clog.c

数学运算库源码

/*							clog.c
 *
 *	Complex natural logarithm
 *
 *
 *
 * SYNOPSIS:
 *
 * void clog();
 * cmplx z, w;
 *
 * 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 = 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 fabs (double);
double sqrt (double);
double pow (double, double);
double log (double);
double exp (double);
double atan2 (double, double);
double cosh (double);
double sinh (double);
double asin (double);
double sin (double);
double cos (double);
double cabs (cmplx *);
void cadd ( cmplx *, cmplx *, cmplx * );
void cmul ( cmplx *, cmplx *, cmplx * );
void csqrt ( cmplx *, cmplx * );
static void cchsh ( double, double *, double * );
static double redupi ( double );
static double ctans ( cmplx * );
void clog ( cmplx *, cmplx * );
void casin ( cmplx *, cmplx * );
void cacos ( cmplx *, cmplx * );
void catan ( cmplx *, cmplx * );
#else
static void cchsh();
static double redupi();
static double ctans();
double cabs(), fabs(), sqrt(), pow();
double log(), exp(), atan2(), cosh(), sinh();
double asin(), sin(), cos();
void cadd(), cmul(), csqrt();
void clog(), casin(), cacos(), catan();
#endif


extern double MAXNUM, MACHEP, PI, PIO2;

void clog( z, w )
register cmplx *z, *w;
{
double p, rr;

/*rr = sqrt( z->r * z->r  +  z->i * z->i );*/
rr = cabs(z);
p = log(rr);
#if ANSIC
rr = atan2( z->i, z->r );
#else
rr = atan2( z->r, z->i );
if( rr > PI )
	rr -= PI + PI;
#endif
w->i = rr;
w->r = p;
}
/*							cexp()
 *
 *	Complex exponential function
 *
 *
 *
 * SYNOPSIS:
 *
 * void cexp();
 * cmplx z, w;
 *
 * cexp( &z, &w );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the exponential of the complex argument z
 * into the complex result w.
 *
 * If
 *     z = x + iy,
 *     r = exp(x),
 *
 * then
 *
 *     w = r cos y + i r 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 cexp( z, w )
register cmplx *z, *w;
{
double r;

r = exp( z->r );
w->r = r * cos( z->i );
w->i = r * sin( z->i );
}
/*							csin()
 *
 *	Complex circular sine
 *
 *
 *
 * SYNOPSIS:
 *
 * void csin();
 * cmplx z, w;
 *
 * csin( &z, &w );
 *
 *
 *
 * DESCRIPTION:
 *
 * If
 *     z = x + iy,
 *
 * then
 *
 *     w = sin x  cosh y  +  i cos x 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 csin(casin(z)) = z.
 *
 */

void csin( z, w )
register cmplx *z, *w;
{
double ch, sh;

cchsh( z->i, &ch, &sh );
w->r = sin( z->r ) * ch;
w->i = cos( z->r ) * sh;
}



/* calculate cosh and sinh */

static void cchsh( x, c, s )
double x, *c, *s;
{
double e, ei;

if( fabs(x) <= 0.5 )
	{
	*c = cosh(x);
	*s = sinh(x);
	}
else
	{
	e = exp(x);
	ei = 0.5/e;
	e = 0.5 * e;
	*s = e - ei;
	*c = e + ei;
	}
}

/*							ccos()
 *
 *	Complex circular cosine
 *
 *
 *
 * SYNOPSIS:
 *
 * void ccos();
 * cmplx z, w;
 *
 * ccos( &z, &w );
 *
 *
 *
 * DESCRIPTION:
 *
 * If
 *     z = x + iy,
 *
 * then
 *
 *     w = cos x  cosh y  -  i sin x 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 ccos( z, w )
register cmplx *z, *w;
{
double ch, sh;

cchsh( z->i, &ch, &sh );
w->r = cos( z->r ) * ch;
w->i = -sin( z->r ) * sh;
}
/*							ctan()
 *
 *	Complex circular tangent
 *
 *
 *
 * SYNOPSIS:
 *
 * void ctan();
 * cmplx z, w;
 *
 * ctan( &z, &w );
 *
 *
 *
 * DESCRIPTION:
 *
 * If
 *     z = x + iy,
 *
 * then
 *
 *           sin 2x  +  i sinh 2y
 *     w  =  --------------------.
 *            cos 2x  +  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 ctan * ccot = 1 and catan(ctan(z))  =  z.
 */

void ctan( z, w )
register cmplx *z, *w;
{
double d;

d = cos( 2.0 * z->r ) + cosh( 2.0 * z->i );

if( fabs(d) < 0.25 )
	d = ctans(z);

if( d == 0.0 )
	{
	mtherr( "ctan", OVERFLOW );
	w->r = MAXNUM;
	w->i = MAXNUM;
	return;
	}

w->r = sin( 2.0 * z->r ) / d;
w->i = 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
 *
 *           sin 2x  -  i sinh 2y
 *     w  =  --------------------.
 *            cosh 2y  -  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 ctan * ccot = 1 + i0.
 */

void ccot( z, w )
register cmplx *z, *w;
{
double d;

d = cosh(2.0 * z->i) - cos(2.0 * z->r);

if( fabs(d) < 0.25 )
	d = ctans(z);

if( d == 0.0 )
	{
	mtherr( "ccot", OVERFLOW );
	w->r = MAXNUM;
	w->i = MAXNUM;
	return;
	}

w->r = sin( 2.0 * z->r ) / d;
w->i = -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 cosh(2y) - cos(2x)	*/

static double ctans(z)
cmplx *z;
{
double f, x, x2, y, y2, rn, t;
double d;

x = fabs( 2.0 * z->r );
y = 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( fabs(t/d) > MACHEP );
return(d);
}
/*							casin()
 *
 *	Complex circular arc sine
 *
 *
 *
 * SYNOPSIS:
 *
 * void casin();
 * cmplx z, w;
 *
 * casin( &z, &w );
 *
 *
 *
 * DESCRIPTION:
 *
 * Inverse complex sine:
 *
 *                               2
 * w = -i clog( iz + 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 csin(casin(z)) = z.
 */