///////////////////////////////////////////////////////////////////////////////
//
/// Finds the zeroes of P(x) in the domain \f$a < x <= b\f$, putting the
/// results into Z
//
///////////////////////////////////////////////////////////////////////////////
void AntisymmetricExpFit::find_zeroes(RealVector &P, const Real &a, const Real &b, RealVector &Z) {
const int N = P.size();
RealMatrix cm(N-1,N-1);
int i,j;
// --- Form the companion matrix
// -----------------------------
cm.fill(0.0);
for(j=0; j<N-1; ++j) {
cm(0,j) = -P(N-2-j)/P(N-1);
}
for(i=1; i<N-1; ++i) {
cm(i,i-1) = 1.0;
}
// --- find eigenvalues of
// --- companion matrix and
// --- extract roots in [0:1]
// --------------------------
EigenSolver<RealMatrix> roots(cm,false);
Z.resize(N);
i = 0;
for(j = 0; j<roots.eigenvalues().size(); ++j) {
if(fabs(roots.eigenvalues()(j).imag()) < 1e-15 &&
roots.eigenvalues()(j).real() > a &&
roots.eigenvalues()(j).real() <= b) {
Z(i) = roots.eigenvalues()(j).real();
++i;
}
}
Z.conservativeResize(i);
}
///////////////////////////////////////////////////////////////////////////////
//
/// Find zeroes of the polynomial \f$y = sum_i e_ix^i\f$,
/// where \f$e\f$ is the \f$c^{th}\f$ eigenvector, and put any zeroes
/// that lie in the interval [0:1] into \c k.
///
/// \param c The identifier of the eigenvector to find zeroes of.
//
///////////////////////////////////////////////////////////////////////////////
void AntisymmetricExpFit::fit_exponents(int c) {
const int N = 2.0*ESolver.eigenvectors().rows();
int i,j;
// --- Construct the eigenpolynomial
// --- from the eigenvalues
// ---------------------------------
RealVector P(N); // eigen polynomial coefficients
if(provable) {
// --- PA = P_q(x) - x^{N/2} P_q(x^{-1})
// --- P = PA/(1-x)
RealVector PA(N+1); // eigen polynomial coefficients
PA(N/2) = 0.0;
PA.topRows(N/2) = ESolver.eigenvectors().col(c).real();
PA.bottomRows(N/2) = -ESolver.eigenvectors().col(c).real().reverse();
P(0) = PA(0);
for(i = 1; i<P.size(); ++i) {
P(i) = P(i-1) + PA(i);
}
} else {
P.topRows(N/2) = ESolver.eigenvectors().col(c).real();
P.bottomRows(N/2) = ESolver.eigenvectors().col(c).real().reverse();
}
RealVector Q; // eigenpolynomial with changed variable
RealVector Z; // scaled positions of zeroes
long double alpha; // scaling factor of polynomial
long double s; // scale
k.resize(c);
i = 0;
alpha = 2.0/3.0;
if(P.size() > 64) {
while(i < c && alpha > 1e-8) {
change_variable(P,alpha,Q);
j = Q.size()-1;
s = pow(2.0,j);
while(j && fabs(Q(j))/s < 1e-18) {
s /= 2.0;
--j;
}
Q.conservativeResize(j+1);
find_zeroes(Q, -0.5, 0.5, Z);
for(j = Z.size()-1; j>=0; --j) {
s = 1.0 - alpha*(1.0 -Z(j));
k(i) = s;
++i;
}
alpha /= 3.0;
}
} else {
find_zeroes(P, 0.0, 1.0, k);
}
}