/* md_clog.c
*
* Complex natural logarithm
*
*
*
* SYNOPSIS:
*
* void md_clog();
* cmplx z, w;
*
* md_clog( &z, &w );
*
*
*
* DESCRIPTION:
*
* Returns complex logarithm to the base e (2.718...) of
* the complex argument x.
*
* If z = x + iy, r = sqrt( x**2 + y**2 ),
* then
* w = md_log(r) + i arctan(y/x).
*
* The arctangent ranges from -PI to +PI.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 7000 8.5e-17 1.9e-17
* IEEE -10,+10 30000 5.0e-15 1.1e-16
*
* Larger relative error can be observed for z near 1 +i0.
* In IEEE arithmetic the peak absolute error is 5.2e-16, rms
* absolute error 1.0e-16.
*/
/*
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1995, 2000 by Stephen L. Moshier
*/
#include "mconf.h"
#ifdef ANSIPROT
static void cchsh ( double x, double *c, double *s );
static double redupi ( double x );
static double ctans ( cmplx *z );
/* These are supposed to be in some standard place. */
double md_fabs (double);
double sqrt (double);
double md_pow (double, double);
double md_log (double);
double md_exp (double);
double md_atan2 (double, double);
double md_cosh (double);
double md_sinh (double);
double md_asin (double);
double md_sin (double);
double md_cos (double);
double md_cabs (cmplx *);
void cadd ( cmplx *, cmplx *, cmplx * );
void cmul ( cmplx *, cmplx *, cmplx * );
void md_csqrt ( cmplx *, cmplx * );
static void cchsh ( double, double *, double * );
static double redupi ( double );
static double ctans ( cmplx * );
void md_clog ( cmplx *, cmplx * );
void md_casin ( cmplx *, cmplx * );
void md_cacos ( cmplx *, cmplx * );
void md_catan ( cmplx *, cmplx * );
#else
static void cchsh();
static double redupi();
static double ctans();
double md_cabs(), md_fabs(), sqrt(), md_pow();
double md_log(), md_exp(), md_atan2(), md_cosh(), md_sinh();
double md_asin(), md_sin(), md_cos();
void cadd(), cmul(), md_csqrt();
void md_clog(), md_casin(), md_cacos(), md_catan();
#endif
extern double MAXNUM, MACHEP, PI, PIO2;
void md_clog( z, w )
register cmplx *z, *w;
{
double p, rr;
/*rr = sqrt( z->r * z->r + z->i * z->i );*/
rr = md_cabs(z);
p = md_log(rr);
#if ANSIC
rr = md_atan2( z->i, z->r );
#else
rr = md_atan2( z->r, z->i );
if( rr > PI )
rr -= PI + PI;
#endif
w->i = rr;
w->r = p;
}
/* md_cexp()
*
* Complex exponential function
*
*
*
* SYNOPSIS:
*
* void md_cexp();
* cmplx z, w;
*
* md_cexp( &z, &w );
*
*
*
* DESCRIPTION:
*
* Returns the exponential of the complex argument z
* into the complex result w.
*
* If
* z = x + iy,
* r = md_exp(x),
*
* then
*
* w = r md_cos y + i r md_sin y.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 8700 3.7e-17 1.1e-17
* IEEE -10,+10 30000 3.0e-16 8.7e-17
*
*/
void md_cexp( z, w )
register cmplx *z, *w;
{
double r;
r = md_exp( z->r );
w->r = r * md_cos( z->i );
w->i = r * md_sin( z->i );
}
/* md_csin()
*
* Complex circular sine
*
*
*
* SYNOPSIS:
*
* void md_csin();
* cmplx z, w;
*
* md_csin( &z, &w );
*
*
*
* DESCRIPTION:
*
* If
* z = x + iy,
*
* then
*
* w = md_sin x md_cosh y + i md_cos x md_sinh y.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 8400 5.3e-17 1.3e-17
* IEEE -10,+10 30000 3.8e-16 1.0e-16
* Also tested by md_csin(md_casin(z)) = z.
