mirror of
https://github.com/vale981/arb
synced 2025-03-04 17:01:40 -05:00
attempt to improve bounds for Airy functions
This commit is contained in:
fredrik
parent
a7d9c0b1c1
commit
d64055fb3f
3 changed files with 362 additions and 143 deletions
|
|
@ -102,8 +102,8 @@ estimate_airy(double x, double y, int ai)
|
|||
|
||||
/* error propagation based on derivatives */
|
||||
void
|
||||
acb_hypgeom_airy_direct_prop(acb_t ai, acb_t aip, acb_t bi, acb_t bip,
|
||||
const acb_t z, slong n, slong prec)
|
||||
acb_hypgeom_airy_prop(acb_t ai, acb_t aip, acb_t bi, acb_t bip,
|
||||
const acb_t z, slong n, int algo, slong prec)
|
||||
{
|
||||
mag_t aib, aipb, bib, bipb, zb, rad;
|
||||
acb_t zz;
|
||||
|
|
@ -124,7 +124,10 @@ acb_hypgeom_airy_direct_prop(acb_t ai, acb_t aip, acb_t bi, acb_t bip,
|
|||
acb_get_mag(zb, z);
|
||||
|
||||
acb_hypgeom_airy_bound(aib, aipb, bib, bipb, z);
|
||||
acb_hypgeom_airy_direct(ai, aip, bi, bip, zz, n, prec);
|
||||
if (algo == 0)
|
||||
acb_hypgeom_airy_direct(ai, aip, bi, bip, zz, n, prec);
|
||||
else
|
||||
acb_hypgeom_airy_asymp(ai, aip, bi, bip, zz, n, prec);
|
||||
|
||||
if (ai != NULL)
|
||||
{
|
||||
|
|
@ -173,6 +176,24 @@ acb_hypgeom_airy_direct_prop(acb_t ai, acb_t aip, acb_t bi, acb_t bip,
|
|||
acb_clear(zz);
|
||||
}
|
||||
|
||||
void
|
||||
acb_hypgeom_airy_direct_prop(acb_t ai, acb_t aip, acb_t bi, acb_t bip,
|
||||
const acb_t z, slong n, slong prec)
|
||||
{
|
||||
acb_hypgeom_airy_prop(ai, aip, bi, bip, z, n, 0, prec);
|
||||
}
|
||||
|
||||
void
|
||||
acb_hypgeom_airy_asymp2(acb_t ai, acb_t aip, acb_t bi, acb_t bip,
|
||||
const acb_t z, slong n, slong prec)
|
||||
{
|
||||
/* avoid singularity in asymptotic expansion near 0 */
|
||||
if (acb_rel_accuracy_bits(z) > 3)
|
||||
acb_hypgeom_airy_asymp(ai, aip, bi, bip, z, n, prec);
|
||||
else
|
||||
acb_hypgeom_airy_prop(ai, aip, bi, bip, z, n, 1, prec);
|
||||
}
|
||||
|
||||
void
|
||||
acb_hypgeom_airy(acb_t ai, acb_t aip, acb_t bi, acb_t bip, const acb_t z, slong prec)
|
||||
{
|
||||
|
|
@ -210,7 +231,10 @@ acb_hypgeom_airy(acb_t ai, acb_t aip, acb_t bi, acb_t bip, const acb_t z, slong
|
|||
n = wp / (-zmag) + 1;
|
||||
}
|
||||
|
||||
acb_hypgeom_airy_direct(ai, aip, bi, bip, z, n, wp);
|
||||
if (acb_is_exact(z))
|
||||
acb_hypgeom_airy_direct(ai, aip, bi, bip, z, n, wp);
|
||||
else
|
||||
acb_hypgeom_airy_direct_prop(ai, aip, bi, bip, z, n, wp);
|
||||
} /* huge input -- use asymptotics and pick n without overflowing */
|
||||
else if ((arf_cmpabs_2exp_si(re, 64) > 0 || arf_cmpabs_2exp_si(im, 64) > 0))
|
||||
{
|
||||
|
|
@ -227,7 +251,7 @@ acb_hypgeom_airy(acb_t ai, acb_t aip, acb_t bi, acb_t bip, const acb_t z, slong
