Accuracy of the Kogbetliantz method for scaled diagonally dominant triangular matrices

Matejaš, Josip; Hari, Vjeran

doi:10.1016/j.amc.2010.09.020

Cited by 9 publications

(6 citation statements)

References 34 publications

(57 reference statements)

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“…For matrix eigenvalue and singular value problems, Jacobi-type methods are known for their accuracy (see [5,6,8,9,30,[42][43][44]54,55]), inherent parallelism [7,15,39,40,50,52,53] and efficiency [3,10,11,28]. The most promising way to further enhance these characteristics is to modify them to become BLAS 3 algorithms (see [12,28,31,33,34,52,53]).…”

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confidence: 99%

Convergence to diagonal form of block Jacobi-type methods

Hari

2014

Numer. Math.

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We provide sufficient conditions for the general sequential block Jacobitype method to converge to the diagonal form for cyclic pivot strategies which are weakly equivalent to the column-cyclic strategy. Given a block-matrix partition (A i j ) of a square matrix A, the paper analyzes the iterative process of the formwhere P (k) and Q (k) are elementary block matrices which differ from the identity matrix in four blocks, two diagonal and the two corresponding off-diagonal blocks. In our analysis of convergence a promising new tool is used, namely, the theory of block Jacobi operators. Typical applications lie in proving the global convergence of block Jacobi-type methods for solving standard and generalized eigenvalue and singular value problems.

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confidence: 99%

Convergence to diagonal form of block Jacobi-type methods

Hari

2014

Numer. Math.

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“…In the Algol routines, and subsequently the Fortran routines of EISPACK, matrix elements were referenced Downloaded 11/30/18 to 130. 88 by row, thus causing great inefficiencies in the Fortran EISPACK software on modern cache-based computer systems.…”

Section: Methodsmentioning

confidence: 99%

“…After the completion of this initial interprocess communication, each process independently Downloaded 11/30/18 to 130. 88 computes the matrix-vector multiplication with the local submatrix. Finally, each process computes the local part of the output vector by gathering and accumulating the partial results from the other processes in the same row of the process grid.…”

Section: Level 25 Blas Implementationmentioning

confidence: 99%

“…As noted earlier, two-sided Jacobi preserves the triangular structure when used with an appropriate cyclic ordering. Hari and Mateja\v s [65,88,89] use the QRP and LQ preprocessing to generate triangular matrices. They prove high relative accuracy results for the two-sided Jacobi method on such triangular matrices, and utilize a parallel ordering due to Sameh [103] that preserves the triangular structure.…”

Section: Dplasma Implementation For Distributed Memorymentioning

confidence: 99%

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The Singular Value Decomposition: Anatomy of Optimizing an Algorithm for Extreme Scale

Dongarra¹,

Gates²,

Haidar³

et al. 2018

SIAM Rev.

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The computation of the singular value decomposition, or SVD, has a long history with many improvements over the years, both in its implementations and algorithmically. Here, we survey the evolution of SVD algorithms for dense matrices, discussing the motivation and performance impacts of changes. There are two main branches of dense SVD methods: bidiagonalization and Jacobi. Bidiagonalization methods started with the implementation by Golub and Reinsch in Algol60, which was subsequently ported to Fortran in the EIS-PACK library, and was later more efficiently implemented in the LINPACK library, targeting contemporary vector machines. To address cache-based memory hierarchies, the SVD algorithm was reformulated to use Level 3 BLAS in the LAPACK library. To address new architectures, ScaLAPACK was introduced to take advantage of distributed computing, and MAGMA was developed for accelerators such as GPUs. Algorithmically, the divide and conquer and MRRR algorithms were developed to reduce the number of operations. Still, these methods remained memory bound, so two-stage algorithms were developed to reduce memory operations and increase the computational intensity, with efficient implementations in PLASMA, DPLASMA, and MAGMA. Jacobi methods started with the two-sided method of Kogbetliantz and the one-sided method of Hestenes. They have likewise had many developments, including parallel and block versions and preconditioning to improve convergence. In this paper, we investigate the impact of these changes by testing various historical and current implementations on a common, modern multicore machine and a distributed computing platform. We show that algorithmic and implementation improvements have increased the speed of the SVD by several orders of magnitude, while using up to 40 times less energy.

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“…Depending on its structure, G might have to be preprocessed into a form more suitable for a simple and accurate [13,14] computation of the transformation parameters than that of a general square 2 × 2 matrix. For the ordinary Kogbetliantz algorithm it is preferred that the pivot matrices throughout the process remain (upper or lower) triangular [4,5] under a cyclic pivot strategy: a serial (e.g., the row or column cyclic, with G 0 triangular) or a parallel one (e.g., the modulus strategy, with G 0 preprocessed into the butterfly form [6]).…”

Section: Computing the Hsvd Of Matrices Of Order Twomentioning

confidence: 99%

A Kogbetliantz-type algorithm for the hyperbolic SVD

Novaković,

Singer

2020

Preprint

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In this paper a two-sided, parallel Kogbetliantz-type algorithm for the hyperbolic singular value decomposition (HSVD) of real and complex square matrices is developed, with a single assumption that the input matrix, of order n, admits such a decomposition into the product of a unitary, a non-negative diagonal, and a J-unitary matrix, where J is a given diagonal matrix of positive and negative signs. When J = ±I, the proposed algorithm computes the ordinary SVD.The paper's most important contribution-a derivation of formulas for the HSVD of 2 × 2 matrices-is presented first, followed by the details of their implementation in floating-point arithmetic. Next, the effects of the hyperbolic transformations on the columns of the iteration matrix are discussed. These effects then guide a redesign of the dynamic pivot ordering, being already a well-established pivot strategy for the ordinary Kogbetliantz algorithm, for the general, n × n HSVD. A heuristic but sound convergence criterion is then proposed, which contributes to high accuracy demonstrated in the numerical testing results. Such a J-Kogbetliantz algorithm as presented here is intrinsically slow, but is nevertheless usable for matrices of small orders. Keywords hyperbolic singular value decomposition • Kogbetliantz algorithm • Hermitian eigenproblem • OpenMP multicore parallelization Mathematics Subject Classification (2010) 65F15 • 65Y05 • 15A18 This work has been supported in part by Croatian Science Foundation under the project IP-2014-09-3670 "Matrix Factorizations and Block Diagonalization Algorithms" (MFBDA).

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Accuracy of the Kogbetliantz method for scaled diagonally dominant triangular matrices

Cited by 9 publications

References 34 publications

Convergence to diagonal form of block Jacobi-type methods

Convergence to diagonal form of block Jacobi-type methods

The Singular Value Decomposition: Anatomy of Optimizing an Algorithm for Extreme Scale

A Kogbetliantz-type algorithm for the hyperbolic SVD

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