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add complex square roots

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This commit is contained in:
Fredrik Johansson 2013-03-22 13:32:48 +01:00
parent 660da34f9d
commit ee9a5eb8fc
4 changed files with 196 additions and 0 deletions
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8
doc/source/fmpcb.rst
View file

@ -314,6 +314,14 @@ Elementary functions
Sets *r* to *x* raised to the power *y*, computed as `x^y = \exp(y \log x)`.
.. function:: void fmpcb_sqrt(fmpcb_t r, const fmpcb_t z, long prec)
Sets *r* to the square root of *z*.
If either the real or imaginary part is exactly zero, only
a single real square root is needed. Generally, we use the formula
`\sqrt{a+bi} = u/2 + ib/u, u = \sqrt{2(|a+bi|+a)}`,
requiring two real square root extractions.
.. function:: void fmpcb_invroot_newton(fmpcb_t r, const fmpcb_t a, ulong m, const fmpcb_t r0, long startprec, long prec)
Given one inverse *m*-th root *r0* (with a valid error bound) of the complex

2
fmpcb.h
View file

@ -533,6 +533,8 @@ void fmpcb_sin_cos_pi(fmpcb_t s, fmpcb_t c, const fmpcb_t z, long prec);
void fmpcb_pow(fmpcb_t r, const fmpcb_t x, const fmpcb_t y, long prec);
void fmpcb_sqrt(fmpcb_t y, const fmpcb_t x, long prec);
void fmpcb_invroot_newton(fmpcb_t r, const fmpcb_t a, ulong m, const fmpcb_t r0, long startprec, long prec);
void fmpcb_root_exp(fmpcb_t r, const fmpcb_t a, long m, long index, long prec);
void fmpcb_root_newton(fmpcb_t r, const fmpcb_t a, long m, long index, long prec);

102
fmpcb/sqrt.c Normal file
View file

@ -0,0 +1,102 @@
/*=============================================================================
This file is part of ARB.
ARB is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
ARB is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with ARB; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
=============================================================================*/
/******************************************************************************
Copyright (C) 2013 Fredrik Johansson
******************************************************************************/
#include "fmpcb.h"
void
fmpcb_sqrt(fmpcb_t y, const fmpcb_t x, long prec)
{
fmprb_t r, t, u;
long wp;
#define a fmpcb_realref(x)
#define b fmpcb_imagref(x)
#define c fmpcb_realref(y)
#define d fmpcb_imagref(y)
if (fmprb_is_zero(b))
{
if (fmprb_is_nonnegative(a))
{
fmprb_sqrt(c, a, prec);
fmprb_zero(d);
return;
}
else if (fmprb_is_nonpositive(a))
{
fmprb_neg(d, a);
fmprb_sqrt(d, d, prec);
fmprb_zero(c);
return;
}
}
if (fmprb_is_zero(a))
{
if (fmprb_is_nonnegative(b))
{
fmprb_mul_2exp_si(c, b, -1);
fmprb_sqrt(c, c, prec);
fmprb_set(d, c);
return;
}
else if (fmprb_is_nonpositive(b))
{
fmprb_mul_2exp_si(c, b, -1);
fmprb_neg(c, c);
fmprb_sqrt(c, c, prec);
fmprb_neg(d, c);
return;
}
}
/* sqrt(a+bi) = sqrt((r+a)/2) + b/sqrt(2*(r+a))*i, r = |a+bi| */
wp = prec + 4;
fmprb_init(r);
fmprb_init(t);
fmprb_init(u);
fmpcb_abs(r, x, wp);
fmprb_add(t, r, a, wp);
fmprb_mul_2exp_si(u, t, 1);
fmprb_sqrt(u, u, wp);
fmprb_div(d, b, u, prec);
fmprb_set_round(c, u, prec);
fmprb_mul_2exp_si(c, c, -1);
fmprb_clear(r);
fmprb_clear(t);
fmprb_clear(u);
#undef a
#undef b
#undef c
#undef d
}

84
fmpcb/test/t-sqrt.c Normal file
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@ -0,0 +1,84 @@
/*=============================================================================
This file is part of ARB.
ARB is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
ARB is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with ARB; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
=============================================================================*/
/******************************************************************************
Copyright (C) 2013 Fredrik Johansson
******************************************************************************/
#include "fmpcb.h"
int main()
{
long iter;
flint_rand_t state;
printf("sqrt....");
fflush(stdout);
flint_randinit(state);
/* check (a^(1/2))^2 = a */
for (iter = 0; iter < 10000; iter++)
{
fmpcb_t a, b, c;
long prec;
fmpcb_init(a);
fmpcb_init(b);
fmpcb_init(c);
fmpcb_randtest(a, state, 1 + n_randint(state, 2000), 10);
fmpcb_randtest(b, state, 1 + n_randint(state, 2000), 10);
prec = 2 + n_randint(state, 2000);
fmpcb_sqrt(b, a, prec);
fmpcb_mul(c, b, b, prec);
if (!fmpcb_contains(c, a))
{
printf("FAIL: containment\n\n");
printf("a = "); fmpcb_print(a); printf("\n\n");
printf("b = "); fmpcb_print(b); printf("\n\n");
printf("c = "); fmpcb_print(c); printf("\n\n");
abort();
}
fmpcb_sqrt(a, a, prec);
if (!fmpcb_equal(a, b))
{
printf("FAIL: aliasing\n\n");
printf("a = "); fmpcb_print(a); printf("\n\n");
printf("b = "); fmpcb_print(b); printf("\n\n");
abort();
}
fmpcb_clear(a);
fmpcb_clear(b);
fmpcb_clear(c);
}
flint_randclear(state);
_fmpz_cleanup();
printf("PASS\n");
return EXIT_SUCCESS;
}