|
|||
n = FLINT_MAX(n, 1);
|
||||
}
|
||||
|
||||
acb_hypgeom_airy_asymp(ai, aip, bi, bip, z, n, wp);
|
||||
acb_hypgeom_airy_asymp2(ai, aip, bi, bip, z, n, wp);
|
||||
}
|
||||
else /* moderate input */
|
||||
{
|
||||
|
|
@ -239,14 +263,18 @@ acb_hypgeom_airy(acb_t ai, acb_t aip, acb_t bi, acb_t bip, const acb_t z, slong
|
|||
|
||||
if (zmag >= 4.0 && (n = asymp_pick_terms(wp, log(zmag))) != -1)
|
||||
{
|
||||
acb_hypgeom_airy_asymp(ai, aip, bi, bip, z, n, wp);
|
||||
acb_hypgeom_airy_asymp2(ai, aip, bi, bip, z, n, wp);
|
||||
}
|
||||
else if (zmag <= 1.5)
|
||||
{
|
||||
t = 3 * (wp * LOG2) / (2 * z15 * EXP1);
|
||||
t = (wp * LOG2) / (2 * d_lambertw(t));
|
||||
n = FLINT_MAX(t + 1, 2);
|
||||
acb_hypgeom_airy_direct(ai, aip, bi, bip, z, n, wp);
|
||||
|
||||
if (acb_is_exact(z))
|
||||
acb_hypgeom_airy_direct(ai, aip, bi, bip, z, n, wp);
|
||||
else
|
||||
acb_hypgeom_airy_direct_prop(ai, aip, bi, bip, z, n, wp);
|
||||
}
|
||||
else
|
||||
{
|
||||
|
|
|
|||
|
|
@ -24,65 +24,29 @@ arb_bound_exp_neg(mag_t b, const arb_t x)
|
|||
arb_clear(t);
|
||||
}
|
||||
|
||||
/* todo -- should be lt in asymp code? */
|
||||
static int
|
||||
arg_le_2pi3(const acb_t z, const acb_t zeta)
|
||||
{
|
||||
if (arb_is_nonnegative(acb_realref(z)))
|
||||
return 1;
|
||||
|
||||
if (arb_is_positive(acb_imagref(z)) &&
|
||||
arb_is_nonnegative(acb_imagref(zeta)))
|
||||
return 1;
|
||||
|
||||
if (arb_is_negative(acb_imagref(z)) &&
|
||||
arb_is_nonpositive(acb_imagref(zeta)))
|
||||
return 1;
|
||||
|
||||
return 0;
|
||||
}
|
||||
|
||||
/* Implements DLMF 9.7.17. We assume |zeta| >= 1/2 and |arg(z)| <= 2 pi/3 here,
|
||||
ignoring the smaller points which must be dealt with separately. */
|
||||
void
|
||||
acb_hypgeom_airy_bound_9_7_17(mag_t bound, const acb_t z, const acb_t zeta)
|
||||
{
|
||||
mag_t D, t, u, v, zeta_lower;
|
||||
mag_t D, t, u, v, zeta_lower, half;
|
||||
|
||||
mag_init(D);
|
||||
mag_init(t);
|
||||
mag_init(u);
|
||||
mag_init(v);
|
||||
mag_init(zeta_lower);
|
||||
mag_init(half);
|
||||
|
||||
mag_one(half);
|
||||
mag_mul_2exp_si(half, half, -1);
|
||||
|
||||
acb_get_mag_lower(zeta_lower, zeta);
|
||||
mag_max(zeta_lower, zeta_lower, half); /* by assumption */
|
||||
|
||||
/* 2 chi(1) exp(7 pi / (72 |zeta|)) * c_1 */
|
||||
/* simplified bound */
|
||||
if (mag_cmp_2exp_si(zeta_lower, -1) >= 0)
|
||||
mag_one(D);
|
||||
else
|
||||
mag_inf(D);
|
||||
|
||||
if (!arg_le_2pi3(z, zeta))
|
||||
{
|
||||
arb_get_mag_lower(u, acb_realref(zeta));
|
||||
arb_get_mag(v, acb_imagref(zeta));
|
||||
|
||||
/* exp(7 pi / (36 u)) < exp(5/(8 u)) */
|
||||
mag_set_ui_2exp_si(t, 5, -3);
|
||||
mag_div(t, t, u);
