/**
* b n _ p o l y _ q u a r t i c _ r o o t s
*@brief
* Uses the quartic formula to find the roots ( in `complex' form )
* of any quartic equation with real coefficients.
*
* @return 1 for success
* @return 0 for fail.
*/
int
bn_poly_quartic_roots(register struct bn_complex *roots, register const struct bn_poly *eqn)
{
struct bn_poly cube, quad1, quad2;
bn_complex_t u[3];
fastf_t U, p, q, q1, q2;
/* something considerably larger than squared floating point fuss */
const fastf_t small = 1.0e-8;
#define Max3(a, b, c) ((c)>((a)>(b)?(a):(b)) ? (c) : ((a)>(b)?(a):(b)))
cube.dgr = 3;
cube.cf[0] = 1.0;
cube.cf[1] = -eqn->cf[2];
cube.cf[2] = eqn->cf[3]*eqn->cf[1]
- 4*eqn->cf[4];
cube.cf[3] = -eqn->cf[3]*eqn->cf[3]
- eqn->cf[4]*eqn->cf[1]*eqn->cf[1]
+ 4*eqn->cf[4]*eqn->cf[2];
if (!bn_poly_cubic_roots( u, &cube )) {
return 0; /* FAIL */
}
if (u[1].im != 0.0) {
U = u[0].re;
} else {
U = Max3( u[0].re, u[1].re, u[2].re );
}
p = eqn->cf[1]*eqn->cf[1]*0.25 + U - eqn->cf[2];
U *= 0.5;
q = U*U - eqn->cf[4];
if (p < 0) {
if (p < -small) {
return 0; /* FAIL */
}
p = 0;
} else {
p = sqrt( p );
}
if (q < 0 ) {
if (q < -small) {
return 0; /* FAIL */
}
q = 0;
} else {
q = sqrt( q );
}
quad1.dgr = quad2.dgr = 2;
quad1.cf[0] = quad2.cf[0] = 1.0;
quad1.cf[1] = eqn->cf[1]*0.5;
quad2.cf[1] = quad1.cf[1] + p;
quad1.cf[1] -= p;
q1 = U - q;
q2 = U + q;
p = quad1.cf[1]*q2 + quad2.cf[1]*q1 - eqn->cf[3];
if (NEAR_ZERO(p, small)) {
quad1.cf[2] = q1;
quad2.cf[2] = q2;
} else {
q = quad1.cf[1]*q1 + quad2.cf[1]*q2 - eqn->cf[3];
if (NEAR_ZERO(q, small)) {
quad1.cf[2] = q2;
quad2.cf[2] = q1;
} else {
return 0; /* FAIL */
}
}
bn_poly_quadratic_roots( &roots[0], &quad1 );
bn_poly_quadratic_roots( &roots[2], &quad2 );
return 1; /* SUCCESS */
}
int
rt_poly_roots(register bn_poly_t *eqn, /* equation to be solved */
register bn_complex_t roots[], /* space to put roots found */
const char *name) /* name of the primitive being checked */
{
register size_t n; /* number of roots found */
fastf_t factor; /* scaling factor for copy */
/* Remove leading coefficients which are too close to zero,
* to prevent the polynomial factoring from blowing up, below.
*/
while (ZERO(eqn->cf[0])) {
for (n=0; n <= eqn->dgr; n++) {
eqn->cf[n] = eqn->cf[n+1];
}
if (--eqn->dgr <= 0)
return 0;
}
/* Factor the polynomial so the first coefficient is one
* for ease of handling.
*/
factor = 1.0 / eqn->cf[0];
(void) bn_poly_scale(eqn, factor);
n = 0; /* Number of roots found */
/* A trailing coefficient of zero indicates that zero
* is a root of the equation.
*/
while (ZERO(eqn->cf[eqn->dgr])) {
roots[n].re = roots[n].im = 0.0;
--eqn->dgr;
++n;
}
while (eqn->dgr > 2) {
if (eqn->dgr == 4) {
if (bn_poly_quartic_roots(&roots[n], eqn)) {
if (rt_poly_checkroots(eqn, &roots[n], 4) == 0) {
return n+4;
}
}
} else if (eqn->dgr == 3) {
if (bn_poly_cubic_roots(&roots[n], eqn)) {
if (rt_poly_checkroots(eqn, &roots[n], 3) == 0) {
return n+3;
}
}
}
/*
* Set initial guess for root to almost zero.
* This method requires a small nudge off the real axis.
*/
bn_cx_cons(&roots[n], 0.0, SMALL);
if ((rt_poly_findroot(eqn, &roots[n], name)) < 0)
return n; /* return those we found, anyways */
if (fabs(roots[n].im) > 1.0e-5* fabs(roots[n].re)) {
/* If root is complex, its complex conjugate is
* also a root since complex roots come in con-
* jugate pairs when all coefficients are real.
*/
++n;
roots[n] = roots[n-1];
bn_cx_conj(&roots[n]);
} else {
/* Change 'practically real' to real */
roots[n].im = 0.0;
}
rt_poly_deflate(eqn, &roots[n]);
++n;
}
/* For polynomials of lower degree, iterative techniques
* are an inefficient way to find the roots.
*/
if (eqn->dgr == 1) {
roots[n].re = -(eqn->cf[1]);
roots[n].im = 0.0;
++n;
} else if (eqn->dgr == 2) {
bn_poly_quadratic_roots(&roots[n], eqn);
n += 2;
}
return n;
}