// finds the complex roots of the polynomial using Bairstow's method
// _p : polynomial array, ascending powers [size: _k x 1]
// _k : polynomials length (poly order = _k - 1)
// _roots : resulting complex roots [size: _k-1 x 1]
void POLY(_findroots_bairstow)(T * _p,
unsigned int _k,
TC * _roots)
{
T p0[_k]; // buffer 0
T p1[_k]; // buffer 1
T * p = NULL; // input polynomial
T * pr = NULL; // output (reduced) polynomial
unsigned int i;
unsigned int k=0; // output roots counter
memmove(p0, _p, _k*sizeof(T));
T u, v;
unsigned int n = _k; // input counter (decrementer)
unsigned int r = _k % 2; // polynomial length odd? (order even?)
unsigned int L = (_k-r)/2; // semi-length
for (i=0; i<L-1+r; i++) {
// set polynomial and reduced polynomial buffer pointers
p = (i % 2) == 0 ? p0 : p1;
pr = (i % 2) == 0 ? p1 : p0;
// initial estimates for u, v
// TODO : ensure no division by zero
if (p[n-1] == 0) {
fprintf(stderr,"warning: poly_findroots_bairstow(), irreducible polynomial");
p[n-1] = 1e-12;
}
u = p[n-2] / p[n-1];
v = p[n-3] / p[n-1];
// compute factor using Bairstow's recursion
POLY(_findroots_bairstow_recursion)(p,n,pr,&u,&v);
// compute complex roots of x^2 + u*x + v
TC r0 = 0.5f*(-u + csqrtf(u*u - 4.0*v));
TC r1 = 0.5f*(-u - csqrtf(u*u - 4.0*v));
// append result to output
_roots[k++] = r0;
_roots[k++] = r1;
#if 0
// print debugging info
unsigned int j;
printf("initial polynomial:\n");
for (j=0; j<n; j++)
printf(" p[%3u] = %12.8f + j*%12.8f\n", j, crealf(p[j]), cimagf(p[j]));
printf("polynomial factor: x^2 + u*x + v\n");
printf(" u : %12.8f + j*%12.8f\n", crealf(u), cimagf(u));
printf(" v : %12.8f + j*%12.8f\n", crealf(v), cimagf(v));
printf("roots:\n");
printf(" r0 : %12.8f + j*%12.8f\n", crealf(r0), cimagf(r0));
printf(" r1 : %12.8f + j*%12.8f\n", crealf(r1), cimagf(r1));
printf("reduced polynomial:\n");
for (j=0; j<n-2; j++)
printf(" pr[%3u] = %12.8f + j*%12.8f\n", j, crealf(pr[j]), cimagf(pr[j]));
#endif
// decrement new (reduced) polynomial size by 2
n -= 2;
}
if (r==0) {
assert(n==2);
_roots[k++] = -pr[0]/pr[1];
}
//assert( k == _k-1 );
}
static long double complex
_csqrtf(long double complex d)
{
return (csqrtf((float complex)d));
}
int main() {
unsigned int n=32; // number of tests
unsigned int d=2; // number of items per line
// data arrays
float complex z[n];
float complex test[n];
float complex err_max = 0.0f;
unsigned int i;
for (i=0; i<n; i++) {
// generate random complex number
z[i] = 2.0f*(2.0f*sandbox_randf() - 1.0f) +
2.0f*(2.0f*sandbox_randf() - 1.0f) * _Complex_I;
test[i] = csqrtf(z[i]);
