The Lanczos algorithm with selective orthogonalization

Parlett, Beresford Ν.; Scott, David

doi:10.1090/s0025-5718-1979-0514820-3

Cited by 329 publications

(75 citation statements)

References 6 publications

(1 reference statement)

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“…[23,24,25,26,27] An advantage of this method is that it automatically includes both rotational and vibrational excitations. However, applying this technique to our system brings technical problems.…”

Section: Energies Of Rovibronic Statesmentioning

confidence: 99%

The effect of cooperative Jahn–Teller interactions on C₆₀⁻anions

Moujaes

Dunn²

2010

J. Phys.: Condens. Matter

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Abstract. Singly-charged buckminster fullerene anions, C ions in a solid through a co-operative JT effect, they will be subject to an additional interaction. There are a number of different mechanisms that can cause this interaction. However, in the molecular field approximation, all can be modelled phenomenologically in terms of a symmetry-lowering interaction written in terms of a linear combination of electronic operators for the h modes involved in the intramolecular JT effect. We will consider the combined effect of this distortion and the intramolecular JT effect. We will analyse the lowest adiabatic potential energy surface, and calculate the energies of the resultant vibronic states. The results are shown to have a complicated dependence on the particular combination of h modes chosen, and the energies of the resultant vibronic states can not easily be deduced from the form of the potential alone.

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Section: Energies Of Rovibronic Statesmentioning

confidence: 99%

The effect of cooperative Jahn–Teller interactions on C₆₀⁻anions

Moujaes

Dunn²

2010

J. Phys.: Condens. Matter

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“…The explicit enforcement of orthogonality implies that the cost increases as the square of the number of eigenfunctions to be determined. Nevertheless, techniques such as the block-Lanczos method with selective orthogonalization 26 can control the linear independence of the Lanczos vectors without the cost of full orthogonalization. This technique has been employed by Braun et al 24 to compute eigenvalues of the examples presented here and some other with very high accuracy.…”

Section: B Higher-energy Eigenfunctionsmentioning

confidence: 99%

Spatially-Discretized High-Temperature Approximations and Their O(n) Implementation on a Grid

Predescu¹

2006

J. Theor. Comput. Chem.

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We consider the problem of performing imaginary-time propagation of wavefunctions on a grid. We demonstrate that spatially-continuous high-temperature approximations can be discretized in such a way that their convergence order is preserved. Requirements of minimal computational work and reutilization of data then uniquely determine the optimal grid, quadrature technique, and propagation method. It is shown that the optimal propagation technique is O(N ) with respect to the grid size. The grid technique is utilized to compare the Monte Carlo efficiency of the Trotter-Suzuki approximation against a recently introduced fourth-order high-temperature approximation, while circumventing the issue of statistical noise, which usually prevents such comparisons from being carried out. We document the appearance of a systematic bias in the Monte Carlo estimators that involve temperature differentiation of the density matrix, bias that is due to the dependence of the eigenvalues on the inverse temperature. This bias is negotiated more successfully by the short-time approximations having higher convergence order, leading to non-trivial computational savings.

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“…He also showed that both partial reorthogonalization and selective orthogonalization introduced by Parlett and Scott [21] or himself [24] are semiorthogonalization strategies. According to the result established by Simon, an adapted Gram-Schmidt orthogonalization routine has been included in the Lanczos step in order to keep the semiorthogonality among the Lanczos vectors during the process.…”

Section: Fischermentioning

confidence: 99%

Numerical solution of eigenvalue problems by means of a wavelet-based Lanczos decomposition

Fischer

2000

Int. J. Quant. Chem.

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ABSTRACT:The simple Lanczos process is very efficient for finding a few extreme eigenvalues of a large symmetric matrix. The main task in each iteration step consists in evaluating a matrix-vector product. It is shown how to apply a fast wavelet-based product in order to speed up computations. Some numerical results are given for three different monodimensional cases: the harmonic oscillator case, the hydrogenlike atoms, and a problem with a pseudo-double-well potential.

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The Lanczos algorithm with selective orthogonalization

Cited by 329 publications

References 6 publications

The effect of cooperative Jahn–Teller interactions on C₆₀⁻anions

The effect of cooperative Jahn–Teller interactions on C₆₀⁻anions

Spatially-Discretized High-Temperature Approximations and Their O(n) Implementation on a Grid

Numerical solution of eigenvalue problems by means of a wavelet-based Lanczos decomposition

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The Lanczos algorithm with selective orthogonalization

Cited by 329 publications

References 6 publications

The effect of cooperative Jahn–Teller interactions on C60−anions

The effect of cooperative Jahn–Teller interactions on C60−anions

Spatially-Discretized High-Temperature Approximations and Their O(n) Implementation on a Grid

Numerical solution of eigenvalue problems by means of a wavelet-based Lanczos decomposition

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Product

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The effect of cooperative Jahn–Teller interactions on C₆₀⁻anions

The effect of cooperative Jahn–Teller interactions on C₆₀⁻anions