Generalizations of Davidson’s Method for Computing Eigenvalues of Sparse Symmetric Matrices

Morgan, Ronald B.; Scott, David

doi:10.1137/0907054

Cited by 146 publications

(107 citation statements)

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“…This dynamic criterion comes from the Newton methods and the use in Jacobi-Davidson is suggested in [Fokkema et al, 1999], and tested in [Genseberger, 2010;. It is well-know that in the first iterations of the Davidson-type methods the extraction method usually produces poor eigenpair approximations and the target τ may be a relatively closer approximation to an exact eigenvalue [Morgan and Scott, 1986]. Therefore until the residual norm associated to the selected eigenpair reaches a threshold value so-called fix (that can be set by EPSJDSetFix), the correction equation is solved with θ = τ [Fokkema et al., 1999].…”

Section: ])mentioning

confidence: 99%

“…The main characteristic of this class of methods is that they expand the subspace with a so-called correction vector t, which is computed from the residual vector r associated to the most wanted eigenpair with the aim of improving it further. This new vector can be computed by simply preconditioning the residual, (6.13) as in the GD method [Davidson, 1975;Morgan and Scott, 1986], or by (approximately) solving the correction equation, (6.14) as in the JD method [Sleijpen and van der Vorst, 2000]. As in (6.13), a preconditioner K can also be introduced in (6.14).…”

Section: Subspace Expansionmentioning

confidence: 99%

“…Subsequent developments [Morgan and Scott, 1986;Natarajan and Vanderbilt, 1989;Morgan, 1990] tried to understand (2.16) as an iterative linear equation solver step of the system…”

Section: Subspace Expansionsmentioning

confidence: 99%

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Trobec¹,

Kosec²

2015

Parallel Scientific Computing

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ResumenEl problema de valores propios (también llamado de autovalores) está presente en multitud dé areas científicas a través de, por ejemplo, la resolución de ecuaciones en derivadas parciales, reducción de modelos y cálculo de funciones matriciales, entre otras muchas aplicaciones. Si los problemas son de dimensión moderada (menor a 10 6 ), pueden ser abordados mediante los llamados métodos directos, como el algoritmo iterativo QR o el método de divide y vencerás. Sin embargo, si el problema es de gran dimensión y sólo se requieren unas pocas soluciones, comparado con el tamaño del problema, los métodos iterativos pueden resultar más eficientes. Además, los métodos iterativos pueden ofrecer mejor rendimiento en arquitecturas de altas prestaciones, como las plataformas paralelas de memoria distribuida, en las que existe un cierto número de nodos computacionales con espacio de memoria propio y sólo pueden compartir información y sincronizarse mediante el paso de mensajes.Esta tesis aborda la implementación de métodos de tipo Davidson, destacando Generalized Davidson y Jacobi-Davidson, una clase de métodos iterativos que pueden ser competitivos en casos especialmente difíciles como calcular valores propios en el interior del espectro o cuando la factorización de matrices es prohibitiva o ineficiente, y sólo es posible una factorización aproximada. La implementación se desarrolla en SLEPc (Scalable Library for Eigenvalue Problem Computations), librería de software libre para la resolución de problemas de gran tamaño de valores propios, problemas cuadráticos de valores propios y problemas de valores singulares, entre otros.Como resultado de esta tesis, SLEPc incorpora una implementación de Generalized Davidson y Jacobi-Davidson para problemas estándares y generalizados, tanto hermitianos como no hermitianos, característica a destacar ya que los problemas no hermitianos no están soportados en otras librerías libres y paralelas con métodos de Davidson. Además de incorporar mejoras en la convergencia de los métodos, como la extracción de pares propios mediante el método de Rayleigh-Ritz armónico, se han realizado otras optimizaciones para mejorar el rendimiento, como evitar la aritmética compleja cuando las matrices son reales (aunque los pares propios puedan tener parte imaginaria) o agrupar las operaciones a bloques para aprovechar la memoria caché. Además se ha presentado un método nuevo para expandir el subespacio, llamado GD2, que ofrece mejores resultados en comparación con Generalized Davidson cuando el precondicionador está muy alejado del ideal.La estabilidad numérica y el rendimiento computacional de la implementación se han evaluado mediante una batería de problemas procedentes de aplicaciones reales, y comparado con otras librerías como PRIMME o Anasazi.Finalmente se presenta la integración de la implementación en dos aplicaciones científicas. Por un lado, Jacobi-Davidson ha mejorado las prestaciones de los cálculos de valores propios y los estudios multiparamétricos que aparecen en GENE, un código que re...

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Section: ])mentioning

confidence: 99%

Section: Subspace Expansionmentioning

confidence: 99%

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Parallel Implementation

Trobec¹,

Kosec²

2015

Parallel Scientific Computing

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“…For instance, Davidson [7] suggested K = diag(A) -{) diag(B) (see also [19,8]). In view of our observations in §3 we expect to create better preconditioners by taking the projections into account.…”

Section: Projecting Preconditionersmentioning

confidence: 99%

“…R.EMARK 7.2. In the Davidson methods [5,7,18,19] the search subspace is expanded by the vector K-1 r, the·vector that appears in the first step of the computation of r' (cf. TH.…”

Section: Projecting Preconditionersmentioning

confidence: 99%

Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems

et al. 1996

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Abstract.In this paper we will show how the Jacobi-Davidson iterative method can be used to solve generalized eigenproblems. Similar ideas as for the standard eigenproblem are used, but the projections, that are required to reduce the given problem to a small manageable size, need more attention. We show that by proper choices for the projection operators quadratic convergence can be achieved. The advantage of our approach is that none of the involved operators needs to be inverted. It turns out that similar projections can be used for the iterative approximation of selected eigenvalues and eigenvectors of polynomial eigenvalue equations. This approach has already been used with great success for the solution of quadratic eigenproblems associated with acoustic problems.

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Vibrational spectrum of Li3 first-excited electronic doublet state: Geometric-phase effects and statistical analysis

Varandas

1999

Int. J. Quant. Chem.

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We present J s 0 calculations of all bound and pseudobound vibrational states of Li in its first-excited electronic doublet state by using a realistic double 3 many-body expansion potential-energy surface and a minimum-residual filter diagonalization technique. The action of the system Hamiltonian on the wave function was evaluated by the spectral transform method in hyperspherical coordinates. Calculations of the vibrational spectra were carried out both without consideration and with consideration of geometric-phase effects. Dynamic Jahn᎐Teller and geometric-phase effects are found to play a significant role, while the calculated fundamental symmetric stretching frequency is larger by 8.3% than its reported experimental value of 326 cm y1 . From the neighbor-spacing distributions of the levels, it is observed that the title vibrational spectrum is quasiregular in the short range and quasi-irregular in the long range. By the ⌬ standard defined in this article, it is found that the spectra are more 2 nonuniform than those of the ''trough'' states for the ground electronic state.

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Generalizations of Davidson’s Method for Computing Eigenvalues of Sparse Symmetric Matrices

Cited by 146 publications

References 6 publications

Parallel Implementation

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Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems

Vibrational spectrum of Li3 first-excited electronic doublet state: Geometric-phase effects and statistical analysis

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