add acb_elliptic_p_prime

acb_elliptic.h

@@ -50,6 +50,8 @@ void acb_elliptic_pi_inc(acb_t res, const acb_t n, const acb_t phi, const acb_t
 
 void acb_elliptic_p(acb_t r, const acb_t z, const acb_t tau, slong prec);
 
+void acb_elliptic_p_prime(acb_t r, const acb_t z, const acb_t tau, slong prec);
+
 void acb_elliptic_p_jet(acb_ptr r, const acb_t z, const acb_t tau, slong len, slong prec);
 
 void _acb_elliptic_p_series(acb_ptr res, acb_srcptr z, slong zlen, const acb_t tau, slong len, slong prec);

acb_elliptic/p_prime.c

@@ -0,0 +1,58 @@
+/*
+    Copyright (C) 2021 Daniel Schultz
+
+    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 "acb_elliptic.h"
+#include "acb_modular.h"
+
+void
+acb_elliptic_p_prime(acb_t r, const acb_t z, const acb_t tau, slong prec)
+{
+    acb_struct tz[4];
+    acb_t t1, t2, t3;
+    int i, real;
+
+    real = acb_is_real(z) && arb_is_int_2exp_si(acb_realref(tau), -1) &&
+                arb_is_positive(acb_imagref(tau));
+
+    acb_init(t1);
+    acb_init(t2);
+    acb_init(t3);
+
+    for (i = 0; i < 4; i++)
+        acb_init(tz + i);
+
+    acb_modular_theta(tz + 0, tz + 1, tz + 2, tz + 3, z, tau, prec);
+
+    /* (-2*pi*eta^2/theta1)^3*theta2*theta3*theta4 */
+    acb_const_pi(t2, prec);
+    acb_mul_2exp_si(t2, t2, 1);
+    acb_neg(t2, t2);
+    acb_modular_eta(t3, tau, prec);
+    acb_mul(t1, t3, t3, prec);
+    acb_mul(t3, t1, t2, prec);
+    acb_div(t1, t3, tz + 0, prec);
+    acb_mul(t2, t1, t1, prec);
+    acb_mul(t3, t1, t2, prec);
+    acb_mul(t1, tz + 1, tz + 2, prec);
+    acb_mul(t2, t1, tz + 3, prec);
+    acb_mul(r, t3, t2, prec);
+
+    if (real)
+        arb_zero(acb_imagref(r));
+
+    acb_clear(t1);
+    acb_clear(t2);
+    acb_clear(t3);
+
+    for (i = 0; i < 4; i++)
+        acb_clear(tz + i);
+}
+

acb_elliptic/test/t-p_p_prime.c

@@ -0,0 +1,73 @@
+/*
+    Copyright (C) 2021 Daniel Schultz
+
+    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 "acb_elliptic.h"
+
+int main()
+{
+    slong iter;
+    flint_rand_t state;
+
+    flint_printf("p_p_prime....");
+    fflush(stdout);
+
+    flint_randinit(state);
+
+    for (iter = 0; iter < 1000 * arb_test_multiplier(); iter++)
+    {
+        acb_struct pj[2];
+        acb_t tau, z, p, pp;
+        slong prec;
+
+        acb_init(tau);
+        acb_init(z);
+        acb_init(p);
+        acb_init(pp);
+        acb_init(pj + 0);
+        acb_init(pj + 1);
+
+        prec = 2 + n_randint(state, 400);
+
+        acb_randtest(z, state, 1 + n_randint(state, 200), 1 + n_randint(state, 10));
+        acb_randtest(tau, state, 1 + n_randint(state, 200), 1 + n_randint(state, 10));
+        if (arf_sgn(arb_midref(acb_imagref(tau))) < 0)
+            acb_neg(tau, tau);
+
+        acb_elliptic_p(p, z, tau, prec);
+        acb_elliptic_p_prime(pp, z, tau, prec);
+        acb_elliptic_p_jet(pj, z, tau, 2, prec);
+
+        if (!acb_overlaps(p, pj + 0) || !acb_overlaps(pp, pj + 1))
+        {
+            flint_printf("FAIL (overlap)\n");
+            flint_printf("tau = "); acb_printd(tau, 30); flint_printf("\n\n");
+            flint_printf("z = "); acb_printd(z, 30); flint_printf("\n\n");
+            flint_printf("p = "); acb_printd(p, 30); flint_printf("\n\n");
+            flint_printf("pp = "); acb_printd(pp, 30); flint_printf("\n\n");
+            flint_printf("pj0 = "); acb_printd(pj + 0, 30); flint_printf("\n\n");
+            flint_printf("pj1 = "); acb_printd(pj + 1, 30); flint_printf("\n\n");
+            flint_abort();
+        }
+
+        acb_clear(tau);
+        acb_clear(z);
+        acb_clear(p);
+        acb_clear(pp);
+        acb_clear(pj + 0);
+        acb_clear(pj + 1);
+    }
+
+    flint_randclear(state);
+    flint_cleanup();
+    flint_printf("PASS\n");
+    return EXIT_SUCCESS;
+}
+

doc/source/acb_elliptic.rst

@@ -299,6 +299,10 @@ The main reference is chapter 23 in [NIST2012]_.
             \frac{\theta_4^2(z,\tau)}{\theta_1^2(z,\tau)} -
             \frac{\pi^2}{3} \left[ \theta_2^4(0,\tau) + \theta_3^4(0,\tau)\right].
 
+.. function:: void acb_elliptic_p_prime(acb_t res, const acb_t z, const acb_t tau, slong prec)
+
+    Computes the derivative `\wp'(z, \tau)` of Weierstrass's elliptic function `\wp(z, \tau)`.
+
 .. function:: void acb_elliptic_p_jet(acb_ptr res, const acb_t z, const acb_t tau, slong len, slong prec)
 
     Computes the formal power series `\wp(z + x, \tau) \in \mathbb{C}[[x]]`,