The Lanczos method

Definition 90 The finite dimensional subspace Ki, of the Euclidean space En, spanned by the vectors c, Ac,., .A c, is called a Krylov space  

Some formulae EUid rules from matrix algebra 

The Lanczos method is based on generating the orthonormal basis in Krylov space Ki =span c, Ac, A c by applying the Gram-Schmidt orthogonaliza-tion process, described in Appendix A. In matrix notations this approach is associated with the reduction of the symmetric matrix A to a tridiagonal matrix and also with the special properties of T/,. This reduction (called also QT decomposition) is described by the formula 

We consider first the tridiagonalization process using Krylov space of dimension N Kl) =span c, Ac, A c. In this case, according to the definition, the matrix Q v is orthogonal, Therefore, the reduction formula (E.21) 

For example, equating the j-th column of each side of (E.23), we obtain a recursive formula, 

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McCormack D A, Kroes G J and Neuhauser D 1998 Resonance affected scattering Comparison of two hybrid methods involving filter diagonalization and the Lanczos method J. Chem. Phys. 109 5177... 

Krylov Approximation of the Matrix Exponential The iterative approximation of the matrix exponential based on Krylov subspaces (via the Lanczos method) has been studied in different contexts [12, 19, 7]. After the iterative construction of the Krylov basis ui,..., Vn j the matrix exponential is approximated by using the representation A oi H(g) in this basis ... 

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. 

Times in Wave Packet Calculations on Scattering with Resonances A Hybrid Approach Involving the Lanczos Method. 

Resonance Affected Scattering Comparison of Two Hybrid Methods Involving Filter Diagonalization and the Lanczos Method. 

Filter Diagonalization Methods with the Lanczos Method for Calculating Vibrational Energy Levels. 

After the SALCs of the Slater determinants have been obtained, the Lanczos method is employed for diagonalization. 

The Lanczos method has been widely applied to the dynamics in Hubbard and Heisenberg model Hamiltonians[39]. The spectral intensity for an operator O is given by... 

The wavepacket dynamics were carried out using the Lanczos method with real wavepackets [121] employing Jacobi coordinates to describe the relative positions of the three nuclei in the body fixed plane. As mentioned the novel results... 

Figure 1. A schematic pictorial representation of the Lanczos method. The eigenvectors of M (here supposed to be Hermitian) are indicatol by m,), and the corresponding eigenvalues are reported oa the real axis. The eigenvectors of the triihagonal matrix generated by xo) are denoted by (,) the eigenvalues selected out by the test state are also indicated for convenience.

We can summarize the Lanczos method by writing down the hierarchy of the best linear combinations generated starting from xo) ... 

As opposed to the Lanczos method, in which the diagonali2 tion of (3.24) is performed, the recursion method focuses on the construction of the diagonal Green s-function matrix element... 

The advantages of this kind of formulation stand out not only in terms of elegance and beauty (the moment method, the Lanczos method, and the recursion method are relevant but particular cases of the memory function equations), but also in the possibility of providing insight into a number of problems, such as the asymptotic behavior of continued fraction parameters and their relationship with moments, the possible inclusion of nonlinear effects, the introduction of the concept of random forces, and so on. 

To verify the equivalence of the memory function approach to the recursion method or the Lanczos method, it is sufficient to note that the state l/ i+i) defined via Eq. (3.39d) coincides with the state... 

It is easily seen by inspection that the biorthogonal basis set definition (3.55) cmnddes with the definifion (3.18) ven in the discussion of the Lanczos method. We recall that the dynamics of operators (liouville equations) or probabilities (Fokker-Planck equations) have a mathematical structure similar to Eq. (3.29) and can thus be treated with the same techniques (see, e.g., Chapter 1) once an appropriate generalization of a scalar product is performed. For instance, this same formalism has been successfully adopted to model phonon thermal baths and to include, in principle, anharmonicity effects in the interesting aspects of lattice dynamics and atom-solid collisions. ... 

The Lanczos Method. The use of classical moment theory involves large determinants, thereby implying delicate problems of numerical stability. This instability arises from very severe cancellations in the determinants, which in turn come from the properties of the moments themselves. Whitehead and Watt bypassed this problem by using the Lanczos al-... 

It appears clear from Chapters I, III, and IV that the Mori theory is the major theoretical tool behind the algorithm illustrated in Section II, which derives the expansion parameters X, and from the moments s . This theory also affords us with a second straightforward way of determining these parameters that of deriving them directly from the biorthogonal basis set of states fi) and j/-) (Eqs. 2.15). As discussed at length in Chapter IV, this is an especially stable way of building up X, and A. The Lanczos method fol-... 

The Lanczos method is based on generating the orthonormal basis in Krylov... 

Note that this technique is equivalent to the Lanczos method, described above. Certainly, after making s substeps, we arrive at a new residual on the (n +1) iteration... 

Note that the most expensive part of the numerical calculations is the determination of the matrix Q using the Lanczos method. This matrix depends only on the coefficients of the matrix and the vector fic. Therefore, due to the fact that matrix does not depend on frequency, we should apply this decomposition only once for all frequency ranges (if also vector c does not depend on frequency, which is typical for many practical problems). The calculation of the inverse of the matrix (T+iujfj,(T) is computationally a much simpler problem, because T is a tri-diagonal matrix, and jj, and diagonal matrices. As a result, one application of SLDM allows us to solve forward problems for the entire frequency range. That is why SLDM increases the speed of solution of the forward problem by an order for multifrequency data. This is the main advantage of this method over any other approach. 

Note that we can use L steps of the Lanczos method to generate matrices Qr, and Ti, L < N, and to introduce a natural approximation to vector d as... 

Fig. 17 Cluster size dependence of the overlap integral between the ground state wave function GS) obtained by the Lanczos method and the trial wave function [6]. In the variational wave function the...

Calculating the matrix elements of the Hamiltonian in this basis set gives a sparse, real, and symmetric M(N) x M(N) matrix at order N. By systematically increasing the order N, one obtained the lowest two eigenvalues at different basis lengths M(N). For example, M(N) = 946 and 20,336 at N = 20 and 60, respectively [11]. The symmetric matrix is represented in a sparse row-wise format [140] and then reordered [141] before triangularizations. The Lanczos method [142] of block-renormalization procedure was employed. 

This diagonalization can be performed by explicit construction of the matrix Haf ) which is then diagonalized by standard methods when the basis set is not too large. For the case of large systems and/or large basis sets, we will prefer iterative techniques, like the Lanczos method [74,143-145], which avoid the explicit construction of the Kohn-Sham matrix it is sufficient in these methods to have a procedure to apply (successively) the Kohn-Sham matrix on vectors cf. Only vectors cf need then to be stored. 

The (Lanczos) method was named the tau method because Lanczos used the letter r to represent the error. 

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