float complex sqrtz_hat = sandbox_csqrtf(z[i]);
float complex err = test[i] - sqrtz_hat;
printf("%3u: z=%6.2f+j%6.2f, sqrt(z)=%6.2f+j%6.2f (%6.2f+j%6.2f) e=%12.4e\n",
i,
crealf(test[i]), cimagf(z[i]),
crealf(test[i]), cimagf(test[i]),
crealf(sqrtz_hat), cimagf(sqrtz_hat),
cabsf(err));
if ( cabsf(err) > cabsf(err_max) )
err_max = err;
}
printf("maximum error: %12.4e;\n", cabsf(err_max));
//
// print autotest lines
//
printf("\n");
printf(" float complex z[%u] = {\n ", n);
for (i=0; i<n; i++) {
printf("%12.4e+_Complex_I*%12.4e", crealf(z[i]), cimagf(z[i]));
if ( i == n-1)
printf(" ");
else if ( ((i+1)%d)==0 )
printf(",\n ");
else
printf(", ");
}
printf("};\n");
printf("\n");
printf(" float complex test[%u] = {\n ", n);
for (i=0; i<n; i++) {
printf("%12.4e+_Complex_I*%12.4e", crealf(test[i]), cimagf(test[i]));
if ( i == n-1)
printf(" ");
else if ( ((i+1)%d)==0 )
printf(",\n ");
else
printf(", ");
}
printf("};\n");
printf("done.\n");
return 0;
}
ld = cexpl(ld); TEST_TRACE(C99 7.3.7.2) d = clog(d); f = clogf(f); ld = clogl(ld); TEST_TRACE(C99 7.3.8.1) d = cabs(d); f = cabsf(f); ld = cabsl(ld); TEST_TRACE(C99 7.3.8.2) d = cpow(d, d); f = cpowf(f, f); ld = cpowl(ld, ld); TEST_TRACE(C99 7.3.8.3) d = csqrt(d); f = csqrtf(f); ld = csqrtl(ld); TEST_TRACE(C99 7.3.9.1) d = carg(d); f = cargf(f); ld = cargl(ld); TEST_TRACE(C99 7.3.9.2) d = cimag(d); f = cimagf(f); ld = cimagl(ld); TEST_TRACE(C99 7.3.9.3) d = conj(d); f = conjf(f); ld = conjl(ld); TEST_TRACE(C99 7.3.9.4) d = cproj(d);
void
docomplexf (void)
{
#ifndef NO_FLOAT
complex float ca, cb, cc;
float f1;
ca = 1.0 + 1.0 * I;
cb = 1.0 - 1.0 * I;
f1 = cabsf (ca);
fprintf (stdout, "cabsf : %f\n", f1);
cc = cacosf (ca);
fprintf (stdout, "cacosf : %f %fi\n", crealf (cc),
cimagf (cc));
cc = cacoshf (ca);
fprintf (stdout, "cacoshf: %f %fi\n", crealf (cc),
cimagf (cc));
f1 = cargf (ca);
fprintf (stdout, "cargf : %f\n", f1);
cc = casinf (ca);
fprintf (stdout, "casinf : %f %fi\n", crealf (cc),
cimagf (cc));
cc = casinhf (ca);
fprintf (stdout, "casinhf: %f %fi\n", crealf (cc),
cimagf (cc));
cc = catanf (ca);
fprintf (stdout, "catanf : %f %fi\n", crealf (cc),
cimagf (cc));
cc = catanhf (ca);
fprintf (stdout, "catanhf: %f %fi\n", crealf (cc),
cimagf (cc));
cc = ccosf (ca);
fprintf (stdout, "ccosf : %f %fi\n", crealf (cc),
cimagf (cc));
cc = ccoshf (ca);
fprintf (stdout, "ccoshf : %f %fi\n", crealf (cc),
cimagf (cc));
cc = cexpf (ca);
fprintf (stdout, "cexpf : %f %fi\n", crealf (cc),
cimagf (cc));
f1 = cimagf (ca);
fprintf (stdout, "cimagf : %f\n", f1);
cc = clogf (ca);
fprintf (stdout, "clogf : %f %fi\n", crealf (cc),
