Tridiagonalization, Lanczos algorithm

In its original form, the Lanczos algorithm was intended as a tridiagonalization method. When used as such, the iterations are continued until termination. The algorithm is most easily described by assuming the tridiagonal result, and by specifying that the basis set should be orthonormal ... 

Once a tridiagonal representation of M is obtained, one can use standard methods to diagonalize the tridiagonal matrix and obtain the eigenvalues (in this particular aspect, the Lanczos algorithm, as commonly used in the literature, differs from the other memory function methods where the Green s... 

We shall set N be the dimension of the finite basis subset used to represent f and v. The calculation can be performed with great efficiency using an iterative algorithm, such as the Lanczos algorithm, that transforms r into a tridiagonalized form. A continued fraction expansion is then obtained ... 

The calculation of the molecular eigenstates with the MVCM model, necessary in traditional time-independent methods, can prove to be very cumbersome or even unfeasible. However, time-independent effective solutions, practicable for reduced-dimensionality models (in practice when the number of relevant normal coordinates is less than 10), may be obtained by taking advantage of the Lanczos iterative tridiagonalization of the Hamiltonian matrix [130]. The Lanczos algorithm proves to be very suitable for the computation of low-resolution spectra however, its effectiveness is better highlighted in a time-dependent framework. In fact, it can be easily realized that Lanczos states are only sequentially coupled, and it is therefore clear that only a limited number of states is necessary to describe short-time dynamics since the latter is the only relevant information for low-resolution spectra (see Chapter 10). 

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