hiro/arb
1
0
Fork 0
mirror of https://github.com/vale981/arb synced 2025-03-04 17:01:40 -05:00
Code Issues Projects Releases Packages Wiki Activity Actions

fix and refactor derivative bounds for legendre polynomials

Browse source
This commit is contained in:
Fredrik Johansson 2017-10-10 22:34:09 +02:00
parent 49e2d28582
commit 8947049b77
4 changed files with 158 additions and 26 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

1
arb_hypgeom.h
View file

@ -104,6 +104,7 @@ void arb_hypgeom_hermite_h(arb_t res, const arb_t nu, const arb_t z, slong prec)
void arb_hypgeom_legendre_p(arb_t res, const arb_t n, const arb_t m, const arb_t z, int type, slong prec);
void arb_hypgeom_legendre_q(arb_t res, const arb_t n, const arb_t m, const arb_t z, int type, slong prec);
void arb_hypgeom_legendre_p_ui_deriv_bound(mag_t dp, mag_t dp2, ulong n, const arb_t x, const arb_t x2sub1);
void arb_hypgeom_legendre_p_ui_asymp(arb_t res, arb_t res2, ulong n, const arb_t x, slong K, slong prec);
void arb_hypgeom_legendre_p_ui_one(arb_t res, arb_t res2, ulong n, const arb_t x, slong K, slong prec);
void arb_hypgeom_legendre_p_ui_zero(arb_t res, arb_t res2, ulong n, const arb_t x, slong K, slong prec);

77
arb_hypgeom/legendre_p_ui.c
View file

@ -26,6 +26,54 @@ static double log2_bin_uiui_fast(ulong n, ulong k)
return n * htab[FLINT_MIN(k, 15)];
}
void
arb_hypgeom_legendre_p_ui_deriv_bound(mag_t dp, mag_t dp2, ulong n, const arb_t x, const arb_t x2sub1)
{
mag_t t, u, xm, x2sub1m;
mag_init(t);
mag_init(u);
mag_init(xm);
mag_init(x2sub1m);
arb_get_mag(xm, x);
arb_get_mag_lower(x2sub1m, x2sub1);
/* |P'(x)| <= min(n(n+1)/2, n/sqrt(1-x^2)) */
mag_set_ui(t, n);
mag_add_ui(t, t, 1);
mag_mul_2exp_si(t, t, -1);
mag_rsqrt(u, x2sub1m);
mag_max(t, t, u);
mag_mul_ui(dp, t, n);
/* |P''(x)| <= min((n+2)(n+1)n(n-1)/8, (2x|P'(x)| + n(n+1))/(1-x^2)) */
mag_mul(dp2, dp, xm);
mag_mul_2exp_si(dp2, dp2, 1);
mag_set_ui(t, n);
mag_add_ui(t, t, 1);
mag_mul_ui(t, t, n);
mag_add(dp2, dp2, t);
mag_div(dp2, dp2, x2sub1m);
mag_set_ui(t, n);
mag_add_ui(t, t, 2);
mag_set_ui(u, n);
mag_add_ui(u, u, 1);
mag_mul(t, t, u);
mag_mul_ui(t, t, n);
mag_mul_ui(t, t, n - 1);
mag_mul_2exp_si(t, t, -3);
mag_min(dp2, dp2, t);
mag_clear(t);
mag_clear(u);
mag_clear(xm);
mag_clear(x2sub1m);
}
void
arb_hypgeom_legendre_p_ui(arb_t res, arb_t res_prime, ulong n, const arb_t x, slong prec)
{
@ -274,11 +322,9 @@ arb_hypgeom_legendre_p_ui(arb_t res, arb_t res_prime, ulong n, const arb_t x, sl
}
else if (FLINT_MIN(cost_zero, cost_one) < (1e6 * prec) * prec && n < UWORD_MAX / 4)
{
mag_t t, u, err1, err2, xrad;
mag_t err1, err2, xrad;
arb_t xmid;
mag_init(t);
mag_init(u);
mag_init(err1);
mag_init(err2);
mag_init(xrad);
@ -288,14 +334,7 @@ arb_hypgeom_legendre_p_ui(arb_t res, arb_t res_prime, ulong n, const arb_t x, sl
mag_zero(arb_radref(xmid));
mag_set(xrad, arb_radref(x));
/* |P'(x)| <= min(n(n+1)/2, n/sqrt(1-x^2)) */
arb_get_mag_lower(u, x2sub1);
mag_rsqrt(u, u);
mag_set_ui(t, n + 1);
mag_mul_2exp_si(t, t, -1);
mag_max(t, t, u);
mag_mul_ui(t, t, n);
mag_mul(err1, t, xrad);
arb_hypgeom_legendre_p_ui_deriv_bound(err1, err2, n, x, x2sub1);
if (cost_zero < cost_one)
arb_hypgeom_legendre_p_ui_zero(res, res_prime, n, xmid, K_zero, wp + cancellation_zero);
@ -304,32 +343,18 @@ arb_hypgeom_legendre_p_ui(arb_t res, arb_t res_prime, ulong n, const arb_t x, sl
if (res != NULL)
{
mag_mul(err1, err1, xrad);
arb_add_error_mag(res, err1);
arb_set_round(res, res, prec);
}
if (res_prime != NULL)
{
/* |P''(x)| <= min((n+2)(n+1)n(n-1)/8, (2x|P'(x)| + n(n+1))/(1-x^2) */
arb_get_mag_lower(u, x2sub1);
mag_set_ui(t, n);
mag_mul_ui(t, t, n + 1);
mag_div(t, t, u);
mag_add(err2, err2, t);
mag_set_ui(t, n + 2);
mag_mul_ui(t, t, n + 1);
mag_mul_ui(t, t, n);
mag_mul_ui(t, t, n - 1);
mag_mul_2exp_si(t, t, -3);
mag_min(err2, err2, t);
mag_mul(err2, err2, xrad);
arb_add_error_mag(res_prime, err2);
arb_set_round(res_prime, res_prime, prec);
}
mag_clear(t);
mag_clear(u);
mag_clear(err1);
mag_clear(err2);
mag_clear(xrad);

