/* 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 fabs ( double ); extern double cabs ( cmplx * ); extern double sqrt ( double ); extern double atan2 ( double, double ); extern double cos ( double ); extern double sin ( double ); extern double sqrt ( double ); extern double frexp ( double, int * ); extern double ldexp ( double, int ); int isnan ( double ); void cdiv ( cmplx *, cmplx *, cmplx * ); void cadd ( cmplx *, cmplx *, cmplx * ); #else double fabs(), cabs(), sqrt(), atan2(), cos(), sin(); double sqrt(), frexp(), 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( (fabs(p) > w) || (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; } /* cabs() * * Complex absolute value * * * * SYNOPSIS: * * double cabs(); * cmplx z; * double a; * * a = 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 cabs( z ) register cmplx *z; { double x, y, b, re, im; int ex, ey, e; #ifdef INFINITIES /* Note, 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 = fabs( z->r ); im = fabs( z->i ); if( re == 0.0 ) return( im ); if( im == 0.0 ) return( re ); /* Get the exponents of the numbers */ x = frexp( re, &ex ); y = 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 = ldexp( re, -e ); y = ldexp( im, -e ); /* Hypotenuse of the right triangle */ b = sqrt( x * x + y * y ); /* Compute the exponent of the answer. */ y = frexp( b, &ey ); ey = e + ey; /* Check it for overflow and underflow. */ if( ey > MAXEXP ) { mtherr( "cabs", OVERFLOW ); return( INFINITY ); } if( ey < MINEXP ) return(0.0); /* Undo the scaling */ b = ldexp( b, e ); return( b ); } /* csqrt() * * Complex square root * * * * SYNOPSIS: * * void csqrt(); * cmplx z, w; * * 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 csqrt( z ) = z, and tested by arguments * close to the real axis. */ void 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 = 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( (fabs(y) < 2.e-4 * fabs(x)) && (x > 0) ) { t = 0.25*y*(y/x); } else { r = 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 hypot( x, y ) double x, y; { cmplx z; z.r = x; z.i = y; return( cabs(&z) ); }