*
*/
void md_csin( z, w )
register cmplx *z, *w;
{
double ch, sh;
cchsh( z->i, &ch, &sh );
w->r = md_sin( z->r ) * ch;
w->i = md_cos( z->r ) * sh;
}
/* calculate md_cosh and md_sinh */
static void cchsh( x, c, s )
double x, *c, *s;
{
double e, ei;
if( md_fabs(x) <= 0.5 )
{
*c = md_cosh(x);
*s = md_sinh(x);
}
else
{
e = md_exp(x);
ei = 0.5/e;
e = 0.5 * e;
*s = e - ei;
*c = e + ei;
}
}
/* md_ccos()
*
* Complex circular cosine
*
*
*
* SYNOPSIS:
*
* void md_ccos();
* cmplx z, w;
*
* md_ccos( &z, &w );
*
*
*
* DESCRIPTION:
*
* If
* z = x + iy,
*
* then
*
* w = md_cos x md_cosh y - i md_sin x md_sinh y.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 8400 4.5e-17 1.3e-17
* IEEE -10,+10 30000 3.8e-16 1.0e-16
*/
void md_ccos( z, w )
register cmplx *z, *w;
{
double ch, sh;
cchsh( z->i, &ch, &sh );
w->r = md_cos( z->r ) * ch;
w->i = -md_sin( z->r ) * sh;
}
/* md_ctan()
*
* Complex circular tangent
*
*
*
* SYNOPSIS:
*
* void md_ctan();
* cmplx z, w;
*
* md_ctan( &z, &w );
*
*
*
* DESCRIPTION:
*
* If
* z = x + iy,
*
* then
*
* md_sin 2x + i md_sinh 2y
* w = --------------------.
* md_cos 2x + md_cosh 2y
*
* On the real axis the denominator is zero at odd multiples
* of PI/2. The denominator is evaluated by its Taylor
* series near these points.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 5200 7.1e-17 1.6e-17
* IEEE -10,+10 30000 7.2e-16 1.2e-16
* Also tested by md_ctan * ccot = 1 and md_catan(md_ctan(z)) = z.
*/
void md_ctan( z, w )
register cmplx *z, *w;
{
double d;
d = md_cos( 2.0 * z->r ) + md_cosh( 2.0 * z->i );
if( md_fabs(d) < 0.25 )
d = ctans(z);
if( d == 0.0 )
{
mtherr( "md_ctan", OVERFLOW );
w->r = MAXNUM;
w->i = MAXNUM;
return;
}
w->r = md_sin( 2.0 * z->r ) / d;
w->i = md_sinh( 2.0 * z->i ) / d;
}
/* ccot()
*
* Complex circular cotangent
*
*
*
* SYNOPSIS:
*
* void ccot();
* cmplx z, w;
*
* ccot( &z, &w );
*
*
*
* DESCRIPTION:
*
* If
* z = x + iy,
*
* then
*
* md_sin 2x - i md_sinh 2y
* w = --------------------.
* md_cosh 2y - md_cos 2x
*
* On the real axis, the denominator has zeros at even
* multiples of PI/2. Near these points it is evaluated
* by a Taylor series.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 3000 6.5e-17 1.6e-17
* IEEE -10,+10 30000 9.2e-16 1.2e-16
* Also tested by md_ctan * ccot = 1 + i0.
