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/* vim: set sw=8: -*- Mode: C; tab-width: 8; indent-tabs-mode: t; c-basic-offset: 8 -*- */
/* complex/math.c (from the GSL 1.1.1)
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000 Jorma Olavi Thtinen, Brian Gough
*
* This program 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.
*
* This program 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 this program; if not, write to the Free Software
* Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
*/
/* Basic complex arithmetic functions
* Original version by Jorma Olavi Thtinen <jotahtin@cc.hut.fi>
*
* Modified for GSL by Brian Gough, 3/2000
*/
/* The following references describe the methods used in these
* functions,
*
* T. E. Hull and Thomas F. Fairgrieve and Ping Tak Peter Tang,
* "Implementing Complex Elementary Functions Using Exception
* Handling", ACM Transactions on Mathematical Software, Volume 20
* (1994), pp 215-244, Corrigenda, p553
*
* Hull et al, "Implementing the complex arcsin and arccosine
* functions using exception handling", ACM Transactions on
* Mathematical Software, Volume 23 (1997) pp 299-335
*
* Abramowitz and Stegun, Handbook of Mathematical Functions, "Inverse
* Circular Functions in Terms of Real and Imaginary Parts", Formulas
* 4.4.37, 4.4.38, 4.4.39
*/
/*
* Gnumeric specific modifications written by Jukka-Pekka Iivonen
* (jiivonen@hutcs.cs.hut.fi)
*
* long double modifications by Morten Welinder.
*/
#include <gnumeric-config.h>
#include <glib/gi18n-lib.h>
#include <gnumeric.h>
#include <func.h>
#include <complex.h>
#include "gsl-complex.h"
#include <parse-util.h>
#include <cell.h>
#include <expr.h>
#include <value.h>
#include <mathfunc.h>
#define GSL_REAL(x) (x)->re
#define GSL_IMAG(x) (x)->im
/***********************************************************************
* Complex arithmetic operators
***********************************************************************/
static inline void
gsl_complex_mul_imag (complex_t const *a, gnm_float y, complex_t *res)
{ /* z=a*iy */
complex_init (res, -y * GSL_IMAG (a), y * GSL_REAL (a));
}
void
gsl_complex_inverse (complex_t const *a, complex_t *res)
{ /* z=1/a */
gnm_float s = 1.0 / complex_mod (a);
complex_init (res, (GSL_REAL (a) * s) * s, -(GSL_IMAG (a) * s) * s);
}
void
gsl_complex_negative (complex_t const *a, complex_t *res)
{
complex_init (res, -GSL_REAL (a), -GSL_IMAG (a));
}
/**********************************************************************
* Inverse Complex Trigonometric Functions
**********************************************************************/
static void
gsl_complex_arcsin_real (gnm_float a, complex_t *res)
{ /* z = arcsin(a) */
if (gnm_abs (a) <= 1.0) {
complex_init (res, gnm_asin (a), 0.0);
} else {
if (a < 0.0) {
complex_init (res, -M_PI_2gnum, gnm_acosh (-a));
} else {
complex_init (res, M_PI_2gnum, -gnm_acosh (a));
}
}
}
void
gsl_complex_arcsin (complex_t const *a, complex_t *res)
{ /* z = arcsin(a) */
gnm_float R = GSL_REAL (a), I = GSL_IMAG (a);
if (I == 0) {
gsl_complex_arcsin_real (R, res);
} else {
gnm_float x = gnm_abs (R), y = gnm_abs (I);
gnm_float r = gnm_hypot (x + 1, y);
gnm_float s = gnm_hypot (x - 1, y);
gnm_float A = 0.5 * (r + s);
gnm_float B = x / A;
gnm_float y2 = y * y;
gnm_float real, imag;
const gnm_float A_crossover = 1.5, B_crossover = 0.6417;
if (B <= B_crossover) {
real = gnm_asin (B);
} else {
if (x <= 1) {
gnm_float D = 0.5 * (A + x) *
(y2 / (r + x + 1) + (s + (1 - x)));
real = gnm_atan (x / gnm_sqrt (D));
} else {
gnm_float Apx = A + x;
gnm_float D = 0.5 * (Apx / (r + x + 1)
+ Apx / (s + (x - 1)));
real = gnm_atan (x / (y * gnm_sqrt (D)));
}
}
if (A <= A_crossover) {
gnm_float Am1;
if (x < 1) {
Am1 = 0.5 * (y2 / (r + (x + 1)) + y2 /
(s + (1 - x)));
} else {
Am1 = 0.5 * (y2 / (r + (x + 1)) +
(s + (x - 1)));
}
imag = gnm_log1p (Am1 + gnm_sqrt (Am1 * (A + 1)));
} else {
imag = gnm_log (A + gnm_sqrt (A * A - 1));
}
complex_init (res, (R >= 0) ? real : -real, (I >= 0) ?