|
||||
mag_exp(t, t);
|
||||
|
||||
/* |1/cos(arg(zeta))| = sqrt(1+(v/u)^2) */
|
||||
mag_div(v, v, u);
|
||||
mag_mul(v, v, v);
|
||||
mag_one(u);
|
||||
mag_add(v, v, u);
|
||||
mag_sqrt(v, v);
|
||||
mag_mul(t, t, v);
|
||||
/* c_1 * 4 chi(1) < 0.62 < 1 -- do nothing */
|
||||
|
||||
mag_max(D, D, t);
|
||||
}
|
||||
/* simplified bound assuming |zeta| >= 1/2 */
|
||||
mag_one(D);
|
||||
|
||||
/* exp(-zeta) / (2 sqrt(pi)) * (1 + D / |zeta|) */
|
||||
|
||||
|
|
@ -105,21 +69,238 @@ acb_hypgeom_airy_bound_9_7_17(mag_t bound, const acb_t z, const acb_t zeta)
|
|||
mag_clear(zeta_lower);
|
||||
}
|
||||
|
||||
void
|
||||
acb_hypgeom_airy_bound_arg_le_2pi3(mag_t A, mag_t B, const acb_t z, slong wp)
|
||||
{
|
||||
acb_t zeta, z1;
|
||||
|
||||
acb_init(zeta);
|
||||
acb_init(z1);
|
||||
|
||||
acb_set_round(zeta, z, wp);
|
||||
acb_sqrt(zeta, zeta, wp);
|
||||
acb_cube(zeta, zeta, wp);
|
||||
acb_mul_2exp_si(zeta, zeta, 1);
|
||||
acb_div_ui(zeta, zeta, 3, wp);
|
||||
|
||||
acb_hypgeom_airy_bound_9_7_17(A, z, zeta);
|
||||
|
||||
/* Use Bi(z) = w_1 Ai(z) + 2 w_2 Ai(z exp(+/- 2pi i / 3)),
|
||||
where w_1, w_2 are roots of unity */
|
||||
if (B != NULL)
|
||||
{
|
||||
arb_sqrt_ui(acb_imagref(z1), 3, wp);
|
||||
arb_set_si(acb_realref(z1), -1);
|
||||
acb_mul_2exp_si(z1, z1, -1);
|
||||
|
||||
/* multiply by exp(-2 pi i / 3) in upper half plane
|
||||
and by exp(2 pi i / 3) in lower half plane, to stay close
|
||||
to positive reals */
|
||||
|
||||
/* in principle, we should be computing the union of both cases,
|
||||
but since Bi is conjugate symmetric with the maximum value
|
||||
attained on the positive real line, it's sufficient to consider
|
||||
the case centered on the midpoint */
|
||||
if (arf_sgn(arb_midref(acb_imagref(z))) >= 0)
|
||||
acb_conj(z1, z1);
|
||||
|
||||
acb_mul(z1, z1, z, wp);
|
||||
acb_neg(zeta, zeta); /* same effect regardless of exp(+/-2 pi i/3) */
|
||||
|
||||
acb_hypgeom_airy_bound_9_7_17(B, z1, zeta);
|
||||
mag_mul_2exp_si(B, B, 1);
|
||||
mag_add(B, B, A);
|
||||
}
|
||||
|
||||
acb_clear(zeta);
|
||||
acb_clear(z1);
|
||||
}
|
||||
|
||||
/* near the negative half line, use
|
||||
|Ai(-z)|, |Bi(-z)| <= |Ai(z exp(pi i/3))| + |Ai(z exp(-pi i/3))| */
|
||||
void
|
||||
acb_hypgeom_airy_bound_arg_ge_2pi3(mag_t A, mag_t B, const acb_t z, slong wp)
|
||||
{
|
||||
acb_t zeta, z1, z2;
|
||||
|
||||
acb_init(zeta);
|
||||
acb_init(z1);
|
||||
acb_init(z2);
|
||||
|
||||
arb_sqrt_ui(acb_imagref(z1), 3, wp);
|
||||
arb_one(acb_realref(z1));
|
||||
acb_mul_2exp_si(z1, z1, -1);
|
||||
acb_conj(z2, z1);
|
||||
|
||||
acb_neg_round(zeta, z, wp);
|
||||
acb_mul(z1, z1, zeta, wp);
|
||||
