Equations for eigenvalues uk via continued fractions

If the set of the couplings {α_n, β_n} is precomputed, it is clear from the preceding chapter, that the LCF is technically more efficient than the PLA. This is because the PLA still needs to generate the Lanczos polynomials {P_K(u), Q_K(u)} whereas LCF does not. The LCF is an accurate, robust and fast processor for computation of shape spectra with an easy way of programming implementations in practice. The modified Newton-Raphson iterative method can be employed to solve F^~(u)=0 for u = u_k. Many quantum-mechanical eigenvalue problems with given potentials can be reduced to diagonalization of triangular matrices. Alternatively, the eigensolutions can be obtained using the corresponding secular determinantal equations with the real variable ω instead of the complex exponential u = exp(-iωτ).