set_ui_lower(t, N);
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mag_pow_ui_lower(t, t, sigma);
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mag_div(TN, TN, t);
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}
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else if (sigma < 0)
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{
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mag_set_ui(t, N);
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mag_pow_ui(t, t, -sigma);
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mag_mul(TN, TN, t);
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}
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/* Bound U(N) */
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mag_set(UN, z);
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/* multiply by (1 + 1/N)**S */
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if (sigma < 0)
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{
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mag_binpow_uiui(t, N, -sigma);
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mag_mul(UN, UN, t);
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}
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/* multiply by (1 + 1/(N log(N)))^d */
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if (d > 0)
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{
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ulong nl;
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/* rounds down */
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nl = mag_d_log_lower_bound(N) * N * (1 - 1e-13);
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mag_binpow_uiui(t, nl, d);
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mag_mul(UN, UN, t);
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}
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/* T(N) / (1 - U(N)) */
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if (mag_cmp_2exp_si(UN, 0) >= 0)
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{
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mag_inf(u);
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}
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else
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{
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mag_one(t);
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mag_sub_lower(t, t, UN);
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mag_div(u, TN, t);
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}
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}
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mag_clear(TN);
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mag_clear(UN);
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mag_clear(t);
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}
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long
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polylog_choose_terms(mag_t err, long sigma, const mag_t z, long d, long prec)
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{
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@ -136,7 +50,7 @@ polylog_choose_terms(mag_t err, long sigma, const mag_t z, long d, long prec)
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for (N = 3; ; N = FLINT_MAX(N+3, N*1.1))
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{
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polylog_remainder_bound(err, z, sigma, d, N);
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mag_polylog_tail(err, z, sigma, d, N);
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/* TODO: do something else when |Li_s(z)| is very small/very large? */
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if (mag_cmp_2exp_si(err, -prec) < 0)
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@ -341,7 +255,7 @@ _acb_poly_polylog_cpx_small(acb_ptr w, const acb_t s, const acb_t z, long len, l
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for (k = 0; k < len; k++)
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{
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polylog_remainder_bound(err, zmag, sigma, k, N);
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mag_polylog_tail(err, zmag, sigma, k, N);
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mag_rfac_ui(errf, k);
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mag_mul(err, err, errf);
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@ -327,3 +327,52 @@ Special functions
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Sets *z* to an upper bound for `|B_n| / n!` where `B_n` denotes
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a Bernoulli number.
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.. function:: void mag_polylog_tail(mag_t u, const mag_t z, long s, ulong d, ulong N)
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Sets *u* to an upper bound for
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.. math ::
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\sum_{k=N}^{\infty} \frac{z^k \log^d(k)}{k^s}.
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Note: in applications where `s` in this formula may be
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real or complex, the user can simply
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substitute any convenient integer `s'` such that `s' \le \operatorname{Re}(s)`.
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Denote the terms by `T(k)`. We pick a nonincreasing function `U(k)` such that
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.. math ::
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\frac{T(k+1)}{T(k)} = z \left(\frac{k}{k+1}\right)^s
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\left( \frac{\log(k+1)}{\log(k)} \right)^d \le U(k).
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Then, as soon as `U(N) < 1`,
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.. math ::
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\sum_{k=N}^{\infty} T(k)
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\le T(N) \sum_{k=0}^{\infty} U(N)^k = \frac{T(N)}{1 - U(N)}.
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In particular, we take
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.. math ::
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U(k) = z \; B(k, \max(0, -s)) \; B(k \log(k), d)
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where `B(m,n) = (1 + 1/m)^n`. This follows from the bounds
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.. math ::
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\left(\frac{k}{k+1}\right)^{s}
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\le \begin{cases}
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1 & \text{if } s \ge 0 \\
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(1 + 1/k)^{-s} & \text{if } s < 0.
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\end{cases}
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and
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.. math ::
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\left( \frac{\log(k+1)}{\log(k)} \right)^d \le
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\left(1 + \frac{1}{k \log(k)}\right)^d.