cimagf (cc));
cc = conjf (ca);
fprintf (stdout, "conjf : %f %fi\n", crealf (cc),
cimagf (cc));
cc = cpowf (ca, cb);
fprintf (stdout, "cpowf : %f %fi\n", crealf (cc),
cimagf (cc));
cc = cprojf (ca);
fprintf (stdout, "cprojf : %f %fi\n", crealf (cc),
cimagf (cc));
f1 = crealf (ca);
fprintf (stdout, "crealf : %f\n", f1);
cc = csinf (ca);
fprintf (stdout, "csinf : %f %fi\n", crealf (cc),
cimagf (cc));
cc = csinhf (ca);
fprintf (stdout, "csinhf : %f %fi\n", crealf (cc),
cimagf (cc));
cc = csqrtf (ca);
fprintf (stdout, "csqrtf : %f %fi\n", crealf (cc),
cimagf (cc));
cc = ctanf (ca);
fprintf (stdout, "ctanf : %f %fi\n", crealf (cc),
cimagf (cc));
cc = ctanhf (ca);
fprintf (stdout, "ctanhf : %f %fi\n", crealf (cc),
cimagf (cc));
#endif
}
int main(int argc, char* argv[])
{
int job;
float x,y, rads, theta;
sf_complex cs, cz;
const int nfat=1, ntheta= 271;
const float eps=0.1, k2=0.05;
vp_init();
for (job=0; job < 5; job++) {
vp_uorig ( -1.75-2.20*job, -1.5 );
vp_uclip (-1.,-1.,1.3,1.1);
vp_fat (nfat);
vp_umove (0.,-.9);
vp_udraw (0.,4.9);
vp_umove (-.5,0.);
vp_udraw (1.2,0.);
vp_utext( .1, .9, 4,0, "Im");
vp_utext( 1.05, -.2, 4,0, "Re");
switch (job) {
case 0:
vp_utext( .1, -.7 , 4,0, "\\F10 w=+p");
vp_utext( .1, .6 , 4,0, "\\F10 w=-p");
vp_utext( .1, .05, 4,0, "\\F10 w=0");
break;
case 1:
vp_utext( .4, -.65, 4,0, "\\F10 w=p/2");
vp_utext( .4, .55, 4,0, "\\F10 w=-p/2");
vp_utext( .1, .05, 4,0, "\\F10 w=0");
break;
default:
break;
}
vp_fat(0);
vp_penup();
for( theta = -180.; theta<180.1; theta += 360./ntheta ) {
rads = 2. * SF_PI * theta / 360.;
cz = cexpf( sf_cmplx(0.,rads) );
#ifdef SF_HAS_COMPLEX_H
cs = (1.+eps - cz )/2.;
#else
cs = sf_crmul(sf_cadd(sf_cmplx(1.+eps,0.),sf_cneg(cz)),0.5);
#endif
switch (job) {
case 0: cs = sf_cmplx( .05, .8*theta/180.); break;
case 1: break;
#ifdef SF_HAS_COMPLEX_H
case 2: cs *= cs; break;
case 3: cs = cs*cs + k2; break;
case 4: cs = csqrtf( cs*cs + k2 ); break;
#else
case 2: cs = sf_cmul(cs,cs);
break;
case 3: cs = sf_cadd(sf_cmul(cs,cs),sf_cmplx(k2,0.));
break;
case 4: cs = csqrtf(sf_cadd(sf_cmul(cs,cs),sf_cmplx(k2,0.)));
break;
#endif
default: break;
}
x = crealf( cs );
y = cimagf( cs );
vp_upendn ( x, -y);
}
}
return 0;
}
complex float
cacoshf (complex float z)
{
return clogf (z + csqrtf (z-1.0f) * csqrtf (z+1.0f));
}
complex float
casinhf (complex float z)
{
return clogf (z + csqrtf (z*z+1.0f));
}
complex float
cacosf (complex float z)
{
return -I*clogf (z + I*csqrtf (1.0f-z*z));
}
complex float
casinf (complex float z)
{
return -I*clogf (I*z + csqrtf (1.0f-z*z));
}
