A Note On Parallel Matrix Inversion

Quintana, Enrique S.; Quintana, Gregorio; Sun, Xiaobai; Geijn, Robert vande

doi:10.1137/s1064827598345679

Cited by 79 publications

(60 citation statements)

References 7 publications

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“…We believe the rest of the notation to be intuitive; for further details, see [7,4]. A description of the unblocked version, called from inside the blocked one, can be found in [8]; for simplicity, we hide the application of pivoting during the factorization, but details can be found there as well.…”

Section: Matrix Inversion Via the Gauss-jordan Eliminationmentioning

confidence: 99%

Using graphics processors to accelerate the computation of the matrix inverse

2011

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We study the use of massively parallel architectures for computing a matrix inverse. Two different algorithms are reviewed, the traditional approach based on Gaussian elimination and the Gauss-Jordan elimination alternative, and several high performance implementations are presented and evaluated. The target architecture is a current general-purpose multi-core processor (CPU) connected to a graphics processor (GPU). Numerical experiments show the efficiency attained by the proposed implementations and how the computation of large-scale inverses, which only a few years ago would have required a distributed-memory cluster, take only a few minutes on a hybrid architecture formed by a multi-core CPU and a GPU.

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Section: Matrix Inversion Via the Gauss-jordan Eliminationmentioning

confidence: 99%

Using graphics processors to accelerate the computation of the matrix inverse

2011

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“…It has been shown that a seemingly lesser algorithm can be much more efficient because it allows parallel processing [12][13][14][15] that is absolutely necessary for the use of the maximum computational power of modern processors, where thousands of tasks can be computed at the same time. The time when a personal computer's central processing unit (CPU) was able to compute only one thread at a time is over.…”

Section: Introductionmentioning

confidence: 99%

Implicit numerical multidimensional heat-conduction algorithm parallelization and acceleration on a graphics card

Pohanka¹,

Ondroušková²

2016

Mater. Tehnol.

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Analytical solutions are much less computationally intensive than numerical ones, and moreover, they are more accurate because they do not contain numerical errors; however, they can only describe a small group of simple heat-conduction problems. A numerical simulation of heat conduction is often used as it is able to describe complex problems, but its computational time is much longer, especially for unsteady multidimensional models with temperature-dependent material properties. After a discretization using the implicit scheme, the heat-conduction problem can be described with N non-linear equations, where N is the large number of the elements of the discretized model. This set of equations can be efficiently solved with an iteration of the line-by-line method, based on the heat-flux superposition, although the computational procedure is strictly serial. This means that no parallel computation can be done, which is strictly required when a graphics card is used to accelerate the computation. This paper describes a multidimensional numerical model of unsteady heat conduction solved with the line-by-line method and a modification of this method for a highly parallel computation. An enormous increase in the speed is demonstrated for the modified line-by-line method accelerated on the graphics card, and the durations of the computations for various mesh sizes are compared with the original line-by-line method. Keywords: heat conduction, numerical simulation, multidimensional numerical model algorithm, acceleration, parallelization, graphics card Analiti~ne re{itve so mnogo manj ra~unsko intenzivne kot numeri~ne, poleg tega pa so bolj natan~ne, ker ne vsebujejo numeri~nih napak, vendar pa lahko opisujejo samo majhno skupino enostavnih problemov prevajanja toplote. Numeri~na simulacija prevajanja toplote se pogosto uporablja, ker je sposobna opisati kompleksne probleme, vendar pa je~as izra~una mnogo dalj{i, {e posebno pri nestabilnih ve~dimenzijskih modelih, z lastnostmi materiala, odvisnimi od temperature. Po diskretizaciji z uporabo implicitne sheme je mogo~e problem prevajanja toplote opisati z N nelinearnimi ena~bami, kjer je N veliko {tevilo elementov diskretiziranega modela. Ta sklop ena~b je mogo~e u~inkovito re{iti s pribli`kom metode vrsta za vrsto, ki temelji na predpostavki toka toplote,~eprav izra~un poteka serijsko. To pomeni, da ni mogo~vzporedni izra~un, kar je striktna zahteva, kadar se uporablja grafi~no kartico za pohitritev izra~una. Ta~lanek opisuje ve~dimenzijski numeri~ni model nestabilnega prevajanja toplote, kar je bilo re{eno z metodo vrsta za vrsto in s pospe{itvijo modifikacije te metode na grafi~ni kartici. Trajanje izra~una je primerjano z osnovno metodo vrsta za vrsto pri razli~nih dimenzijah mre`e.

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“…We will show that there are multiple loop-based algorithms for each of these three operations, all of which can be orchestrated so that the result overwrites the input without requiring temporary space. Also presented will be two algorithms that overwrite A by its inverse without the explicit computation of these intermediate results, requiring only a single sweep through the matrix, as was already briefly mentioned in [Quintana et al 2001]. The performance benefit of the single-sweep algorithm for a distributed memory archi-· Paolo Bientinesi et al tecture is illustrated in Fig.…”

Section: Introductionmentioning

confidence: 99%

Families of algorithms related to the inversion of a Symmetric Positive Definite matrix

Bientinesi

Gunter

Geijn

2008

ACM Trans. Math. Softw.

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We study the high-performance implementation of the inversion of a Symmetric Positive Definite (SPD) matrix on architectures ranging from sequential processors to Symmetric MultiProcessors to distributed memory parallel computers. This inversion is traditionally accomplished in three "sweeps": a Cholesky factorization of the SPD matrix, the inversion of the resulting triangular matrix, and finally the multiplication of the inverted triangular matrix by its own transpose. We state different algorithms for each of these sweeps as well as algorithms that compute the result in a single sweep. One algorithm outperforms the current ScaLAPACK implementation by 20-30 percent due to improved load-balance on a distributed memory architecture.

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A Note On Parallel Matrix Inversion

Cited by 79 publications

References 7 publications

Using graphics processors to accelerate the computation of the matrix inverse

Using graphics processors to accelerate the computation of the matrix inverse

Implicit numerical multidimensional heat-conduction algorithm parallelization and acceleration on a graphics card

Families of algorithms related to the inversion of a Symmetric Positive Definite matrix

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