imag : -imag);
}
}
static void
gsl_complex_arccos_real (gnm_float a, complex_t *res)
{ /* z = arccos(a) */
if (gnm_abs (a) <= 1.0) {
complex_init (res, gnm_acos (a), 0);
} else {
if (a < 0.0) {
complex_init (res, M_PIgnum, -gnm_acosh (-a));
} else {
complex_init (res, 0, gnm_acosh (a));
}
}
}
void
gsl_complex_arccos (complex_t const *a, complex_t *res)
{ /* z = arccos(a) */
gnm_float R = GSL_REAL (a), I = GSL_IMAG (a);
if (I == 0) {
gsl_complex_arccos_real (R, res);
} else {
gnm_float x = gnm_abs (R);
gnm_float y = gnm_abs (I);
gnm_float r = gnm_hypot (x + 1, y);
gnm_float s = gnm_hypot (x - 1, y);
gnm_float A = 0.5 * (r + s);
gnm_float B = x / A;
gnm_float y2 = y * y;
gnm_float real, imag;
const gnm_float A_crossover = 1.5;
const gnm_float B_crossover = 0.6417;
if (B <= B_crossover) {
real = gnm_acos (B);
} else {
if (x <= 1) {
gnm_float D = 0.5 * (A + x) *
(y2 / (r + x + 1) + (s + (1 - x)));
real = gnm_atan (gnm_sqrt (D) / x);
} else {
gnm_float Apx = A + x;
gnm_float D = 0.5 * (Apx / (r + x + 1) + Apx /
(s + (x - 1)));
real = gnm_atan ((y * gnm_sqrt (D)) / x);
}
}
if (A <= A_crossover) {
gnm_float Am1;
if (x < 1) {
Am1 = 0.5 * (y2 / (r + (x + 1)) + y2 /
(s + (1 - x)));
} else {
Am1 = 0.5 * (y2 / (r + (x + 1)) +
(s + (x - 1)));
}
imag = gnm_log1p (Am1 + gnm_sqrt (Am1 * (A + 1)));
} else {
imag = gnm_log (A + gnm_sqrt (A * A - 1));
}
complex_init (res, (R >= 0) ? real : M_PIgnum - real, (I >= 0) ?
-imag : imag);
}
}
void
gsl_complex_arctan (complex_t const *a, complex_t *res)
{ /* z = arctan(a) */
gnm_float R = GSL_REAL (a), I = GSL_IMAG (a);
if (I == 0) {
complex_init (res, gnm_atan (R), 0);
} else {
/* FIXME: This is a naive implementation which does not fully
* take into account cancellation errors, overflow, underflow
* etc. It would benefit from the Hull et al treatment. */
gnm_float r = gnm_hypot (R, I);
gnm_float imag;
gnm_float u = 2 * I / (1 + r * r);
/* FIXME: the following cross-over should be optimized but 0.1
* seems to work ok */
if (gnm_abs (u) < 0.1) {
imag = 0.25 * (gnm_log1p (u) - gnm_log1p (-u));
} else {
gnm_float A = gnm_hypot (R, I + 1);
gnm_float B = gnm_hypot (R, I - 1);
imag = 0.5 * gnm_log (A / B);
}
if (R == 0) {
if (I > 1) {
complex_init (res, M_PI_2gnum, imag);
} else if (I < -1) {
complex_init (res, -M_PI_2gnum, imag);
} else {
complex_init (res, 0, imag);
}
} else {
complex_init (res, 0.5 * gnm_atan2 (2 * R,
((1 + r) * (1 - r))),
imag);
}
}
}
void
gsl_complex_arcsec (complex_t const *a, complex_t *res)
{ /* z = arcsec(a) */
gsl_complex_inverse (a, res);
gsl_complex_arccos (res, res);
}
void
gsl_complex_arccsc (complex_t const *a, complex_t *res)
{ /* z = arccsc(a) */
gsl_complex_inverse (a, res);
gsl_complex_arcsin (res, res);
}
void
gsl_complex_arccot (complex_t const *a, complex_t *res)
{ /* z = arccot(a) */
if (GSL_REAL (a) == 0.0 && GSL_IMAG (a) == 0.0) {
complex_init (res, M_PI_2gnum, 0);
} else {
gsl_complex_inverse (a, res);
gsl_complex_arctan (res, res);
}
}
/**********************************************************************
