/* cmplx.c
*
* Complex number arithmetic
*
*
*
* SYNOPSIS:
*
* typedef struct {
* double r; real part
* double i; imaginary part
* }cmplx;
*
* cmplx *a, *b, *c;
*
* cadd( a, b, c ); c = b + a
* csub( a, b, c ); c = b - a
* cmul( a, b, c ); c = b * a
* cdiv( a, b, c ); c = b / a
* cneg( c ); c = -c
* cmov( b, c ); c = b
*
*
*
* DESCRIPTION:
*
* Addition:
* c.r = b.r + a.r
* c.i = b.i + a.i
*
* Subtraction:
* c.r = b.r - a.r
* c.i = b.i - a.i
*
* Multiplication:
* c.r = b.r * a.r - b.i * a.i
* c.i = b.r * a.i + b.i * a.r
*
* Division:
* d = a.r * a.r + a.i * a.i
* c.r = (b.r * a.r + b.i * a.i)/d
* c.i = (b.i * a.r - b.r * a.i)/d
* ACCURACY:
*
* In DEC arithmetic, the test (1/z) * z = 1 had peak relative
* error 3.1e-17, rms 1.2e-17. The test (y/z) * (z/y) = 1 had
* peak relative error 8.3e-17, rms 2.1e-17.
*
* Tests in the rectangle {-10,+10}:
* Relative error:
* arithmetic function # trials peak rms
* DEC cadd 10000 1.4e-17 3.4e-18
* IEEE cadd 100000 1.1e-16 2.7e-17
* DEC csub 10000 1.4e-17 4.5e-18
* IEEE csub 100000 1.1e-16 3.4e-17
* DEC cmul 3000 2.3e-17 8.7e-18
* IEEE cmul 100000 2.1e-16 6.9e-17
* DEC cdiv 18000 4.9e-17 1.3e-17
* IEEE cdiv 100000 3.7e-16 1.1e-16
*/
/* cmplx.c
* complex number arithmetic
*/
/*
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1995, 2000 by Stephen L. Moshier
*/
#include "mconf.h"
#ifdef ANSIPROT
extern double md_fabs ( double );
extern double md_cabs ( cmplx * );
extern double sqrt ( double );
extern double md_atan2 ( double, double );
extern double md_cos ( double );
extern double md_sin ( double );
extern double sqrt ( double );
extern double md_frexp ( double, int * );
extern double md_ldexp ( double, int );
int isnan ( double );
void cdiv ( cmplx *, cmplx *, cmplx * );
void cadd ( cmplx *, cmplx *, cmplx * );
#else
double md_fabs(), md_cabs(), sqrt(), md_atan2(), md_cos(), md_sin();
double sqrt(), md_frexp(), md_ldexp();
int isnan();
void cdiv(), cadd();
#endif
extern double MAXNUM, MACHEP, PI, PIO2, INFINITY, NAN;
/*
typedef struct
{
double r;
double i;
}cmplx;
*/
cmplx czero = {0.0, 0.0};
extern cmplx czero;
cmplx cone = {1.0, 0.0};
extern cmplx cone;
/* c = b + a */
void cadd( a, b, c )
register cmplx *a, *b;
cmplx *c;
{
c->r = b->r + a->r;
c->i = b->i + a->i;
}
/* c = b - a */
void csub( a, b, c )
register cmplx *a, *b;
cmplx *c;
{
c->r = b->r - a->r;
c->i = b->i - a->i;
}
/* c = b * a */
void cmul( a, b, c )
register cmplx *a, *b;
cmplx *c;
{
double y;
y = b->r * a->r - b->i * a->i;
c->i = b->r * a->i + b->i * a->r;
c->r = y;
}
/* c = b / a */
void cdiv( a, b, c )
register cmplx *a, *b;
cmplx *c;
{
double y, p, q, w;
y = a->r * a->r + a->i * a->i;
p = b->r * a->r + b->i * a->i;
q = b->i * a->r - b->r * a->i;
if( y < 1.0 )
{
w = MAXNUM * y;
if( (md_fabs(p) > w) || (md_fabs(q) > w) || (y == 0.0) )
{
c->r = MAXNUM;
c->i = MAXNUM;
mtherr( "cdiv", OVERFLOW );
return;
}
}
c->r = p/y;
c->i = q/y;
}
/* b = a
Caution, a `short' is assumed to be 16 bits wide. */
void cmov( a, b )
void *a, *b;
{
register short *pa, *pb;
int i;
pa = (short *) a;
pb = (short *) b;
i = 8;
do
*pb++ = *pa++;
while( --i );
}
void cneg( a )
register cmplx *a;
{
a->r = -a->r;
a->i = -a->i;
}
/* md_cabs()
*
* Complex absolute value
*
*
*
* SYNOPSIS:
*
* double md_cabs();
* cmplx z;
* double a;
*
* a = md_cabs( &z );
*
*
*
* DESCRIPTION:
*
*
* If z = x + iy
*
* then
*
* a = sqrt( x**2 + y**2 ).
