/*
Description:
Find the eigen values and eigen vectors of a real symmetric
3x3 matrix
A D F
D B E
F E C
Parameters:
A - [in] matrix entry
B - [in] matrix entry
C - [in] matrix entry
D - [in] matrix entry
E - [in] matrix entry
F - [in] matrix entry
e1 - [out] eigen value
E1 - [out] eigen vector with eigen value e1
e2 - [out] eigen value
E2 - [out] eigen vector with eigen value e2
e3 - [out] eigen value
E3 - [out] eigen vector with eigen value e3
Returns:
True if successful.
*/
bool ON_Sym3x3EigenSolver( double A, double B, double C,
double D, double E, double F,
double* e1, ON_3dVector E1,
double* e2, ON_3dVector E2,
double* e3, ON_3dVector E3
)
{
// STEP 1: reduce to tri-diagonal form
double cos_phi = 1.0;
double sin_phi = 0.0;
double AA = A, BB = B, CC = C, DD = D, EE = E, FF = F;
if ( F != 0.0 )
{
double theta = 0.5*(C-A)/F;
double t;
if ( fabs(theta) > 1.0e154 )
{
t = 0.5/fabs(theta);
}
else if ( fabs(theta) > 1.0 )
{
t = 1.0/(fabs(theta)*(1.0 + sqrt(1.0 + 1.0/(theta*theta))));
}
else
{
t = 1.0/(fabs(theta) + sqrt(1.0+theta*theta));
}
if ( theta < 0.0 )
t = -t;
if ( fabs(t) > 1.0 )
{
double tt = 1.0/t;
cos_phi = 1.0/(fabs(t)*sqrt(1.0 + tt*tt));
}
else
cos_phi = 1.0/sqrt(1.0 + t*t);
sin_phi = t*cos_phi;
double tau = sin_phi/(1.0 + cos_phi);
AA = A - t*F;
BB = B;
CC = C + t*F;
DD = D - sin_phi*(E + tau*D);
EE = E + sin_phi*(D + tau*E);
// debugging test - FF should be close to zero.
// cos_phi*cos_phi + sin_phi*size_phi should be close to 1
FF = (cos_phi*cos_phi - sin_phi*sin_phi)*F + sin_phi*cos_phi*(A-C);
}
double ee1, ee2, ee3;
ON_3dVector EE1, EE2, EE3;
bool rc = ON_SymTriDiag3x3EigenSolver( AA, BB, CC, DD, EE,
&ee1, EE1,
&ee2, EE2,
&ee3, EE3 );
E1.Set(cos_phi*EE1.x - sin_phi*EE1.z, EE1.y, sin_phi*EE1.x + cos_phi*EE1.z );
E2.Set(cos_phi*EE2.x - sin_phi*EE2.z, EE2.y, sin_phi*EE2.x + cos_phi*EE2.z );
E3.Set(cos_phi*EE3.x - sin_phi*EE3.z, EE3.y, sin_phi*EE3.x + cos_phi*EE3.z );
if ( e1 )
*e1 = ee1;
if ( e2 )
*e2 = ee2;
if ( e3 )
*e3 = ee3;
// debugging check of eigen values
ON_3dVector err1, err2, err3;
{
err1.x = (A*E1.x + D*E1.y + F*E1.z) - ee1*E1.x;
err1.y = (D*E1.x + B*E1.y + E*E1.z) - ee1*E1.y;
err1.z = (F*E1.x + E*E1.y + C*E1.z) - ee1*E1.z;
err2.x = (A*E2.x + D*E2.y + F*E2.z) - ee2*E2.x;
err2.y = (D*E2.x + B*E2.y + E*E2.z) - ee2*E2.y;
err2.z = (F*E2.x + E*E2.y + C*E2.z) - ee2*E2.z;
err3.x = (A*E3.x + D*E3.y + F*E3.z) - ee3*E3.x;
err3.y = (D*E3.x + B*E3.y + E*E3.z) - ee3*E3.y;
err3.z = (F*E3.x + E*E3.y + C*E3.z) - ee3*E3.z;
}
return rc;
}