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@ -30,52 +30,8 @@ where
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T(k) = \frac{z^k \log^d(k)}{k^s}.
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Let `U(k)` be a nonincreasing bound for the magnitude of the term ratio
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.. math ::
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\frac{T(k+1)}{T(k)} = z \left(\frac{k}{k+1}\right)^s
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\left( \frac{\log(k+1)}{\log(k)} \right)^d.
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Then the remainder after the `k = N-1` term is bounded by
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.. math ::
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\left| \sum_{k=N}^{\infty} T(k) \right|
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\le |T(N)| \sum_{k=0}^{\infty} U(N)^k = \frac{|T(N)|}{1 - U(N)}
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whenever `U(N) < 1`.
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If `s = \sigma + i \tau` where `\sigma, \tau \in \mathbb{R}`, we can take
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.. math ::
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U(k) = |z| \; B(k, \max(0, -\sigma)) \; B(k \log(k), d)
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wherein `B(m,n) = (1 + 1/m)^n`. This follows from the bounds
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.. math ::
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\left| \left(\frac{k}{k+1}\right)^s \right|
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= \left(\frac{k}{k+1}\right)^{\sigma}
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\le \begin{cases}
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1 & \text{if } \sigma \ge 0 \\
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(1 + 1/k)^{-\sigma} & \text{if } \sigma < 0.
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\end{cases}
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and
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.. math ::
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\left( \frac{\log(k+1)}{\log(k)} \right)^d \le
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\left(1 + \frac{1}{k \log(k)}\right)^d.
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We can replace `\sigma` with any lower bound, conveniently
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the nearest integer in the direction of `-\infty`, and `|z|` with any
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upper bound.
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To bound `B(m,n)` when `m` is large, it is useful to note that
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`B(m,n) = \exp(n \log(1+1/m)) \le \exp(n/m)`).
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The remainder term `\left| \sum_{k=N}^{\infty} T(k) \right|` is bounded
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via :func:`mag_polylog_tail`.
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Expansion for general z
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-------------------------------------------------------------------------------
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2
mag.h
2
mag.h
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@ -585,6 +585,8 @@ void mag_hypot(mag_t z, const mag_t x, const mag_t y);
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void mag_binpow_uiui(mag_t b, ulong m, ulong n);
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void mag_polylog_tail(mag_t u, const mag_t z, long sigma, ulong d, ulong N);
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void mag_set_ui(mag_t z, ulong x);
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void mag_set_ui_lower(mag_t z, ulong x);
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113
mag/polylog_tail.c
Normal file
113
mag/polylog_tail.c
Normal file
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@ -0,0 +1,113 @@
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/*=============================================================================
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This file is part of ARB.
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ARB is free software; you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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ARB is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with ARB; if not, write to the Free Software
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Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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=============================================================================*/
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/******************************************************************************
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Copyright (C) 2014 Fredrik Johansson
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******************************************************************************/
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#include "mag.h"
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void
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mag_polylog_tail(mag_t u, const mag_t z, long sigma, ulong d, ulong N)
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{
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mag_t TN, UN, t;
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if (N < 2)
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{
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mag_inf(u);
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return;
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}
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mag_init(TN);
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mag_init(UN);
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mag_init(t);
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if (mag_cmp_2exp_si(z, 0) >= 0)
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{
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mag_inf(u);
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}
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else
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{
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/* Bound T(N) */
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mag_pow_ui(TN, z, N);
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/* multiply by log(N)^d */
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if (d > 0)
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{
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mag_log_ui(t, N);
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mag_pow_ui(t, t, d);
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mag_mul(TN, TN, t);
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}
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/* multiply by 1/k^s */
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if (sigma > 0)
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{
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mag_set_ui_lower(t, N);
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mag_pow_ui_lower(t, t, sigma);
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mag_div(TN, TN, t);
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}
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else if (sigma < 0)
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{
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mag_set_ui(t, N);
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mag_pow_ui(t, t, -sigma);
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mag_mul(TN, TN, t);
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}
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/* Bound U(N) */
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mag_set(UN, z);
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/* multiply by (1 + 1/N)**S */
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if (sigma < 0)
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{
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mag_binpow_uiui(t, N, -sigma);
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mag_mul(UN, UN, t);
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}
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/* multiply by (1 + 1/(N log(N)))^d */
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if (d > 0)
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{
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ulong nl;
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/* rounds down */
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nl = mag_d_log_lower_bound(N) * N * (1 - 1e-13);
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mag_binpow_uiui(t, nl, d);
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mag_mul(UN, UN, t);
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}
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/* T(N) / (1 - U(N)) */
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if (mag_cmp_2exp_si(UN, 0) >= 0)
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{
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mag_inf(u);
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}
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else
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{
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mag_one(t);
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mag_sub_lower(t, t, UN);
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mag_div(u, TN, t);
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}
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}
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mag_clear(TN);
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mag_clear(UN);
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mag_clear(t);
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}
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130
mag/test/t-polylog_tail.c
Normal file
130
mag/test/t-polylog_tail.c
Normal file
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@ -0,0 +1,130 @@
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/*=============================================================================
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This file is part of ARB.
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ARB is free software; you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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ARB is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with ARB; if not, write to the Free Software
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Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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=============================================================================*/
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/******************************************************************************
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Copyright (C) 2014 Fredrik Johansson
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******************************************************************************/
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#include "mag.h"
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#include "arb.h"
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int main()
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{
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long iter;
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flint_rand_t state;
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printf("polylog_tail....");
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fflush(stdout);
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flint_randinit(state);
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for (iter = 0; iter < 2000; iter++)
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{
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mag_t xb, yb;
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ulong N, k, d;
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long s, prec;
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arb_t z, t, u, r;
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mag_init(xb);
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mag_init(yb);
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arb_init(z);
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arb_init(t);
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arb_init(u);
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arb_init(r);
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mag_randtest_special(xb, state, 6);
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mag_randtest_special(yb, state, 6);
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N = n_randint(state, 100);
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d = n_randint(state, 100);
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s = n_randint(state, 100) - 50;
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prec = 4 * MAG_BITS;
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mag_polylog_tail(yb, xb, s, d, N);
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arb_zero(z);
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arf_set_mag(arb_midref(z), xb);
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arb_zero(r);
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for (k = N; k < N + 100; k++)
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{
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arb_pow_ui(t, z, k, prec);
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arb_log_ui(u, k, prec);
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arb_pow_ui(u, u, d, prec);
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arb_mul(t, t, u, prec);
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arb_set_ui(u, k);
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if (s >= 0)
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{
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arb_pow_ui(u, u, s, prec);
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arb_div(t, t, u, prec);
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}
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else
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{
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arb_pow_ui(u, u, -s, prec);
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arb_mul(t, t, u, prec);
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}
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arb_add(r, r, t, prec);
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}
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MAG_CHECK_BITS(xb)
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MAG_CHECK_BITS(yb)
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arb_zero(z);
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mag_set(arb_radref(z), yb);
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if (!arb_is_finite(z))
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arb_indeterminate(z);
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if (!arb_contains(z, r))
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{
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printf("FAIL\n\n");
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printf("N = %lu\n\n", N);
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printf("d = %lu\n\n", d);
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printf("s = %ld\n\n", s);
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printf("xb = "); mag_printd(xb, 15); printf("\n\n");
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printf("yb = "); mag_printd(yb, 15); printf("\n\n");
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printf("z = "); arb_printd(z, 15); printf("\n\n");
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printf("r = "); arb_printd(r, 15); printf("\n\n");
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abort();
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}
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mag_polylog_tail(xb, xb, s, d, N);
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if (!mag_equal(xb, yb))
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{
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printf("FAIL (aliasing)\n\n");
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abort();
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}
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mag_clear(xb);
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mag_clear(yb);
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arb_clear(z);
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arb_clear(t);
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arb_clear(u);
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arb_clear(r);
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}
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flint_randclear(state);
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flint_cleanup();
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printf("PASS\n");
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return EXIT_SUCCESS;
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}
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