|
||||
acb_sqrt(zeta, zeta, wp);
|
||||
acb_cube(zeta, zeta, wp);
|
||||
acb_mul_2exp_si(zeta, zeta, 1);
|
||||
acb_div_ui(zeta, zeta, 3, wp);
|
||||
acb_mul_onei(zeta, zeta);
|
||||
|
||||
acb_hypgeom_airy_bound_9_7_17(A, z1, zeta);
|
||||
|
||||
/* conjugate symmetry */
|
||||
if (arb_is_zero(acb_imagref(z)))
|
||||
{
|
||||
mag_mul_2exp_si(A, A, 1);
|
||||
}
|
||||
else
|
||||
{
|
||||
mag_t D;
|
||||
mag_init(D);
|
||||
acb_mul(z2, z2, zeta, wp);
|
||||
acb_neg(zeta, zeta);
|
||||
acb_hypgeom_airy_bound_9_7_17(D, z2, zeta);
|
||||
mag_add(A, A, D);
|
||||
mag_clear(D);
|
||||
}
|
||||
|
||||
if (B != NULL)
|
||||
mag_set(B, A);
|
||||
|
||||
acb_clear(zeta);
|
||||
acb_clear(z1);
|
||||
acb_clear(z2);
|
||||
}
|
||||
|
||||
static int
|
||||
arg_lt_2pi3_fast(const acb_t z)
|
||||
{
|
||||
arf_t t;
|
||||
mag_t x, y, s;
|
||||
int res;
|
||||
|
||||
if (arb_is_zero(acb_imagref(z)) && arb_is_nonnegative(acb_realref(z)))
|
||||
return 1;
|
||||
|
||||
arf_init(t);
|
||||
mag_init(x);
|
||||
mag_init(y);
|
||||
mag_init(s);
|
||||
|
||||
arf_set_mag(t, arb_radref(acb_realref(z)));
|
||||
arf_sub(t, arb_midref(acb_realref(z)), t, MAG_BITS, ARF_RND_FLOOR);
|
||||
|
||||
if (arf_sgn(t) >= 0)
|
||||
{
|
||||
res = 1;
|
||||
}
|
||||
else
|
||||
{
|
||||
arf_get_mag(x, t);
|
||||
arb_get_mag_lower(y, acb_imagref(z));
|
||||
mag_set_ui(s, 3);
|
||||
mag_sqrt(s, s);
|
||||
mag_mul(s, s, x);
|
||||
res = mag_cmp(s, y) <= 0;
|
||||
}
|
||||
|
||||
arf_clear(t);
|
||||
mag_clear(x);
|
||||
mag_clear(y);
|
||||
mag_clear(s);
|
||||
|
||||
return res;
|
||||
}
|
||||
|
||||
static int
|
||||
arg_gt_2pi3_fast(const acb_t z)
|
||||
{
|
||||
arf_t t;
|
||||
mag_t x, y, s;
|
||||
int res;
|
||||
|
||||
if (arb_is_zero(acb_imagref(z)) && arb_is_negative(acb_realref(z)))
|
||||
return 1;
|
||||
|
||||
arf_init(t);
|
||||
mag_init(x);
|
||||
mag_init(y);
|
||||
mag_init(s);
|
||||
|
||||
arf_set_mag(t, arb_radref(acb_realref(z)));
|
||||
arf_add(t, arb_midref(acb_realref(z)), t, MAG_BITS, ARF_RND_CEIL);
|
||||
|
||||
if (arf_sgn(t) >= 0)
|
||||
{
|
||||
res = 0;
|
||||
}
|
||||
else
|
||||
{
|
||||
arf_get_mag_lower(x, t);
|
||||
arb_get_mag(y, acb_imagref(z));
|
||||
mag_set_ui_lower(s, 3);
|
||||
mag_sqrt_lower(s, s);
|
||||
mag_mul_lower(s, s, x);
|
||||
res = mag_cmp(s, y) >= 0;
|
||||
}
|
||||
|
||||
arf_clear(t);
|
||||
mag_clear(x);
|
||||
mag_clear(y);
|
||||
mag_clear(s);
|
||||
|
||||
return res;
|
||||
}
|
||||
|
||||
void
|
||||
acb_hypgeom_airy_bound(mag_t ai, mag_t aip, mag_t bi, mag_t bip, const acb_t z)
|
||||
{
|
||||
acb_t zeta;
|
||||
acb_t z1, z2;
|
||||
arf_srcptr zre, zim;
|
||||
mag_t A, B, D, zlo, zhi;
|
||||
