/* cmplxf.c * * Complex number arithmetic * * * * SYNOPSIS: * * typedef struct { * float r; real part * float i; imaginary part * }cmplxf; * * cmplxf *a, *b, *c; * * caddf( a, b, c ); c = b + a * csubf( a, b, c ); c = b - a * cmulf( a, b, c ); c = b * a * cdivf( a, b, c ); c = b / a * cnegf( c ); c = -c * cmovf( 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 * IEEE cadd 30000 5.9e-8 2.6e-8 * IEEE csub 30000 6.0e-8 2.6e-8 * IEEE cmul 30000 1.1e-7 3.7e-8 * IEEE cdiv 30000 2.1e-7 5.7e-8 */ /* cmplx.c * complex number arithmetic */ /* Cephes Math Library Release 2.1: December, 1988 Copyright 1984, 1987, 1988 by Stephen L. Moshier Direct inquiries to 30 Frost Street, Cambridge, MA 02140 */ #include "mconf.h" extern float MAXNUMF, MACHEPF, PIF, PIO2F; #define fabsf(x) ( (x) < 0 ? -(x) : (x) ) #ifdef ANSIC float sqrtf(float), frexpf(float, int *); float ldexpf(float, int); float cabsf(cmplxf *), atan2f(float, float), cosf(float), sinf(float); #else float sqrtf(), frexpf(), ldexpf(); float cabsf(), atan2f(), cosf(), sinf(); #endif /* typedef struct { float r; float i; }cmplxf; */ cmplxf czerof = {0.0, 0.0}; extern cmplxf czerof; cmplxf conef = {1.0, 0.0}; extern cmplxf conef; /* c = b + a */ void caddf( a, b, c ) register cmplxf *a, *b; cmplxf *c; { c->r = b->r + a->r; c->i = b->i + a->i; } /* c = b - a */ void csubf( a, b, c ) register cmplxf *a, *b; cmplxf *c; { c->r = b->r - a->r; c->i = b->i - a->i; } /* c = b * a */ void cmulf( a, b, c ) register cmplxf *a, *b; cmplxf *c; { register float 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 cdivf( a, b, c ) register cmplxf *a, *b; cmplxf *c; { float 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 < 1.0f ) { w = MAXNUMF * y; if( (fabsf(p) > w) || (fabsf(q) > w) || (y == 0.0f) ) { c->r = MAXNUMF; c->i = MAXNUMF; mtherrf( "cdivf", OVERFLOW ); return; } } c->r = p/y; c->i = q/y; } /* b = a */ void cmovf( a, b ) register short *a, *b; { int i; i = 8; do *b++ = *a++; while( --i ); } void cnegf( a ) register cmplxf *a; { a->r = -a->r; a->i = -a->i; } /* cabsf() * * Complex absolute value * * * * SYNOPSIS: * * float cabsf(); * cmplxf z; * float a; * * a = cabsf( &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 * IEEE -10,+10 30000 1.2e-7 3.4e-8 */ /* 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 { float r; float i; }cmplxf; */ /* square root of max and min numbers */ #define SMAX 1.3043817825332782216E+19 #define SMIN 7.6664670834168704053E-20 #define PREC 12 #define MAXEXPF 128 #define SMAXT (2.0f * SMAX) #define SMINT (0.5f * SMIN) float cabsf( z ) register cmplxf *z; { float x, y, b, re, im; int ex, ey, e; re = fabsf( z->r ); im = fabsf( z->i ); if( re == 0.0f ) { return( im ); } if( im == 0.0f ) { return( re ); } /* Get the exponents of the numbers */ x = frexpf( re, &ex ); y = frexpf( 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 = ldexpf( re, -e ); y = ldexpf( im, -e ); /* Hypotenuse of the right triangle */ b = sqrtf( x * x + y * y ); /* Compute the exponent of the answer. */ y = frexpf( b, &ey ); ey = e + ey; /* Check it for overflow and underflow. */ if( ey > MAXEXPF ) { mtherrf( "cabsf", OVERFLOW ); return( MAXNUMF ); } if( ey < -MAXEXPF ) return(0.0f); /* Undo the scaling */ b = ldexpf( b, e ); return( b ); } /* csqrtf() * * Complex square root * * * * SYNOPSIS: * * void csqrtf(); * cmplxf z, w; * * csqrtf( &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 solution * reported 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 * IEEE -10,+10 100000 1.8e-7 4.2e-8 * */ void csqrtf( z, w ) cmplxf *z, *w; { cmplxf q, s; float x, y, r, t; x = z->r; y = z->i; if( y == 0.0f ) { if( x < 0.0f ) { w->r = 0.0f; w->i = sqrtf(-x); return; } else { w->r = sqrtf(x); w->i = 0.0f; return; } } if( x == 0.0f ) { r = fabsf(y); r = sqrtf(0.5f*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( (fabsf(y) < fabsf(0.015f*x)) && (x > 0) ) { t = 0.25f*y*(y/x); } else { r = cabsf(z); t = 0.5f*(r - x); } r = sqrtf(t); q.i = r; q.r = 0.5f*y/r; /* Heron iteration in complex arithmetic: * q = (q + z/q)/2 */ cdivf( &q, z, &s ); caddf( &q, &s, w ); w->r *= 0.5f; w->i *= 0.5f; }