On the global convergence of the block Jacobi method for the positive definite generalized eigenvalue problem

Hari, Vjeran

doi:10.1007/s10092-021-00415-8

Cited by 7 publications

(8 citation statements)

References 43 publications

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“…There are several Jacobi methods for solving PGEP, such as the complex Falk-Langemeyer (FL) method [15], Hari-Zimmermann (HZ) method [8] and Cholesky-Jacobi (CJ) method [12]. Preliminary numerical tests indicate that they all have excellent accuracy properties.…”

Section: An Application To a Jacobi Methods For Pgepmentioning

confidence: 99%

On the convergence of complex Jacobi methods

Hari

Kovač

2019

Linear and Multilinear Algebra

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In this paper we prove the global convergence of the complex Jacobi method for Hermitian matrices for a large class of generalized serial pivot strategies. For a given Hermitian matrix A of order n we find a constant γ < 1 depending on n, such that S(A ′ ) ≤ γS(A), where A ′ is obtained from A by applying one or more cycles of the Jacobi method and S(·) stands for the off-norm. Using the theory of complex Jacobi operators, the result is generalized so it can be used for proving convergence of more general Jacobi-type processes. In particular, we use it to prove the global convergence of Cholesky-Jacobi method for solving the positive definite generalized eigenvalue problem.

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Section: An Application To a Jacobi Methods For Pgepmentioning

confidence: 99%

On the convergence of complex Jacobi methods

Hari

Kovač

2019

Linear and Multilinear Algebra

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“…There are different approaches to solving the generalized eigenvalue problems. One group is Jacobi-type sequential algorithms which come from the Falk-Langemeyer method [9] and the Hari-Zimmermann algorithm [17]. The other group of algorithms is based on the generalization of the Kogbetliantz method [24].…”

Section: Gepmentioning

confidence: 99%

One-shot Distributed Algorithm for Generalized Eigenvalue Problem

Lv¹,

He²,

Huang³

et al. 2020

Preprint

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Nowadays, more and more datasets are stored in a distributed way for the sake of memory storage or data privacy. The generalized eigenvalue problem (GEP) plays a vital role in a large family of high-dimensional statistical models. However, the existing distributed method for eigenvalue decomposition cannot be applied in GEP for the divergence of the empirical covariance matrix.Here we propose a general distributed GEP framework with one-shot communication for GEP. If the symmetric data covariance has repeated eigenvalues, e.g., in canonical component analysis, we further modify the method for better convergence. The theoretical analysis on approximation error is conducted and the relation to the divergence of the data covariance, the eigenvalues of the empirical data covariance, and the number of local servers is analyzed. Numerical experiments also show the effectiveness of the proposed algorithms.

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“…This method is a slight modification (at least from the mathematical point of view) of the real method presented in [21]. The complex transformations, for the two-sided method, have been derived in the PhD thesis of Vjeran Hari [12]. The software distribution † of all phases presented in this paper is written mostly in Fortran, with some auxiliary parts in C, while the parallelization relies on the OpenMP constructs.…”

Section: And Tmentioning

confidence: 99%

The LAPW method with eigendecomposition based on the Hari--Zimmermann generalized hyperbolic SVD

Singer,

Di Napoli,

Novaković

et al. 2019

Preprint

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In this paper we propose an accurate, highly parallel algorithm for the generalized eigendecomposition of a matrix pair (H, S), given in a factored form (F * JF, G * G). Matrices H and S are generally complex and Hermitian, and S is positive definite. These type of matrices emerge from the representation of the Hamiltonian of a quantum mechanical system in terms of an overcomplete set of basis functions. This expansion is part of a class of models within the broad field of Density Functional Theory, which is considered the golden standard in Condensed Matter Physics. The overall algorithm consists of four phases, the second and the fourth being optional, where the two last phases are computation of the generalized hyperbolic SVD of a complex matrix pair (F, G), according to a given matrix J defining the hyperbolic scalar product. If J = I, then these two phases compute the GSVD in parallel very accurately and efficiently.

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On the global convergence of the block Jacobi method for the positive definite generalized eigenvalue problem

Cited by 7 publications

References 43 publications

On the convergence of complex Jacobi methods

On the convergence of complex Jacobi methods

One-shot Distributed Algorithm for Generalized Eigenvalue Problem

The LAPW method with eigendecomposition based on the Hari--Zimmermann generalized hyperbolic SVD

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