* Complex Hyperbolic Functions
**********************************************************************/
void
gsl_complex_sinh (complex_t const *a, complex_t *res)
{ /* z = sinh(a) */
gnm_float R = GSL_REAL (a), I = GSL_IMAG (a);
complex_init (res, gnm_sinh (R) * gnm_cos (I), cosh (R) * gnm_sin (I));
}
void
gsl_complex_cosh (complex_t const *a, complex_t *res)
{ /* z = cosh(a) */
gnm_float R = GSL_REAL (a), I = GSL_IMAG (a);
complex_init (res, cosh (R) * gnm_cos (I), gnm_sinh (R) * gnm_sin (I));
}
void
gsl_complex_tanh (complex_t const *a, complex_t *res)
{ /* z = tanh(a) */
gnm_float R = GSL_REAL (a), I = GSL_IMAG (a);
if (gnm_abs (R) < 1.0) {
gnm_float D =
gnm_pow (gnm_cos (I), 2.0) +
gnm_pow (gnm_sinh (R), 2.0);
complex_init (res, gnm_sinh (R) * cosh (R) / D,
0.5 * gnm_sin (2 * I) / D);
} else {
gnm_float D =
gnm_pow (gnm_cos (I), 2.0) +
gnm_pow (gnm_sinh (R), 2.0);
gnm_float F = 1 + gnm_pow (gnm_cos (I) / gnm_sinh (R), 2.0);
complex_init (res, 1.0 / (gnm_tanh (R) * F),
0.5 * gnm_sin (2 * I) / D);
}
}
void
gsl_complex_sech (complex_t const *a, complex_t *res)
{ /* z = sech(a) */
gsl_complex_cosh (a, res);
gsl_complex_inverse (res, res);
}
void
gsl_complex_csch (complex_t const *a, complex_t *res)
{ /* z = csch(a) */
gsl_complex_sinh (a, res);
gsl_complex_inverse (res, res);
}
void
gsl_complex_coth (complex_t const *a, complex_t *res)
{ /* z = coth(a) */
gsl_complex_tanh (a, res);
gsl_complex_inverse (res, res);
}
/**********************************************************************
* Inverse Complex Hyperbolic Functions
**********************************************************************/
void
gsl_complex_arcsinh (complex_t const *a, complex_t *res)
{ /* z = arcsinh(a) */
gsl_complex_mul_imag (a, 1.0, res);
gsl_complex_arcsin (res, res);
gsl_complex_mul_imag (res, -1.0, res);
}
void
gsl_complex_arccosh (complex_t const *a, complex_t *res)
{ /* z = arccosh(a) */
gsl_complex_arccos (a, res);
gsl_complex_mul_imag (res, GSL_IMAG (res) > 0 ? -1.0 : 1.0, res);
}
static void
gsl_complex_arctanh_real (gnm_float a, complex_t *res)
{ /* z = arctanh(a) */
if (a > -1.0 && a < 1.0) {
complex_init (res, gnm_atanh (a), 0);
} else {
complex_init (res, gnm_atanh (1 / a),
(a < 0) ? M_PI_2gnum : -M_PI_2gnum);
}
}
void
gsl_complex_arctanh (complex_t const *a, complex_t *res)
{ /* z = arctanh(a) */
if (GSL_IMAG (a) == 0.0) {
gsl_complex_arctanh_real (GSL_REAL (a), res);
} else {
gsl_complex_mul_imag (a, 1.0, res);
gsl_complex_arctan (res, res);
gsl_complex_mul_imag (res, -1.0, res);
}
}
void
gsl_complex_arcsech (complex_t const *a, complex_t *res)
{ /* z = arcsech(a); */
gsl_complex_inverse (a, res);
gsl_complex_arccosh (res, res);
}
void
gsl_complex_arccsch (complex_t const *a, complex_t *res)
{ /* z = arccsch(a); */
gsl_complex_inverse (a, res);
gsl_complex_arcsinh (res, res);
}
void
gsl_complex_arccoth (complex_t const *a, complex_t *res)
{ /* z = arccoth(a); */
gsl_complex_inverse (a, res);
gsl_complex_arctanh (res, res);
}
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