Lanczos recursion

The vectors generated by the Lanczos recursion differ from the Krylov vectors in that the former are mutually orthogonal and properly normalized, at least in exact arithmetic. In fact, the Lanczos vectors can be considered as the Gram-Schmidt orthogonalized Krylov vectors.27 Because the orthogonalization is performed implicitly along the recursion, the numerical costs are minimal. 

In fact, the Lanczos reduction was originally proposed as a tridiagonalization scheme, predating the Givens and Householder methods. Unlike the latter methods, however, the Lanczos method is recursive. This means that the dimensionality of the T matrix is determined by the number of steps of the Lanczos recursion (K), which is usually much smaller than the dimensionality of the Hamiltonian matrix (N) in real calculations. 

A commonly used approach for computing the transition amplitudes is to approximate the propagator in the Krylov subspace, in a similar spirit to the time-dependent wave packet approach.7 For example, the Lanczos-based QMR has been used for U(H) = (E — H)-1 when calculating S-matrix elements from an initial channel (%m )-93 97 The transition amplitudes to all final channels (Xm) can be computed from the cross-correlation functions, namely their overlaps with the recurring vectors. Since the initial vector is given by xmo only a column of the S-matrix can be obtained from a single Lanczos recursion. 

The two vectors in Eq. [43] are used to initiate two Lanczos recursions that yield not only converged eigenvalues but also the residues R j . In particular, Wyatt and Scott have shown that these residues are simply the squared first elements of the Lanczos eigenvectors in a K-step recursion 46... 

To illustrate the principles of SLP, we note that a Lanczos recursion initiated by an arbitrary vector qj can, in exact arithmetic, yield not only the eigenvalues of H, but also overlaps of prespecified vectors with eigenvectors, as shown below ... 

The filtering, namely the construction of energy local bases, can also be carried out using the Lanczos recursion or similar recursive methods. However, filtered vectors at E/ can only be obtained using the Green filter ... 

Unfortunately, the symmetry adaptation scheme described above for the Chebyshev recursion cannot be applied directly to the Lanczos recursion. 

Because of round-off errors, symmetry contamination is often present even when the initial vector is properly symmetrized. To circumvent this problem, an effective scheme to reinforce the symmetry at every Lanczos recursion step has been proposed independently by Chen and Guo100 and by Wang and Carrington.195 Specifically, the Lanczos recursion is executed with symmetry-adapted vectors, but the matrix-vector multiplication is performed at every Lanczos step with the unsymmetrized vector. In other words, the symmetrized vectors are combined just before the operation Hq, and the resultant vector is symmetrized using the projection operators ... 

The orthogonal characteristic polynomials or eigenpolynomials Qn(u) play one of the central roles in spectral analysis since they form a basis due to the completeness relation (163). They can be computed either via the Lanczos recursion (84) or from the power series representation (114). The latter method generates the expansion coefficients q , -r through the recursion (117). Alternatively, these coefficients can be deduced from the Lanczos recursion (97) for the rth derivative Q /r(0) since we have qni r = (l/r )Q r(0) as in Eq. (122). The polynomial set Qn(u) is the basis comprised of scalar functions in the Lanczos vector space C from Eq. (135). In Eq. (135), the definition (142) of the inner product implies that the polynomials Qn(u) and Qm(u) are orthogonal to each other (for n= m) with respect to the complex weight function dk, as per (166). The completeness (163) of the set Q (u) enables expansion of every function f(u) e C in a series in terms of the... 

Alternatively, band peak shifts can be computed from at) initio absorption spectra for SO2 isotopologues [28], which is the method I ve used here. The ah initio spectra were computed by time domain methods (Lanczos recursion and Cheby-shev propagation) using a new potential energy surface for SO2. The ah initio... 

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