mirror of
https://github.com/vale981/arb
synced 2025-03-05 09:21:38 -05:00
add complex log, exp, pow
This commit is contained in:
Fredrik Johansson
parent
1dd795be42
commit
45ad953f69
6 changed files with 295 additions and 1 deletions
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@ -123,6 +123,15 @@ Precision and comparisons
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Returns nonzero iff zero is contained in *x*.
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Complex parts
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-------------------------------------------------------------------------------
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.. function:: void fmpcb_arg(fmprb_t r, const fmpcb_t z, long prec)
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Sets *r* to a real interval containing the complex argument of *z*. We
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define the complex argument have a discontinuity on `(-\infty,0]`, with
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the special value `\operatorname{arg}(0) = 0`, and
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`\operatorname{arg}(x+0i) = \pi` for `x < 0`.
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Arithmetic
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-------------------------------------------------------------------------------
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@ -175,9 +184,27 @@ Arithmetic
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Sets *z* to the quotient of *x* and *y*.
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Elementary functions
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-------------------------------------------------------------------------------
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.. function:: void fmpcb_log(fmpcb_t y, const fmpcb_t z, long prec)
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Sets *y* to the principal branch of the natural logarithm of *z*,
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computed as
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`\log(a+bi) = \frac{1}{2} \log(a^2 + b^2) + i \operatorname{arg}(a+bi)`.
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.. function:: void fmpcb_exp(fmpcb_t y, const fmpcb_t z, long prec)
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Sets *y* to the exponential function of *z*, computed as
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`\exp(a+bi) = \exp(a) \left( \cos(b) + \sin(b) i \right)`.
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.. function:: void fmpcb_pow_fmpz(fmpcb_t y, const fmpcb_t b, const fmpz_t e, long prec)
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.. function:: void fmpcb_pow_ui(fmpcb_t y, const fmpcb_t b, ulong e, long prec)
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Sets *y* to *b* raised to the power *e*.
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Sets *y* to *b* raised to the power *e*, computed using binary exponentiation.
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.. function:: void fmpcb_pow(fmpcb_t r, const fmpcb_t x, const fmpcb_t y, long prec)
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Sets *r* to *x* raised to the power *y*, computed as `x^y = \exp(y \log x)`.
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8
fmpcb.h
8
fmpcb.h
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@ -198,6 +198,8 @@ fmpcb_get_abs_ubound_fmpr(fmpr_t u, const fmpcb_t z, long prec)
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fmpr_clear(v);
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}
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void fmpcb_arg(fmprb_t r, const fmpcb_t z, long prec);
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static __inline__ void
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fmpcb_add(fmpcb_t z, const fmpcb_t x, const fmpcb_t y, long prec)
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{
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@ -314,6 +316,12 @@ void fmpcb_pow_fmpz(fmpcb_t y, const fmpcb_t b, const fmpz_t e, long prec);
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void fmpcb_pow_ui(fmpcb_t y, const fmpcb_t b, ulong e, long prec);
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void fmpcb_log(fmpcb_t r, const fmpcb_t z, long prec);
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void fmpcb_exp(fmpcb_t r, const fmpcb_t z, long prec);
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void fmpcb_pow(fmpcb_t r, const fmpcb_t x, const fmpcb_t y, long prec);
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static __inline__ void
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_fmpcb_vec_zero(fmpcb_struct * A, long n)
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{
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137
fmpcb/arg.c
Normal file
137
fmpcb/arg.c
Normal file
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@ -0,0 +1,137 @@
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/*=============================================================================
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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
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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ARB is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with ARB; if not, write to the Free Software
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Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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=============================================================================*/
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/******************************************************************************
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Copyright (C) 2012 Fredrik Johansson
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******************************************************************************/
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#include "fmpcb.h"
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void
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fmpcb_arg(fmprb_t r, const fmpcb_t z, long prec)
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{
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#define a fmpcb_realref(z)
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#define b fmpcb_imagref(z)
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#define am fmprb_midref(a)
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#define ar fmprb_radref(a)
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#define bm fmprb_midref(b)
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#define br fmprb_radref(b)
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/* a real number */
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if (fmprb_is_zero(b))
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{
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/* exactly zero */
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if (fmprb_is_zero(a))
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{
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/* define arg(0) = 0 by convention */
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fmprb_zero(r);
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}
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/* interval contains only nonnegative numbers */
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else if (fmpr_sgn(am) > 0 && fmpr_cmpabs(am, ar) >= 0)
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{
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fmprb_zero(r);
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}
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/* interval contains only positive numbers */
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else if (fmpr_sgn(am) < 0 && fmpr_cmpabs(am, ar) > 0)
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{
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fmprb_const_pi(r, prec);
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}
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else
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{
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/* both positive and negative -- argument will be in [0, pi] */
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fmprb_t t;
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fmprb_init(t);
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fmprb_const_pi(t, prec);
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fmprb_mul_2exp_si(t, t, -1);
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fmprb_zero(r);
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fmprb_add_error(r, t);
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fmprb_clear(t);
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}
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}
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/* an imaginary number */
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else if (fmprb_is_zero(a))
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{
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/* interval contains only positive numbers */
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if (fmpr_sgn(bm) > 0 && fmpr_cmpabs(bm, br) > 0)
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{
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fmprb_const_pi(r, prec);
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fmprb_mul_2exp_si(r, r, -1);
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}
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/* interval contains only negative numbers */
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else if (fmpr_sgn(bm) < 0 && fmpr_cmpabs(bm, br) > 0)
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{
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fmprb_const_pi(r, prec);
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fmprb_neg(r, r);
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fmprb_mul_2exp_si(r, r, -1);
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}
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else
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{
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/* both positive and negative -- argument will be in 0 +/- pi/2 */