slong wp;
|
||||
int near_zero;
|
||||
|
||||
if (acb_contains_zero(z))
|
||||
/* Fast and slightly tighter bounds for real z <= 0 */
|
||||
/* Todo: could implement bounds for z >= 0 too */
|
||||
if (acb_is_real(z) && arb_is_nonpositive(acb_realref(z)))
|
||||
{
|
||||
if (ai != NULL) mag_inf(ai);
|
||||
if (aip != NULL) mag_inf(aip);
|
||||
if (bi != NULL) mag_inf(bi);
|
||||
if (bip != NULL) mag_inf(bip);
|
||||
mag_init(zlo);
|
||||
mag_init(zhi);
|
||||
mag_init(A);
|
||||
mag_init(B);
|
||||
mag_init(D);
|
||||
|
||||
if (ai != NULL || bi != NULL)
|
||||
{
|
||||
acb_get_mag_lower(zlo, z);
|
||||
mag_rsqrt(A, zlo);
|
||||
mag_sqrt(A, A);
|
||||
mag_mul_ui(A, A, 150);
|
||||
mag_set_ui(D, 160);
|
||||
mag_min(A, A, D);
|
||||
mag_mul_2exp_si(A, A, -8);
|
||||
|
||||
if (ai != NULL) mag_set(ai, A);
|
||||
if (bi != NULL) mag_set(bi, A);
|
||||
}
|
||||
|
||||
if (aip != NULL || bip != NULL)
|
||||
{
|
||||
acb_get_mag(zhi, z);
|
||||
mag_sqrt(A, zhi);
|
||||
mag_sqrt(A, A);
|
||||
mag_mul_ui(A, A, 150);
|
||||
mag_set_ui(D, 160);
|
||||
mag_max(B, A, D);
|
||||
mag_mul_2exp_si(B, B, -8);
|
||||
mag_set_ui(D, 67);
|
||||
mag_max(A, A, D);
|
||||
mag_mul_2exp_si(A, A, -8);
|
||||
|
||||
if (aip != NULL) mag_set(aip, A);
|
||||
if (bip != NULL) mag_set(bip, B);
|
||||
}
|
||||
|
||||
mag_clear(zlo);
|
||||
mag_clear(zhi);
|
||||
mag_clear(A);
|
||||
mag_clear(B);
|
||||
mag_clear(D);
|
||||
return;
|
||||
}
|
||||
|
||||
|
|
@ -131,103 +312,92 @@ acb_hypgeom_airy_bound(mag_t ai, mag_t aip, mag_t bi, mag_t bip, const acb_t z)
|
|||
mag_init(zhi);
|
||||
|
||||
wp = MAG_BITS * 2;
|
||||
zre = arb_midref(acb_realref(z));
|
||||
zim = arb_midref(acb_imagref(z));
|
||||
|
||||
/* near the negative half line, use
|
||||
|Ai(-z)|, |Bi(-z)| <= |Ai(z exp(pi i/3))| + |Ai(z exp(-pi i/3))| */
|
||||
if (arf_sgn(zre) < 0 && arf_cmpabs(zre, zim) > 0)
|
||||
acb_get_mag_lower(zlo, z);
|
||||
acb_get_mag(zhi, z);
|
||||
|
||||
if (mag_cmp_2exp_si(zhi, 0) <= 0)
|
||||
{
|
||||
acb_init(z1);
|
||||
acb_init(z2);
|
||||
|
||||
arb_sqrt_ui(acb_imagref(z1), 3, wp);
|
||||
arb_one(acb_realref(z1));
|
||||
acb_mul_2exp_si(z1, z1, -1);
|
||||
acb_conj(z2, z1);
|
||||
|
||||
acb_neg_round(zeta, z, wp);
|
||||
acb_mul(z1, z1, zeta, wp);
|
||||
|
||||
acb_sqrt(zeta, zeta, wp);
|
||||
acb_cube(zeta, zeta, wp);
|
||||
acb_mul_2exp_si(zeta, zeta, 1);
|
||||
acb_div_ui(zeta, zeta, 3, wp);
|
||||
acb_mul_onei(zeta, zeta);
|
||||
|
||||
acb_hypgeom_airy_bound_9_7_17(A, z1, zeta);
|
||||
|
||||
/* conjugate symmetry */
|
||||
if (arb_is_zero(acb_imagref(z)))
|
||||
{
|
||||
mag_mul_2exp_si(A, A, 1);
|
||||
}
|
||||
else
|
||||
{
|
||||
acb_mul(z2, z2, zeta, wp);