98
arb_hypgeom/test/t-legendre_p_ui_deriv_bound.c Normal file
View file

@ -0,0 +1,98 @@
/*
Copyright (C) 2017 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_hypgeom.h"
int main()
{
slong iter;
flint_rand_t state;
flint_printf("legendre_p_ui_deriv_bound....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 10000 * arb_test_multiplier(); iter++)
{
arb_t x, x2sub1, y, z;
mag_t m1, m2, exact;
ulong n;
slong prec;
arb_init(x);
arb_init(x2sub1);
arb_init(y);
arb_init(z);
mag_init(m1);
mag_init(m2);
mag_init(exact);
n = n_randtest(state) % 10000;
prec = 64;
arb_randtest(x, state, 2 * prec, 0);
mag_zero(arb_radref(x));
while (arf_cmpabs_2exp_si(arb_midref(x), 0) >= 0)
arb_mul_2exp_si(x, x, -1);
arb_mul_2exp_si(x, x, -n_randint(state, 10));
arb_mul(x2sub1, x, x, 2 * prec);
arb_sub_ui(x2sub1, x2sub1, 1, 2 * prec);
arb_neg(x2sub1, x2sub1);
arb_hypgeom_legendre_p_ui_deriv_bound(m1, m2, n, x, x2sub1);
arb_hypgeom_legendre_p_ui(y, z, n, x, prec);
arb_get_mag_lower(exact, z);
if (mag_cmp(m1, exact) < 0)
{
flint_printf("FAIL: first derivative\n\n");
flint_printf("n = %wu\n\n", n);
flint_printf("x = "); arb_printn(x, 50, 0); flint_printf("\n\n");
flint_printf("z = "); arb_printn(z, 50, 0); flint_printf("\n\n");
flint_printf("m1 = "); mag_printd(m1, 15); flint_printf("\n\n");
flint_abort();
}
arb_mul(z, z, x, prec);
arb_mul_2exp_si(z, z, 1);
arb_mul_ui(y, y, n, prec);
arb_mul_ui(y, y, n + 1, prec);
arb_sub(z, z, y, prec);
arb_div(z, z, x2sub1, prec);
arb_get_mag_lower(exact, z);
if (mag_cmp(m2, exact) < 0)
{
flint_printf("FAIL: second derivative\n\n");
flint_printf("n = %wu\n\n", n);
flint_printf("x = "); arb_printn(x, 50, 0); flint_printf("\n\n");
flint_printf("z = "); arb_printn(z, 50, 0); flint_printf("\n\n");
flint_printf("m2 = "); mag_printd(m2, 15); flint_printf("\n\n");
flint_abort();
}
arb_clear(x);
arb_clear(x2sub1);
arb_clear(y);
arb_clear(z);
mag_clear(m1);
mag_clear(m2);
mag_clear(exact);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}

8
doc/source/arb_hypgeom.rst
View file

@ -336,6 +336,14 @@ Orthogonal polynomials and functions
See :func:`acb_hypgeom_legendre_p` and :func:`acb_hypgeom_legendre_q`
for definitions.
.. function:: void arb_hypgeom_legendre_p_ui_deriv_bound(mag_t dp, mag_t dp2, ulong n, const arb_t x, const arb_t x2sub1)
Sets *dp* to an upper bound for `P'_n(x)` and *dp2* to an upper
bound for `P''_n(x)` on given *x* assumed to represent a real
number with `|x| \le 1`. The variable *x2sub1* must contain
the precomputed value `1-x^2` (or `x^2-1`). This method is used
internally to bound the propagated error for Legendre polynomials.
.. function:: void arb_hypgeom_legendre_p_ui_zero(arb_t res, arb_t res_prime, ulong n, const arb_t x, slong K, slong prec)
.. function:: void arb_hypgeom_legendre_p_ui_one(arb_t res, arb_t res_prime, ulong n, const arb_t x, slong K, slong prec)