Lanczos method

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 ... 

Abstract. We present novel time integration schemes for Newtonian dynamics whose fastest oscillations are nearly harmonic, for constrained Newtonian dynamics including the Car-Parrinello equations of ab initio molecular dynamics, and for mixed quantum-classical molecular dynamics. The methods attain favorable properties by using matrix-function vector products which are computed via Lanczos method. This permits to take longer time steps than in standard integrators. 

We here describe the alternative of approximating <,c(S)b via Lanczos method. The Lanczos process [18, 22] recursively generates an orthonormal basis Qm = [qi,.., qm] of the mth Krylov subspace... 

Equation (4) is a three-term recursion for propagating a wave packet, and, assuming one starts out with some 4>(0) and (r) consistent with Eq. (1), then the iterations of Eq. (4) will generate the correct wave packet. The difficulty, of course, is that the action of the cosine operator in Eq. (4) is of the same difficulty as evaluating the action of the exponential operator in Eq. (1), requiring many evaluations of H on the current wave packet. Gray [8], for example, employed a short iterative Lanczos method [9] to evaluate the cosine operator. However, there is a numerical simplification if the representation of H is real. In this case, if we decompose the wave packet into real and imaginary parts. 

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. 

PIST distinguishes itself from other spectral transform Lanczos methods by using two important innovations. First, the linear equation Eq. [38] is solved by QMR but not to a high degree of accuracy. In practice, the QMR recursion is terminated once a prespecified (and relatively large) tolerance is reached. Consequently, the resulting Lanczos vectors are only approximately filtered. This inexact spectral transform is efficient because many less matrix-vector multiplications are needed, and its deficiencies can subsequently... 

G. CharronandT. Carrington Jr., Molec. Phys., 79,13 (1993). A Fourier-Lanczos Method for... 

Vibrational Eigenstates of Four-Atom Molecules A Parallel Strategy Employing the Implicitly Restarted Lanczos Method. 

New York, 1977, pp. 361-377. The Block Lanczos Method for Computing Eigenvalues. 

Lanczos Method for the Numerical Solution of Large Sparse Generalized Symmetric Eigenvalue Problems. 

Vibrational Eigenfunctions Using a Modified Single Lanczos Method Application to Acetylene (HCCH). 

Lanczos Method for Calculating Energy Levels with Different Symmetries from a Single Sequence of Iterations. 

Resonance Energies, Widths, and Wave Functions Using a Lanczos Method in Real Arithmetic. 

Rovibrational Energy Levels of a Triatomic Molecule with a Simple Lanczos Method. 

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... 

Phatak, A. and De Hoog, F., Exploiting the connection between PLS, Lanczos methods and conjugate gradients alternative proofs of some properties of PLS, J. Chemom., 16, 361-367, 2002. 

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... 

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