clog.c

数学运算库源码


void 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( fabs(x) > 1.0 )
		{
		w->r = PIO2;
		w->i = 0.0;
		mtherr( "casin", DOMAIN );
		}
	else
		{
		w->r = asin(x);
		w->i = 0.0;
		}
	return;
	}

/* Power series expansion */
/*
b = 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 = fabs(ct.r) + 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;
csqrt( &zz, &z2 );

cadd( &z2, &ct, &zz );
clog( &zz, &zz );
w->r = zz.i;	/* mult by 1/i = -i */
w->i = -zz.r;
return;
}
/*							cacos()
 *
 *	Complex circular arc cosine
 *
 *
 *
 * SYNOPSIS:
 *
 * void cacos();
 * cmplx z, w;
 *
 * 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 cacos( z, w )
cmplx *z, *w;
{

casin( z, w );
w->r = PIO2  -  w->r;
w->i = -w->i;
}
/*							catan()
 *
 *	Complex circular arc tangent
 *
 *
 *
 * SYNOPSIS:
 *
 * void catan();
 * cmplx z, w;
 *
 * 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  =  - 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 catan( ctan(z) )  =  z, with |x| and |y| < PI/2,
 * had peak relative error 1.5e-16, rms relative error
 * 2.9e-17.  See also clog().
 */

void 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 = atan2( 2.0 * x, a )/2.0;
#else
t = 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 = log(a)/4.0;
return;

ovrf:
mtherr( "catan", OVERFLOW );
w->r = MAXNUM;
w->i = MAXNUM;
}


/*							csinh
 *
 *	Complex hyperbolic sine
 *
 *
 *
 * SYNOPSIS:
 *
 * void csinh();
 * cmplx z, w;
 *
 * csinh( &z, &w );
 *
 *
 * DESCRIPTION:
 *
 * csinh z = (cexp(z) - cexp(-z))/2
 *         = sinh x * cos y  +  i cosh x * sin y .
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      -10,+10     30000       3.1e-16     8.2e-17
 *
 */

void
csinh (z, w)
     cmplx *z, *w;
{
  double x, y;

  x = z->r;
  y = z->i;
  w->r = sinh (x) * cos (y);
  w->i = cosh (x) * sin (y);
}


/*							casinh
 *
 *	Complex inverse hyperbolic sine
 *
 *
 *
 * SYNOPSIS:
 *
 * void casinh();
 * cmplx z, w;
 *
 * casinh (&z, &w);
 *
 *
 *
 * DESCRIPTION:
 *
 * casinh z = -i casin iz .
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      -10,+10     30000       1.8e-14     2.6e-15
 *
 */

void
casinh (z, w)
     cmplx *z, *w;
{
  cmplx u;

  u.r = 0.0;
  u.i = 1.0;
  cmul( z, &u, &u );
  casin( &u, w );
  u.r = 0.0;
  u.i = -1.0;
  cmul( &u, w, w );
}

/*							ccosh
 *
 *	Complex hyperbolic cosine
 *
 *
 *
 * SYNOPSIS:
 *
 * void ccosh();
 * cmplx z, w;
 *
 * ccosh (&z, &w);
 *
 *
 *
 * DESCRIPTION:
 *
 * ccosh(z) = cosh x  cos y + i sinh x sin y .
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      -10,+10     30000       2.9e-16     8.1e-17
 *
 */

void
ccosh (z, w)
     cmplx *z, *w;
{
  double x, y;

  x = z->r;
  y = z->i;
  w->r = cosh (x) * cos (y);
  w->i = sinh (x) * sin (y);
}


/*							cacosh
 *
 *	Complex inverse hyperbolic cosine
 *
 *
 *
 * SYNOPSIS:
 *
 * void cacosh();
 * cmplx z, w;
 *
 * cacosh (&z, &w);
 *
 *
 *
 * DESCRIPTION:
 *
 * acosh z = i acos z .
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      -10,+10     30000       1.6e-14     2.1e-15
 *
 */

void
cacosh (z, w)
     cmplx *z, *w;
{
  cmplx u;

  cacos( z, w );
  u.r = 0.0;
  u.i = 1.0;
  cmul( &u, w, w );
}


/*							ctanh
 *
 *	Complex hyperbolic tangent
 *
 *
 *
 * SYNOPSIS:
 *
 * void ctanh();
 * cmplx z, w;
 *
 * ctanh (&z, &w);
 *
 *
 *
 * DESCRIPTION:
 *
 * tanh z = (sinh 2x  +  i sin 2y) / (cosh 2x + 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
ctanh (z, w)
     cmplx *z, *w;
{
  double x, y, d;

  x = z->r;
  y = z->i;
  d = cosh (2.0 * x) + cos (2.0 * y);
  w->r = sinh (2.0 * x) / d;
  w->i = sin (2.0 * y) / d;
  return;
}


/*							catanh
 *
 *	Complex inverse hyperbolic tangent
 *
 *
 *
 * SYNOPSIS:
 *
 * void catanh();
 * cmplx z, w;
 *
 * catanh (&z, &w);
 *
 *
 *
 * DESCRIPTION:
 *
 * Inverse tanh, equal to  -i catan (iz);
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      -10,+10     30000       2.3e-16     6.2e-17
 *
 */

void
catanh (z, w)
     cmplx *z, *w;
{
  cmplx u;

  u.r = 0.0;
  u.i = 1.0;
  cmul (z, &u, &u);  /* i z */
  catan (&u, w);
  u.r = 0.0;
  u.i = -1.0;
  cmul (&u, w, w);  /* -i catan iz */
  return;
}


/*							cpow
 *
 *	Complex power function
 *
 *
 *
 * SYNOPSIS:
 *
 * void cpow();
 * cmplx a, z, w;
 *
 * cpow (&a, &z, &w);
 *
 *
 *
 * DESCRIPTION:
 *
 * Raises complex A to the complex Zth power.
 * Definition is per AMS55 # 4.2.8,
 * analytically equivalent to cpow(a,z) = cexp(z clog(a)).
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      -10,+10     30000       9.4e-15     1.5e-15
 *
 */


void
cpow (a, z, w)
     cmplx *a, *z, *w;
{
  double x, y, r, theta, absa, arga;

  x = z->r;
  y = z->i;
  absa = cabs (a);
  if (absa == 0.0)
    {
      w->r = 0.0;
      w->i = 0.0;
      return;
    }
  arga = atan2 (a->i, a->r);
  r = pow (absa, x);
  theta = x * arga;
  if (y != 0.0)
    {
      r = r * exp (-y * arga);
      theta = theta + y * log (absa);
    }
  w->r = r * cos (theta);
  w->i = r * sin (theta);
  return;
}