|
||||
acb_neg(zeta, zeta);
|
||||
acb_hypgeom_airy_bound_9_7_17(D, z2, zeta);
|
||||
mag_add(A, A, D);
|
||||
}
|
||||
|
||||
mag_set(B, A);
|
||||
|
||||
acb_clear(z1);
|
||||
acb_clear(z2);
|
||||
/* inside unit circle -- don't look at asymptotics */
|
||||
if (ai != NULL) mag_set_ui_2exp_si(ai, 159, -8);
|
||||
if (aip != NULL) mag_set_ui_2exp_si(aip, 125, -8);
|
||||
if (bi != NULL) mag_set_ui_2exp_si(bi, 310, -8);
|
||||
if (bip != NULL) mag_set_ui_2exp_si(bip, 239, -8);
|
||||
}
|
||||
else
|
||||
{
|
||||
acb_set_round(zeta, z, wp);
|
||||
acb_sqrt(zeta, zeta, wp);
|
||||
acb_cube(zeta, zeta, wp);
|
||||
acb_mul_2exp_si(zeta, zeta, 1);
|
||||
acb_div_ui(zeta, zeta, 3, wp);
|
||||
/* look at asymptotics outside unit circle */
|
||||
near_zero = mag_cmp_2exp_si(zlo, 0) <= 0;
|
||||
if (near_zero)
|
||||
mag_one(zlo);
|
||||
|
||||
acb_hypgeom_airy_bound_9_7_17(A, z, zeta);
|
||||
|
||||
/* Use Bi(z) = w_1 Ai(z) + 2 w_2 Ai(z exp(+/- 2pi i / 3)),
|
||||
where w_1, w_2 are roots of unity */
|
||||
if (bi != NULL || bip != NULL)
|
||||
if (arg_lt_2pi3_fast(z))
|
||||
{
|
||||
acb_init(z1);
|
||||
acb_hypgeom_airy_bound_arg_le_2pi3(A, (bi != NULL || bip != NULL) ? B : NULL, z, wp);
|
||||
}
|
||||
else if (arg_gt_2pi3_fast(z))
|
||||
{
|
||||
acb_hypgeom_airy_bound_arg_ge_2pi3(A, (bi != NULL || bip != NULL) ? B : NULL, z, wp);
|
||||
}
|
||||
else
|
||||
{
|
||||
mag_t A2, B2;
|
||||
|
||||
arb_sqrt_ui(acb_imagref(z1), 3, wp);
|
||||
arb_set_si(acb_realref(z1), -1);
|
||||
acb_mul_2exp_si(z1, z1, -1);
|
||||
mag_init(A2);
|
||||
mag_init(B2);
|
||||
|
||||
/* multiply by exp(-2 pi i / 3) in upper half plane
|
||||
and by exp(2 pi i / 3) in lower half plane, to stay close
|
||||
to positive reals */
|
||||
if (arf_sgn(zim) >= 0)
|
||||
acb_conj(z1, z1);
|
||||
acb_hypgeom_airy_bound_arg_le_2pi3(A, (bi != NULL || bip != NULL) ? B : NULL, z, wp);
|
||||
acb_hypgeom_airy_bound_arg_ge_2pi3(A2, (bi != NULL || bip != NULL) ? B2 : NULL, z, wp);
|
||||
|
||||
acb_mul(z1, z1, z, wp);
|
||||
acb_neg(zeta, zeta); /* same effect regardless of exp(+/-2 pi i/3) */
|
||||
mag_max(A, A, A2);
|
||||
mag_max(B, B, A2);
|
||||
|
||||
acb_hypgeom_airy_bound_9_7_17(B, z1, zeta);
|
||||
mag_mul_2exp_si(B, B, 1);
|
||||
mag_add(B, B, A);
|
||||
mag_clear(A2);
|
||||
mag_clear(B2);
|
||||
}
|
||||
|
||||
acb_clear(z1);
|
||||
/* bound |z|^(1/4) */
|
||||
mag_sqrt(zhi, zhi);
|
||||
mag_sqrt(zhi, zhi);
|
||||
|
||||
/* bound |z|^(-1/4) */
|
||||
mag_rsqrt(zlo, zlo);
|
||||
mag_sqrt(zlo, zlo);
|
||||
|
||||
if (ai != NULL) mag_mul(ai, A, zlo);
|
||||
if (aip != NULL) mag_mul(aip, A, zhi);
|
||||
if (bi != NULL) mag_mul(bi, B, zlo);
