void CRationalApproximation::compute_shifts_weights_const()
{
float64_t PI=M_PI;
// eigenvalues are always real in this case
float64_t max_eig=m_eigen_solver->get_max_eigenvalue();
float64_t min_eig=m_eigen_solver->get_min_eigenvalue();
// we need $(\frac{\lambda_{M}}{\lambda_{m}})^{frac{1}{4}}$ and
// $(\lambda_{M}*\lambda_{m})^{frac{1}{4}}$ for the rest of the
// calculation where $lambda_{M}$ and $\lambda_{m} are the maximum
// and minimum eigenvalue respectively
float64_t m_div=CMath::pow(max_eig/min_eig, 0.25);
float64_t m_mult=CMath::pow(max_eig*min_eig, 0.25);
// k=$\frac{(\frac{\lambda_{M}}{\lambda_{m}})^\frac{1}{4}-1}
// {(\frac{\lambda_{M}}{\lambda_{m}})^\frac{1}{4}+1}$
float64_t k=(m_div-1)/(m_div+1);
float64_t L=-CMath::log(k)/PI;
// compute K and K'
float64_t K=0.0, Kp=0.0;
CJacobiEllipticFunctions::ellipKKp(L, K, Kp);
// compute constant multiplier
m_constant_multiplier=-8*K*m_mult/(k*PI*m_num_shifts);
// compute Jacobi elliptic functions sn, cn, dn and compute shifts, weights
// using conformal mapping of the quadrature rule for discretization of the
// contour integral
float64_t m=CMath::sq(k);
for (index_t i=0; i<m_num_shifts; ++i)
{
complex64_t t=complex64_t(0.0, 0.5*Kp)-K+(0.5+i)*2*K/m_num_shifts;
complex64_t sn, cn, dn;
CJacobiEllipticFunctions::ellipJC(t, m, sn, cn, dn);
complex64_t w=m_mult*(1.0+k*sn)/(1.0-k*sn);
complex64_t dzdt=cn*dn/CMath::sq(1.0/k-sn);
// compute shifts and weights as per Hale et. al. (2008) [2]
m_shifts[i]=CMath::sq(w);
switch (m_function_type)
{
case OF_SQRT:
m_weights[i]=dzdt;
break;
case OF_LOG:
m_weights[i]=2.0*CMath::log(w)*dzdt/w;
break;
case OF_POLY:
SG_NOTIMPLEMENTED
m_weights[i]=complex64_t(0.0);
break;
case OF_UNDEFINED:
SG_WARNING("Operator function is undefined!\n")
m_weights[i]=complex64_t(0.0);
break;
}
}
}
void SGVector<complex64_t>::zero()
{
if (vector && vlen)
set_const(complex64_t(0.0));
}
