The analysis of restart DGMRES for solving singular linear systems

Zhou, Jieyong; Wei, Yimin

doi:10.1016/j.amc.2005.09.063

Cited by 5 publications

(2 citation statements)

References 7 publications

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“…However, detailed analysis of these phenomena is out of the scope of this paper. Similar phenomena for "DGMRES algorithm" have recently been reported in (Zhou & Wei 2006). Generally speaking, we may take around 20, 30, 40, or 50 as a proper choice of the Krylov dimension m of the GMRES(m) solver for both cases of ε = 10 −6 and ε = 10 −2 , while m = 50 is our preference since relatively bigger m can make the iterative solver more stable, and it appears more fair to set m = 50 when comparing the convergence speed of GMRES(m, 10 −6 ) and GMRES(m, 10 −2 ).…”

Section: Efficiency and Accuracymentioning

confidence: 55%

The GMRES method applied to the BEM extrapolation of solar force-free magnetic fields

Yan²,

Devel³

et al. 2007

A&A

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Context. Since the 1990's, Yan and colleagues have formulated a kind of boundary integral formulation for the linear or non-linear solar force-free magnetic fields with finite energy in semi-infinite space, and developed a computational procedure by virtue of the boundary element method (BEM) to extrapolate the magnetic fields above the photosphere. Aims. In this paper, the generalized minimal residual method (GMRES) is introduced into the BEM extrapolation of the solar forcefree magnetic fields, in order to efficiently solve the associated BEM system of linear equations, which was previously solved by the Gauss elimination method with full pivoting. Methods. Being a modern iterative method for non-symmetric linear systems, the GMRES method reduces the computational cost for the BEM system from O(N 3 ) to O(N 2 ), where N is the number of unknowns in the linear system. Intensive numerical experiments are conducted on a well-known analytical model of the force-free magnetic field to reveal the convergence behaviour of the GMRES method subjected to the BEM systems. The impacts of the relevant parameters on the convergence speed are investigated in detail. Results. Taking the Krylov dimension to be 50 and the relative residual bound to be 10 −6 (or 10 −2 ), the GMRES method is at least 1000 (or 9000) times faster than the full pivoting Gauss elimination method used in the original BEM extrapolation code, when N is greater than 12 321, according to the CPU timing information measured on a common desktop computer (CPU speed 2.8 GHz; RAM 1 GB) for the model problem.

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Section: Efficiency and Accuracymentioning

confidence: 55%

The GMRES method applied to the BEM extrapolation of solar force-free magnetic fields

Yan²,

Devel³

et al. 2007

A&A

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“…It has been further developed in several papers, say [28,29]. There are many possible variations of this algorithm.…”

Section: Computing the Group Inverse Using The Dgmres Methodsmentioning

confidence: 98%

The stability of formulae of the Gohberg–Semencul–Trench type for Moore–Penrose and group inverses of Toeplitz matrices

Xie

Wei

2016

Linear Algebra and its Applications

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In memory of Professor Hans Schneider MSC: 15A18 15A22 65F15 Keywords: Toeplitz matrix Moore-Penrose inverse Group inverse LSQR DGMRESWe present a stability analysis of Gohberg-Semencul-Trench type formulae for the Moore-Penrose and group inverses of singular Toeplitz matrices. We develop a fast algorithm for the computation of the Moore-Penrose inverse based on a Gohberg-Semencul-Trench type formula and the LSQR method. For the group inverse, the DGMRES method is used to perform the fast computation. Numerical tests show that the fast algorithms designed here are at least as good as the known Newton iteration.

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DGMRES method augmented with eigenvectors for computing the Drazin-inverse solution of singular linear systems

Meng

2016

Acta Math. Appl. Sin. Engl. Ser.

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The analysis of restart DGMRES for solving singular linear systems

Cited by 5 publications

References 7 publications

The GMRES method applied to the BEM extrapolation of solar force-free magnetic fields

The GMRES method applied to the BEM extrapolation of solar force-free magnetic fields

The stability of formulae of the Gohberg–Semencul–Trench type for Moore–Penrose and group inverses of Toeplitz matrices

DGMRES method augmented with eigenvectors for computing the Drazin-inverse solution of singular linear systems

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