clog.c

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/*							clog.c
 *
 *	Complex natural logarithm
 *
 *
 *
 * SYNOPSIS:
 *
 * double complex clog();
 * double complex z, w;
 *
 * w = clog (z);
 *
 *
 *
 * 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.
 */

#include "complex.h"
#include "mconf.h"

#ifdef ANSIPROT
static void cchsh ( double x, double *c, double *s );
static double redupi ( double x );
static double ctans ( double complex z );
#else
static void cchsh();
static double redupi();
static double ctans();
double cabs(), fabs(), sqrt();
double log(), exp(), atan2(), cosh(), sinh();
double asin(), sin(), cos();
#endif

extern double MAXNUM, MACHEP, PI, PIO2;

double complex
clog (z)
     double complex z;
{
  double complex w;
  double p, rr;

  /*rr = sqrt( z->r * z->r  +  z->i * z->i );*/
  rr = cabs(z);
  p = log(rr);
  rr = atan2 (cimag (z), creal (z));
  w = p + rr * I;
  return (w);
}
/*							cexp()
 *
 *	Complex exponential function
 *
 *
 *
 * SYNOPSIS:
 *
 * double complex cexp ();
 * double complex z, w;
 *
 * w = cexp (z);
 *
 *
 *
 * 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
 *
 */

double complex
cexp(z)
     double complex z;
{
  double complex w;
  double r, x, y;

  x = creal (z);
  y = cimag (z);
  r = exp (x);
  w = r * cos (y) + r * sin (y) * I;
  return (w);
}
/*							csin()
 *
 *	Complex circular sine
 *
 *
 *
 * SYNOPSIS:
 *
 * double complex csin();
 * double complex z, w;
 *
 * w = csin (z);
 *
 *
 *
 * DESCRIPTION:
 *
 * If
 *     z = x + iy,
 *
 * then
 *
 *     w = sin x  cosh y  +  i cos x sinh y.
 *
 * csin(z) = -i csinh(iz).
 *
 * 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.
 *
 */

double complex
csin (z)
     double complex z;
{
  double complex w;
  double ch, sh;

  cchsh( cimag (z), &ch, &sh );
  w = sin (creal(z)) * ch + (cos (creal(z)) * sh) * I;
  return (w);
}



/* 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:
 *
 * double complex ccos();
 * double complex z, w;
 *
 * w = ccos (z);
 *
 *
 *
 * 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
 */

double complex
ccos (z)
     double complex z;
{
  double complex w;
  double ch, sh;

  cchsh( cimag(z), &ch, &sh );
  w = cos(creal (z)) * ch - (sin (creal (z)) * sh) * I;
  return (w);
}
/*							ctan()
 *
 *	Complex circular tangent
 *
 *
 *
 * SYNOPSIS:
 *
 * double complex ctan();
 * double complex z, w;
 *
 * w = ctan (z);
 *
 *
 *
 * 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.
 *
 * ctan(z) = -i ctanh(iz).
 *
 * 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.
 */

double complex
ctan (z)
     double complex z;
{
  double complex w;
  double d;

  d = cos (2.0 * creal (z)) + cosh (2.0 * cimag (z));

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

  if (d == 0.0 )
    {
      mtherr ("ctan", OVERFLOW);
      w = MAXNUM + MAXNUM * I;
      return (w);
    }

  w = sin (2.0 * creal(z)) / d + (sinh (2.0 * cimag(z)) / d) * I;
  return (w);
}
/*							ccot()
 *
 *	Complex circular cotangent
 *
 *
 *
 * SYNOPSIS:
 *
 * double complex ccot();
 * double complex z, w;
 *
 * w = ccot (z);
 *
 *
 *
 * 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.
 */

double complex
ccot (z)
     double complex z;
{
  double complex w;
  double d;

  d = cosh (2.0 * cimag (z)) - cos (2.0 * creal(z));

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

  if (d == 0.0)
    {
      mtherr ("ccot", OVERFLOW);
      w = MAXNUM + MAXNUM * I;
      return (w);
    }

  w = sin (2.0 * creal(z)) / d - (sinh (2.0 * cimag(z)) / d) * I;
  return w;
}

/* 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)
     double complex z;
{
  double f, x, x2, y, y2, rn, t;
  double d;

  x = fabs (2.0 * creal (z));
  y = fabs (2.0 * cimag(z));

  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:
 *
 * double complex casin();
 * double complex z, w;
 *
 * w = casin (z);
 *
 *
 *
 * DESCRIPTION:
 *
 * Inverse complex sine:
 *
 *                               2
 * w = -i clog( iz + csqrt( 1 - z ) ).
 *
 * casin(z) = -i casinh(iz)
 *
 * 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.