*/
void ccot( z, w )
register cmplx *z, *w;
{
double d;
d = md_cosh(2.0 * z->i) - md_cos(2.0 * z->r);
if( md_fabs(d) < 0.25 )
d = ctans(z);
if( d == 0.0 )
{
mtherr( "ccot", OVERFLOW );
w->r = MAXNUM;
w->i = MAXNUM;
return;
}
w->r = md_sin( 2.0 * z->r ) / d;
w->i = -md_sinh( 2.0 * z->i ) / d;
}
/* Program to subtract nearest integer multiple of PI */
/* extended precision value of PI: */
#ifdef UNK
static double DP1 = 3.14159265160560607910E0;
static double DP2 = 1.98418714791870343106E-9;
static double DP3 = 1.14423774522196636802E-17;
#endif
#ifdef DEC
static unsigned short P1[] = {0040511,0007732,0120000,0000000,};
static unsigned short P2[] = {0031010,0055060,0100000,0000000,};
static unsigned short P3[] = {0022123,0011431,0105056,0001560,};
#define DP1 *(double *)P1
#define DP2 *(double *)P2
#define DP3 *(double *)P3
#endif
#ifdef IBMPC
static unsigned short P1[] = {0x0000,0x5400,0x21fb,0x4009};
static unsigned short P2[] = {0x0000,0x1000,0x0b46,0x3e21};
static unsigned short P3[] = {0xc06e,0x3145,0x6263,0x3c6a};
#define DP1 *(double *)P1
#define DP2 *(double *)P2
#define DP3 *(double *)P3
#endif
#ifdef MIEEE
static unsigned short P1[] = {
0x4009,0x21fb,0x5400,0x0000
};
static unsigned short P2[] = {
0x3e21,0x0b46,0x1000,0x0000
};
static unsigned short P3[] = {
0x3c6a,0x6263,0x3145,0xc06e
};
#define DP1 *(double *)P1
#define DP2 *(double *)P2
#define DP3 *(double *)P3
#endif
static double redupi(x)
double x;
{
double t;
long i;
t = x/PI;
if( t >= 0.0 )
t += 0.5;
else
t -= 0.5;
i = t; /* the multiple */
t = i;
t = ((x - t * DP1) - t * DP2) - t * DP3;
return(t);
}
/* Taylor series expansion for md_cosh(2y) - md_cos(2x) */
static double ctans(z)
cmplx *z;
{
double f, x, x2, y, y2, rn, t;
double d;
x = md_fabs( 2.0 * z->r );
y = md_fabs( 2.0 * z->i );
x = redupi(x);
x = x * x;
y = y * y;
x2 = 1.0;
y2 = 1.0;
f = 1.0;
rn = 0.0;
d = 0.0;
do
{
rn += 1.0;
f *= rn;
rn += 1.0;
f *= rn;
x2 *= x;
y2 *= y;
t = y2 + x2;
t /= f;
d += t;
rn += 1.0;
f *= rn;
rn += 1.0;
f *= rn;
x2 *= x;
y2 *= y;
t = y2 - x2;
t /= f;
d += t;
}
while( md_fabs(t/d) > MACHEP );
return(d);
}
/* md_casin()
*
* Complex circular arc sine
*
*
*
* SYNOPSIS:
*
* void md_casin();
* cmplx z, w;
*
* md_casin( &z, &w );
*
*
*
* DESCRIPTION:
*
* Inverse complex sine:
*
* 2
* w = -i md_clog( iz + md_csqrt( 1 - z ) ).
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 10100 2.1e-15 3.4e-16
* IEEE -10,+10 30000 2.2e-14 2.7e-15
* Larger relative error can be observed for z near zero.
* Also tested by md_csin(md_casin(z)) = z.
*/
void md_casin( z, w )
cmplx *z, *w;
{
static cmplx ca, ct, zz, z2;
double x, y;
x = z->r;
y = z->i;
if( y == 0.0 )
{
if( md_fabs(x) > 1.0 )
{
w->r = PIO2;
w->i = 0.0;
mtherr( "md_casin", DOMAIN );
}
else
{
w->r = md_asin(x);
w->i = 0.0;
}
return;
}
/* Power series expansion */
/*
b = md_cabs(z);
if( b < 0.125 )
{
z2.r = (x - y) * (x + y);
z2.i = 2.0 * x * y;
cn = 1.0;
n = 1.0;
ca.r = x;
ca.i = y;
sum.r = x;
sum.i = y;
do
{
ct.r = z2.r * ca.r - z2.i * ca.i;
ct.i = z2.r * ca.i + z2.i * ca.r;
ca.r = ct.r;
ca.i = ct.i;
cn *= n;
n += 1.0;
cn /= n;
n += 1.0;
b = cn/n;
ct.r *= b;
ct.i *= b;
sum.r += ct.r;
sum.i += ct.i;
b = md_fabs(ct.r) + md_fabs(ct.i);
}
while( b > MACHEP );
w->r = sum.r;
w->i = sum.i;
return;
}
*/
ca.r = x;
ca.i = y;
ct.r = -ca.i; /* iz */
ct.i = ca.r;
/* sqrt( 1 - z*z) */
/* cmul( &ca, &ca, &zz ) */
zz.r = (ca.r - ca.i) * (ca.r + ca.i); /*x * x - y * y */
zz.i = 2.0 * ca.r * ca.i;
zz.r = 1.0 - zz.r;
zz.i = -zz.i;
md_csqrt( &zz, &z2 );
cadd( &z2, &ct, &zz );
md_clog( &zz, &zz );
w->r = zz.i; /* mult by 1/i = -i */
w->i = -zz.r;
return;
}
/* md_cacos()
*
* Complex circular arc cosine
*
*
*
* SYNOPSIS:
*
* void md_cacos();
* cmplx z, w;
*
* md_cacos( &z, &w );
*
*
*
* DESCRIPTION:
*
*
* w = arccos z = PI/2 - arcsin z.