|
||||
if (bip != NULL) mag_mul(bip, B, zhi);
|
||||
|
||||
if (near_zero)
|
||||
{
|
||||
/* max by bounds on the unit circle */
|
||||
if (ai != NULL)
|
||||
{
|
||||
mag_set_ui_2exp_si(D, 159, -8);
|
||||
mag_max(ai, ai, D);
|
||||
}
|
||||
|
||||
if (aip != NULL)
|
||||
{
|
||||
mag_set_ui_2exp_si(D, 125, -8);
|
||||
mag_max(aip, aip, D);
|
||||
}
|
||||
|
||||
if (bi != NULL)
|
||||
{
|
||||
mag_set_ui_2exp_si(D, 310, -8);
|
||||
mag_max(bi, bi, D);
|
||||
}
|
||||
|
||||
if (bip != NULL)
|
||||
{
|
||||
mag_set_ui_2exp_si(D, 239, -8);
|
||||
mag_max(bip, bip, D);
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
acb_get_mag(zhi, z);
|
||||
acb_get_mag_lower(zlo, z);
|
||||
|
||||
/* bound |z|^(1/4) */
|
||||
mag_sqrt(zhi, zhi);
|
||||
mag_sqrt(zhi, zhi);
|
||||
|
||||
/* bound |z|^(-1/4) */
|
||||
mag_rsqrt(zlo, zlo);
|
||||
mag_sqrt(zlo, zlo);
|
||||
|
||||
if (ai != NULL) mag_mul(ai, A, zlo);
|
||||
if (aip != NULL) mag_mul(aip, A, zhi);
|
||||
if (bi != NULL) mag_mul(bi, B, zlo);
|
||||
if (bip != NULL) mag_mul(bip, B, zhi);
|
||||
|
||||
acb_clear(zeta);
|
||||
mag_clear(A);
|
||||
mag_clear(B);
|
||||
|
|
|
|||
|
|
@ -542,11 +542,25 @@ f_erf_bent(acb_ptr res, const acb_t z, void * param, slong order, slong prec)
|
|||
return 0;
|
||||
}
|
||||
|
||||
/* f(z) = Ai(z) */
|
||||
int
|
||||
f_airy_ai(acb_ptr res, const acb_t z, void * param, slong order, slong prec)
|
||||
{
|
||||
if (order > 1)
|
||||
flint_abort(); /* Would be needed for Taylor method. */
|
||||
|
||||
|
||||
acb_hypgeom_airy(res, NULL, NULL, NULL, z, prec);
|
||||
|
||||
return 0;
|
||||
}
|
||||
|
||||
|
||||
/* ------------------------------------------------------------------------- */
|
||||
/* Main test program */
|
||||
/* ------------------------------------------------------------------------- */
|
||||
|
||||
#define NUM_INTEGRALS 26
|
||||
#define NUM_INTEGRALS 27
|
||||
|
||||
const char * descr[NUM_INTEGRALS] =
|
||||
{
|
||||
|
|
@ -576,6 +590,7 @@ const char * descr[NUM_INTEGRALS] =
|
|||
"int_0^{1000} W_0(x) dx",
|
||||
"int_0^pi max(sin(x), cos(x)) dx",
|
||||
"int_{-1}^1 erf(x/sqrt(0.0002)*0.5+1.5)*exp(-x) dx",
|
||||
"int_{-10}^10 Ai(x) dx",
|
||||
};
|
||||
|
||||
int main(int argc, char *argv[])
|
||||
|
|
@ -975,6 +990,12 @@ int main(int argc, char *argv[])
|
|||
acb_calc_integrate(s, f_erf_bent, NULL, a, b, goal, tol, options, prec);
|
||||
break;
|
||||
|
||||
case 26:
|
||||
acb_set_si(a, -10);
|
||||
acb_set_si(b, 10);
|
||||
acb_calc_integrate(s, f_airy_ai, NULL, a, b, goal, tol, options, prec);
|
||||
break;
|
||||
|
||||
default:
|
||||
abort();
|
||||
}
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue