mirror of
https://github.com/vale981/arb
synced 2025-03-06 01:41:39 -05:00
270 lines
8.1 KiB
C
270 lines
8.1 KiB
C
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/*
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Copyright (C) 2018 Fredrik Johansson
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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 it under
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the terms of the GNU Lesser General Public License (LGPL) as published
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by the Free Software Foundation; either version 2.1 of the License, or
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(at your option) any later version. See <http://www.gnu.org/licenses/>.
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*/
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#include "arb_hypgeom.h"
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/*
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https://dlmf.nist.gov/9.9
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a_k ~ -T(3/8 pi (4k-1))
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a'_k ~ -U(3/8 pi (4k-3))
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b_k ~ -T(3/8 pi (4k-3))
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b'_k ~ -U(3/8 pi (4k-1))
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For a_k and b_k, the u^8 and u^10 truncations are known to give lower
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bounds. [G. Pittaluga and L. Sacripante (1991) Inequalities for the
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zeros of the Airy functions. SIAM J. Math. Anal. 22 (1), pp. 260–267.]
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We don't have proofs for a'_k and b'_k. However, in that case, we can just
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do a single interval Newton step to verify that we have isolated a
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zero (the enclosure must be for the correct zero due to sandwiching).
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*/
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#define AI 0
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#define BI 1
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#define AI_PRIME 2
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#define BI_PRIME 3
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static const double initial[4][10] = {{
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-658118728906175.0,-1150655474581104.0,-1553899449042978.0,-1910288501594969.0,
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-2236074816421182.0,-2539650438812533.0,-2826057838960988.0,
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-3098624122012011.0,-3359689702679955.0,-3610979637739094.0},{
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-330370902027041.0,-920730911234245.0,-1359731821477101.0,-1736658984124319.0,
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-2076373934490092.0,-2390271103799312.0,-2684763040788193.0,
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-2963907159065113.0,-3230475233555475.0,-3486466475611047.0},{
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-286764727967452.0,-914286338795679.0,-1356737313209586.0,-1734816794389239.0,
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-2075083421171399.0,-2389296605766914.0,-2683990299959380.0,
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-2963272965051282.0,-3229941298662311.0,-3486008018531685.0},{
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-645827356227815.0,-1146491233835383.0,-1551601459626981.0,-1908764696253222.0,
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-2234961611612173.0,-2538787015856429.0,-2825360342097020.0,
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-3098043823061022.0,-3359196018589429.0,-3610552233837226.0,
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}};
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void
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_arb_hypgeom_airy_zero(arb_t res, const fmpz_t n, int which, slong prec)
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{
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slong asymp_accuracy, wp;
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if (fmpz_cmp_ui(n, 10) <= 0)
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{
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if (fmpz_sgn(n) <= 0)
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{
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flint_printf("Airy zero only defined for index >= 1\n");
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flint_abort();
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}
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/* The asymptotic expansions work well except when n == 1, so
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use precomputed starting intervals (also for the first
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few larger n as a small optimization). */
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arf_set_d(arb_midref(res), ldexp(initial[which][fmpz_get_ui(n)-1], -48));
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mag_set_d(arb_radref(res), ldexp(1.0, -48));
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asymp_accuracy = 48;
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}
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else
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{
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arb_t z, u, u2, u4, s;
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fmpz_t c;
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arb_init(z);
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arb_init(u);
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arb_init(u2);
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arb_init(u4);
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arb_init(s);
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fmpz_init(c);
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if (which == AI || which == BI_PRIME)
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asymp_accuracy = 13 + 10 * (fmpz_bits(n) - 1);
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else
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{
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fmpz_sub_ui(c, n, 1);
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asymp_accuracy = 13 + 10 * (fmpz_bits(c) - 1);
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}
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wp = asymp_accuracy + 8;
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/* Reduce precision since we may not need to do any Newton steps. */
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if (which == AI || which == BI)
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wp = FLINT_MIN(wp, prec + 8);
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arb_const_pi(z, wp);
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fmpz_mul_2exp(c, n, 2);
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if (which == AI || which == BI_PRIME)
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fmpz_sub_ui(c, c, 1);
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else
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fmpz_sub_ui(c, c, 3);
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fmpz_mul_ui(c, c, 3);
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arb_mul_fmpz(z, z, c, wp);
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arb_mul_2exp_si(z, z, -3);
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arb_inv(u, z, wp);
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arb_mul(u2, u, u, wp);
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arb_mul(u4, u2, u2, wp);
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if (which == AI || which == BI)
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{
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/* u^8 truncation gives lower bound */
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arb_mul_si(s, u4, -108056875, wp);
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arb_addmul_si(s, u2, 6478500, wp);
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arb_add_si(s, s, -967680, wp);
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arb_mul(s, s, u4, wp);
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arb_addmul_si(s, u2, 725760, wp);
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arb_div_ui(s, s, 6967296, wp);
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/* u^10 term gives upper bound */
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arb_mul(u4, u4, u4, 10);
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arb_mul(u4, u4, u2, 10);
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arb_mul_ui(u4, u4, 486, 10);
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}
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else
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{
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/* u^8 truncation gives upper bound */
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arb_mul_si(s, u4, 18683371, wp);
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arb_addmul_si(s, u2, -1087338, wp);
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arb_add_si(s, s, 151200, wp);
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arb_mul(s, s, u4, wp);
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arb_addmul_si(s, u2, -181440, wp);
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arb_div_ui(s, s, 1244160, wp);
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/* u^10 term gives lower bound */
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arb_mul(u4, u4, u4, 10);
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arb_mul(u4, u4, u2, 10);
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arb_mul_ui(u4, u4, 477, 10);
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arb_neg(u4, u4);
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}
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arb_mul_2exp_si(u4, u4, -1);
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arb_add(s, s, u4, wp);
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arb_add_error(s, u4);
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arb_add_ui(s, s, 1, wp);
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arb_root_ui(z, z, 3, wp);
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arb_mul(z, z, z, wp);
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arb_mul(res, z, s, wp);
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arb_neg(res, res);
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arb_clear(z);
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arb_clear(u);
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arb_clear(u2);
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arb_clear(u4);
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arb_clear(s);
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fmpz_clear(c);
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}
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/* Do interval Newton steps for refinement. Important: for the
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primed zeros, we need to do at least one interval Newton step to
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validate the initial (tentative) inclusion. */
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if (asymp_accuracy < prec || (which == AI_PRIME || which == BI_PRIME))
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{
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arb_t f, fprime, root;
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mag_t C, r;
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slong * steps;
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slong step, extraprec;
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arb_init(f);
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arb_init(fprime);
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arb_init(root);
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mag_init(C);
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mag_init(r);
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steps = flint_malloc(sizeof(slong) * FLINT_BITS);
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extraprec = 0.25 * fmpz_bits(n) + 8;
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wp = asymp_accuracy + extraprec;
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/* C = |f''| or |f'''| on the initial interval given by res */
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/* f''(x) = xf(x) */
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/* f'''(x) = xf'(x) + f(x) */
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if (which == AI || which == AI_PRIME)
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arb_hypgeom_airy(f, fprime, NULL, NULL, res, wp);
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else
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arb_hypgeom_airy(NULL, NULL, f, fprime, res, wp);
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if (which == AI || which == BI)
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arb_mul(f, f, res, wp);
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else
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arb_addmul(f, fprime, res, wp);
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arb_get_mag(C, f);
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step = 0;
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steps[step] = prec;
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while (steps[step] / 2 > asymp_accuracy - extraprec)
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{
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steps[step + 1] = steps[step] / 2;
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step++;
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}
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arb_set(root, res);
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for ( ; step >= 0; step--)
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{
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wp = steps[step] + extraprec;
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wp = FLINT_MAX(wp, arb_rel_accuracy_bits(root) + extraprec);
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/* store radius, set root to the midpoint */
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mag_set(r, arb_radref(root));
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mag_zero(arb_radref(root));
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if (which == AI || which == AI_PRIME)
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arb_hypgeom_airy(f, fprime, NULL, NULL, root, wp);
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else
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arb_hypgeom_airy(NULL, NULL, f, fprime, root, wp);
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/* f, f' = f', xf */
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if (which == AI_PRIME || which == BI_PRIME)
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{
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arb_mul(f, f, root, wp);
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arb_swap(f, fprime);
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}
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/* f'([m+/-r]) = f'(m) +/- f''([m +/- r]) * r */
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mag_mul(r, C, r);
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arb_add_error_mag(fprime, r);
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arb_div(f, f, fprime, wp);
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arb_sub(root, root, f, wp);
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/* Verify inclusion so that C is still valid, and for the
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primed zeros also to make sure that the initial
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intervals really were correct. */
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if (!arb_contains(res, root))
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{
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flint_printf("unexpected: no containment of Airy zero\n");
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arb_indeterminate(root);
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break;
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}
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}
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arb_set(res, root);
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arb_clear(f);
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arb_clear(fprime);
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arb_clear(root);
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mag_clear(C);
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mag_clear(r);
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flint_free(steps);
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}
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arb_set_round(res, res, prec);
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}
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void
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arb_hypgeom_airy_zero(arb_t ai, arb_t aip, arb_t bi, arb_t bip, const fmpz_t n, slong prec)
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{
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if (ai != NULL)
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_arb_hypgeom_airy_zero(ai, n, AI, prec);
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if (aip != NULL)
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_arb_hypgeom_airy_zero(aip, n, AI_PRIME, prec);
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if (bi != NULL)
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_arb_hypgeom_airy_zero(bi, n, BI, prec);
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if (bip != NULL)
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_arb_hypgeom_airy_zero(bip, n, BI_PRIME, prec);
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}
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