*
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 5200 1.6e-15 2.8e-16
* IEEE -10,+10 30000 1.8e-14 2.2e-15
*/
void md_cacos( z, w )
cmplx *z, *w;
{
md_casin( z, w );
w->r = PIO2 - w->r;
w->i = -w->i;
}
/* md_catan()
*
* Complex circular arc tangent
*
*
*
* SYNOPSIS:
*
* void md_catan();
* cmplx z, w;
*
* md_catan( &z, &w );
*
*
*
* DESCRIPTION:
*
* If
* z = x + iy,
*
* then
* 1 ( 2x )
* Re w = - arctan(-----------) + k PI
* 2 ( 2 2)
* (1 - x - y )
*
* ( 2 2)
* 1 (x + (y+1) )
* Im w = - md_log(------------)
* 4 ( 2 2)
* (x + (y-1) )
*
* Where k is an arbitrary integer.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 5900 1.3e-16 7.8e-18
* IEEE -10,+10 30000 2.3e-15 8.5e-17
* The check md_catan( md_ctan(z) ) = z, with |x| and |y| < PI/2,
* had peak relative error 1.5e-16, rms relative error
* 2.9e-17. See also md_clog().
*/
void md_catan( z, w )
cmplx *z, *w;
{
double a, t, x, x2, y;
x = z->r;
y = z->i;
if( (x == 0.0) && (y > 1.0) )
goto ovrf;
x2 = x * x;
a = 1.0 - x2 - (y * y);
if( a == 0.0 )
goto ovrf;
#if ANSIC
t = md_atan2( 2.0 * x, a )/2.0;
#else
t = md_atan2( a, 2.0 * x )/2.0;
#endif
w->r = redupi( t );
t = y - 1.0;
a = x2 + (t * t);
if( a == 0.0 )
goto ovrf;
t = y + 1.0;
a = (x2 + (t * t))/a;
w->i = md_log(a)/4.0;
return;
ovrf:
mtherr( "md_catan", OVERFLOW );
w->r = MAXNUM;
w->i = MAXNUM;
}
/* md_csinh
*
* Complex hyperbolic sine
*
*
*
* SYNOPSIS:
*
* void md_csinh();
* cmplx z, w;
*
* md_csinh( &z, &w );
*
*
* DESCRIPTION:
*
* md_csinh z = (md_cexp(z) - md_cexp(-z))/2
* = md_sinh x * md_cos y + i md_cosh x * md_sin y .
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -10,+10 30000 3.1e-16 8.2e-17
*
*/
void
md_csinh (z, w)
cmplx *z, *w;
{
double x, y;
x = z->r;
y = z->i;
w->r = md_sinh (x) * md_cos (y);
w->i = md_cosh (x) * md_sin (y);
}
/* md_casinh
*
* Complex inverse hyperbolic sine
*
*
*
* SYNOPSIS:
*
* void md_casinh();
* cmplx z, w;
*
* md_casinh (&z, &w);
*
*
*
* DESCRIPTION:
*
* md_casinh z = -i md_casin iz .
