/*
* highly inefficient computation of the imaginary part of complex
* conjugate eigenvalues of a 3x3 non-symmetric matrix
*/
double
gage_imaginary_part_eigenvalues( gage_t *M ) {
double A, B, C, scale, frob, m[9], _eval[3];
double beta, gamma;
int roots;
frob = ELL_3M_FROB(M);
scale = frob > 10 ? 10.0/frob : 1.0;
ELL_3M_SCALE(m, scale, M);
/*
** from gordon with mathematica; these are the coefficients of the
** cubic polynomial in x: det(x*I - M). The full cubic is
** x^3 + A*x^2 + B*x + C.
*/
A = -m[0] - m[4] - m[8];
B = m[0]*m[4] - m[3]*m[1]
+ m[0]*m[8] - m[6]*m[2]
+ m[4]*m[8] - m[7]*m[5];
C = (m[6]*m[4] - m[3]*m[7])*m[2]
+ (m[0]*m[7] - m[6]*m[1])*m[5]
+ (m[3]*m[1] - m[0]*m[4])*m[8];
roots = ell_cubic(_eval, A, B, C, AIR_TRUE);
if ( roots != ell_cubic_root_single )
return 0.;
/* 2 complex conjuguate eigenvalues */
beta = A + _eval[0];
gamma = -C/_eval[0];
return sqrt( 4.*gamma - beta*beta );
}
/*
******** ell_3m_eigenvalues_d()
**
** finds eigenvalues of given matrix.
**
** returns information about the roots according to ellCubeRoot enum,
** see header for ellCubic for details.
**
** given matrix is NOT modified
**
** This does NOT use biff
**
** Doing the frobenius normalization proved successfull in avoiding the
** the creating of NaN eigenvalues when the coefficients of the matrix
** were really large (> 50000). Also, when the matrix norm was really
** small, the comparison to "epsilon" in ell_cubic mistook three separate
** roots for a single and a double, with this matrix in particular:
** 1.7421892 0.0137642 0.0152975
** 0.0137642 1.7565432 -0.0062296
** 0.0152975 -0.0062296 1.7700019
** (actually, this is prior to tenEigensolve's isotropic removal)
**
** HEY: tenEigensolve_d and tenEigensolve_f start by removing the
** isotropic part of the tensor. It may be that that smarts should
** be migrated here, but GLK is uncertain how it would change the
** handling of non-symmetric matrices.
*/
int
ell_3m_eigenvalues_d(double _eval[3], const double _m[9], const int newton) {
double A, B, C, scale, frob, m[9], eval[3];
int roots;
frob = ELL_3M_FROB(_m);
scale = frob ? 1.0/frob : 1.0;
ELL_3M_SCALE(m, scale, _m);
/*
printf("!%s: m = %g %g %g; %g %g %g; %g %g %g\n", "ell_3m_eigenvalues_d",
m[0], m[1], m[2], m[3], m[4], m[5], m[6], m[7], m[8]);
*/
/*
** from gordon with mathematica; these are the coefficients of the
** cubic polynomial in x: det(x*I - M). The full cubic is
** x^3 + A*x^2 + B*x + C.
*/
A = -m[0] - m[4] - m[8];
B = m[0]*m[4] - m[3]*m[1]
+ m[0]*m[8] - m[6]*m[2]
+ m[4]*m[8] - m[7]*m[5];
C = (m[6]*m[4] - m[3]*m[7])*m[2]
+ (m[0]*m[7] - m[6]*m[1])*m[5]
+ (m[3]*m[1] - m[0]*m[4])*m[8];
/*
printf("!%s: A B C = %g %g %g\n", "ell_3m_eigenvalues_d", A, B, C);
*/
roots = ell_cubic(eval, A, B, C, newton);
/* no longer need to sort here */
ELL_3V_SCALE(_eval, 1.0/scale, eval);
return roots;
}
int
main(int argc, char **argv) {
char buf[512];
double ans0, ans1, ans2, A, B, C;
int ret;
double r[3];
me = argv[0];
if (argc != 4) {
usage();
}
sprintf(buf, "%s %s %s", argv[1], argv[2], argv[3]);
if (3 != sscanf(buf, "%lf %lf %lf", &A, &B, &C)) {
fprintf(stderr, "%s: couldn't parse 3 floats from command line\n", me);
exit(1);
}
ell_debug = AIR_TRUE;
ret = ell_cubic(r, A, B, C, AIR_TRUE);
ans0 = C + r[0]*(B + r[0]*(A + r[0]));
switch(ret) {
case ell_cubic_root_single:
printf("1 single root: %f -> %f\n", r[0], ans0);
break;
case ell_cubic_root_triple:
printf("1 triple root: %f -> %f\n", r[0], ans0);
break;
case ell_cubic_root_single_double:
ans1 = C + r[1]*(B + r[1]*(A + r[1]));
printf("1 single root %f -> %f, 1 double root %f -> %f\n",
r[0], ans0, r[1], ans1);
break;
case ell_cubic_root_three:
ans1 = C + r[1]*(B + r[1]*(A + r[1]));
ans2 = C + r[2]*(B + r[2]*(A + r[2]));
printf("3 distinct roots:\n %f -> %f\n %f -> %f\n %f -> %f\n",
r[0], ans0, r[1], ans1, r[2], ans2);
break;
default:
printf("%s: something fatally wacky happened\n", me);
exit(1);
}
exit(0);
}