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fmprb_t t;
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fmprb_const_pi(t, FMPRB_RAD_PREC);
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fmprb_mul_2exp_si(t, t, -1);
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fmprb_zero(r);
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fmprb_add_error(r, t);
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fmprb_clear(t);
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}
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}
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/* strictly in the right half-plane -- atan(b/a) */
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else if (fmpr_sgn(am) > 0 && fmpr_cmpabs(am, ar) > 0)
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{
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fmprb_div(r, b, a, prec);
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fmprb_atan(r, r, prec);
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}
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/* strictly in the upper half-plane -- pi/2 - atan(a/b) */
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else if (fmpr_sgn(bm) > 0 && fmpr_cmpabs(bm, br) > 0)
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{
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fmprb_t t;
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fmprb_init(t);
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fmprb_div(r, a, b, prec);
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fmprb_atan(r, r, prec);
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fmprb_const_pi(t, prec);
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fmprb_mul_2exp_si(t, t, -1);
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fmprb_sub(r, t, r, prec);
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fmprb_clear(t);
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}
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/* strictly in the lower half-plane -- -pi/2 - atan(a/b) */
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else if (fmpr_sgn(bm) < 0 && fmpr_cmpabs(bm, br) > 0)
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{
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fmprb_t t;
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fmprb_init(t);
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fmprb_div(r, a, b, prec);
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fmprb_atan(r, r, prec);
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fmprb_const_pi(t, prec);
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fmprb_mul_2exp_si(t, t, -1);
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fmprb_add(r, t, r, prec);
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fmprb_neg(r, r);
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fmprb_clear(t);
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}
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/* overlaps the nonpositive half-axis -- [-pi, pi] */
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else
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{
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fmprb_t t;
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fmprb_const_pi(t, FMPRB_RAD_PREC);
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fmprb_zero(r);
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fmprb_add_error(r, t);
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fmprb_clear(t);
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}
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}
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50
fmpcb/exp.c
Normal file
50
fmpcb/exp.c
Normal file
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@ -0,0 +1,50 @@
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/*=============================================================================
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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
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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ARB is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with ARB; if not, write to the Free Software
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Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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=============================================================================*/
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/******************************************************************************
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Copyright (C) 2012 Fredrik Johansson
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******************************************************************************/
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#include "fmpcb.h"
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void
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fmpcb_exp(fmpcb_t r, const fmpcb_t z, long prec)
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{
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#define a fmpcb_realref(z)
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#define b fmpcb_imagref(z)
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fmprb_t t, u, v;
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fmprb_init(t);
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fmprb_init(u);
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fmprb_init(v);
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fmprb_exp(t, a, prec);
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fmprb_sin_cos(u, v, b, prec);
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fmprb_mul(fmpcb_realref(r), t, v, prec);
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fmprb_mul(fmpcb_imagref(r), t, u, prec);
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fmprb_clear(t);
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fmprb_clear(u);
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fmprb_clear(v);
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}
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61
fmpcb/log.c
Normal file
61
fmpcb/log.c
Normal file
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@ -0,0 +1,61 @@
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/*=============================================================================
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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
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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ARB is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with ARB; if not, write to the Free Software
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Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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=============================================================================*/
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/******************************************************************************
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Copyright (C) 2012 Fredrik Johansson
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******************************************************************************/
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#include "fmpcb.h"
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void
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fmpcb_log(fmpcb_t r, const fmpcb_t z, long prec)
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{
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#define a fmpcb_realref(z)
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#define b fmpcb_imagref(z)
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fmprb_t t, u;
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fmprb_init(t);
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fmprb_init(u);
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fmprb_mul(t, a, a, prec);
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fmprb_mul(u, b, b, prec);
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fmprb_add(t, t, u, prec);
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if (fmprb_contains_zero(t) || fmpr_sgn(fmprb_midref(t)) < 0)
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{
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fmpr_zero(fmprb_midref(t));
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fmpr_pos_inf(fmprb_radref(t));
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}
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else
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{
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fmprb_log(t, t, prec);
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}
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fmpcb_arg(u, z, prec);
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fmprb_mul_2exp_si(fmpcb_realref(r), t, -1);
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fmprb_set(fmpcb_imagref(r), u);
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fmprb_clear(t);
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fmprb_clear(u);
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}
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11
fmpcb/pow.c
11
fmpcb/pow.c
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@ -78,3 +78,14 @@ fmpcb_pow_ui(fmpcb_t y, const fmpcb_t b, ulong e, long prec)
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fmpz_clear(f);
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}
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void
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fmpcb_pow(fmpcb_t r, const fmpcb_t x, const fmpcb_t y, long prec)
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{
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fmpcb_t t;
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fmpcb_init(t);
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fmpcb_log(t, x, prec);
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fmpcb_mul(t, t, y, prec);
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fmpcb_exp(r, t, prec);
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fmpcb_clear(t);
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}
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