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -10,+10 30000 1.8e-14 2.6e-15
*
*/
void
md_casinh (z, w)
cmplx *z, *w;
{
cmplx u;
u.r = 0.0;
u.i = 1.0;
cmul( z, &u, &u );
md_casin( &u, w );
u.r = 0.0;
u.i = -1.0;
cmul( &u, w, w );
}
/* md_ccosh
*
* Complex hyperbolic cosine
*
*
*
* SYNOPSIS:
*
* void md_ccosh();
* cmplx z, w;
*
* md_ccosh (&z, &w);
*
*
*
* DESCRIPTION:
*
* md_ccosh(z) = md_cosh x md_cos y + i md_sinh x md_sin y .
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -10,+10 30000 2.9e-16 8.1e-17
*
*/
void
md_ccosh (z, w)
cmplx *z, *w;
{
double x, y;
x = z->r;
y = z->i;
w->r = md_cosh (x) * md_cos (y);
w->i = md_sinh (x) * md_sin (y);
}
/* md_cacosh
*
* Complex inverse hyperbolic cosine
*
*
*
* SYNOPSIS:
*
* void md_cacosh();
* cmplx z, w;
*
* md_cacosh (&z, &w);
*
*
*
* DESCRIPTION:
*
* md_acosh z = i md_acos z .
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -10,+10 30000 1.6e-14 2.1e-15
*
*/
void
md_cacosh (z, w)
cmplx *z, *w;
{
cmplx u;
md_cacos( z, w );
u.r = 0.0;
u.i = 1.0;
cmul( &u, w, w );
}
/* md_ctanh
*
* Complex hyperbolic tangent
*
*
*
* SYNOPSIS:
*
* void md_ctanh();
* cmplx z, w;
*
* md_ctanh (&z, &w);
*
*
*
* DESCRIPTION:
*
* md_tanh z = (md_sinh 2x + i md_sin 2y) / (md_cosh 2x + md_cos 2y) .
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -10,+10 30000 1.7e-14 2.4e-16
*
*/
/* 5.253E-02,1.550E+00 1.643E+01,6.553E+00 1.729E-14 21355 */
void
md_ctanh (z, w)
cmplx *z, *w;
{
double x, y, d;
x = z->r;
y = z->i;
d = md_cosh (2.0 * x) + md_cos (2.0 * y);
w->r = md_sinh (2.0 * x) / d;
w->i = md_sin (2.0 * y) / d;
return;
}
/* md_catanh
*
* Complex inverse hyperbolic tangent
*
*
*
* SYNOPSIS:
*
* void md_catanh();
* cmplx z, w;
*
* md_catanh (&z, &w);
*
*
*
* DESCRIPTION:
*
* Inverse md_tanh, equal to -i md_catan (iz);
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -10,+10 30000 2.3e-16 6.2e-17
*
*/
void
md_catanh (z, w)
cmplx *z, *w;
{
cmplx u;
u.r = 0.0;
u.i = 1.0;
cmul (z, &u, &u); /* i z */
md_catan (&u, w);
u.r = 0.0;
u.i = -1.0;
cmul (&u, w, w); /* -i md_catan iz */
return;
}
/* md_cpow
*
* Complex power function
*
*
*
* SYNOPSIS:
*
* void md_cpow();
* cmplx a, z, w;
*
* md_cpow (&a, &z, &w);
*
*
*
* DESCRIPTION:
*
* Raises complex A to the complex Zth power.
* Definition is per AMS55 # 4.2.8,
* analytically equivalent to md_cpow(a,z) = md_cexp(z md_clog(a)).
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -10,+10 30000 9.4e-15 1.5e-15
*
*/
void
md_cpow (a, z, w)
cmplx *a, *z, *w;
{
double x, y, r, theta, absa, arga;
x = z->r;
y = z->i;
absa = md_cabs (a);
if (absa == 0.0)
{
w->r = 0.0;
w->i = 0.0;
return;
}
arga = md_atan2 (a->i, a->r);
r = md_pow (absa, x);
theta = x * arga;
if (y != 0.0)
{
r = r * md_exp (-y * arga);
theta = theta + y * md_log (absa);
}
w->r = r * md_cos (theta);
w->i = r * md_sin (theta);
return;
}
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