*
* Overflow and underflow are avoided by testing the magnitudes
* of x and y before squaring. If either is outside half of
* the floating point full scale range, both are rescaled.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -30,+30 30000 3.2e-17 9.2e-18
* IEEE -10,+10 100000 2.7e-16 6.9e-17
*/
/*
Cephes Math Library Release 2.1: January, 1989
Copyright 1984, 1987, 1989 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
/*
typedef struct
{
double r;
double i;
}cmplx;
*/
#ifdef UNK
#define PREC 27
#define MAXEXP 1024
#define MINEXP -1077
#endif
#ifdef DEC
#define PREC 29
#define MAXEXP 128
#define MINEXP -128
#endif
#ifdef IBMPC
#define PREC 27
#define MAXEXP 1024
#define MINEXP -1077
#endif
#ifdef MIEEE
#define PREC 27
#define MAXEXP 1024
#define MINEXP -1077
#endif
double md_cabs( z )
register cmplx *z;
{
double x, y, b, re, im;
int ex, ey, e;
#ifdef INFINITIES
/* Note, md_cabs(INFINITY,NAN) = INFINITY. */
if( z->r == INFINITY || z->i == INFINITY
|| z->r == -INFINITY || z->i == -INFINITY )
return( INFINITY );
#endif
#ifdef NANS
if( isnan(z->r) )
return(z->r);
if( isnan(z->i) )
return(z->i);
#endif
re = md_fabs( z->r );
im = md_fabs( z->i );
if( re == 0.0 )
return( im );
if( im == 0.0 )
return( re );
/* Get the exponents of the numbers */
x = md_frexp( re, &ex );
y = md_frexp( im, &ey );
/* Check if one number is tiny compared to the other */
e = ex - ey;
if( e > PREC )
return( re );
if( e < -PREC )
return( im );
/* Find approximate exponent e of the geometric mean. */
e = (ex + ey) >> 1;
/* Rescale so mean is about 1 */
x = md_ldexp( re, -e );
y = md_ldexp( im, -e );
/* Hypotenuse of the right triangle */
b = sqrt( x * x + y * y );
/* Compute the exponent of the answer. */
y = md_frexp( b, &ey );
ey = e + ey;
/* Check it for overflow and underflow. */
if( ey > MAXEXP )
{
mtherr( "md_cabs", OVERFLOW );
return( INFINITY );
}
if( ey < MINEXP )
return(0.0);
/* Undo the scaling */
b = md_ldexp( b, e );
return( b );
}
/* md_csqrt()
*
* Complex square root
*
*
*
* SYNOPSIS:
*
* void md_csqrt();
* cmplx z, w;
*
* md_csqrt( &z, &w );
*
*
*
* DESCRIPTION:
*
*
* If z = x + iy, r = |z|, then
*
* 1/2
* Im w = [ (r - x)/2 ] ,
*
* Re w = y / 2 Im w.
*
*
* Note that -w is also a square root of z. The root chosen
* is always in the upper half plane.
*
* Because of the potential for cancellation error in r - x,
* the result is sharpened by doing a Heron iteration
* (see sqrt.c) in complex arithmetic.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 25000 3.2e-17 9.6e-18
* IEEE -10,+10 100000 3.2e-16 7.7e-17
*
* 2
* Also tested by md_csqrt( z ) = z, and tested by arguments
* close to the real axis.
*/
void md_csqrt( z, w )
cmplx *z, *w;
{
cmplx q, s;
double x, y, r, t;
x = z->r;
y = z->i;
if( y == 0.0 )
{
if( x < 0.0 )
{
w->r = 0.0;
w->i = sqrt(-x);
return;
}
else
{
w->r = sqrt(x);
w->i = 0.0;
return;
}
}
if( x == 0.0 )
{
r = md_fabs(y);
r = sqrt(0.5*r);
if( y > 0 )
w->r = r;
else
w->r = -r;
w->i = r;
return;
}
/* Approximate sqrt(x^2+y^2) - x = y^2/2x - y^4/24x^3 + ... .
* The relative error in the first term is approximately y^2/12x^2 .
*/
if( (md_fabs(y) < 2.e-4 * md_fabs(x))
&& (x > 0) )
{
t = 0.25*y*(y/x);
}
else
{
r = md_cabs(z);
t = 0.5*(r - x);
}
r = sqrt(t);
q.i = r;
q.r = y/(2.0*r);
/* Heron iteration in complex arithmetic */
cdiv( &q, z, &s );
cadd( &q, &s, w );
w->r *= 0.5;
w->i *= 0.5;
}
double md_hypot( x, y )
double x, y;
{
cmplx z;
z.r = x;
z.i = y;
return( md_cabs(&z) );
}