Generalized global conjugate gradient squared algorithm

Zhang, Jianhua; Dai, Hua; Zhao, Jing

doi:10.1016/j.amc.2010.05.026

Cited by 9 publications

(14 citation statements)

References 25 publications

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“…The first global methods are global full orthogonalization (Gl-FOM) and global generalized minimal residual (Gl-GMRES), introduced in [20]. Gl-BiCG and Gl-BiCGStab were suggested in [21], and global variants of less well-known Krylov subspace methods were subsequently proposed in [18] (Gl-CMRH), [34] (Gl-CGS), [15] (Gl-SCD) and [35] (Gl-BiCR and its variants).…”

Section: Discussion Of the Literaturementioning

confidence: 99%

On short recurrence Krylov type methods for linear systems with many right-hand sides

Rashedi

Ebadi

Birk

et al. 2016

Journal of Computational and Applied Mathematics

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Block and global Krylov subspace methods have been proposed as methods adapted to the situation where one iteratively solves systems with the same matrix and several right hand sides. These methods are advantageous, since they allow to cast the major part of the arithmetic in terms of matrix-block vector products, and since, in the block case, they take their iterates from a potentially richer subspace. In this paper we consider the most established Krylov subspace methods which rely on short recurrencies, i.e. BiCG, QMR and BiCGStab. We propose modifications of their block variants which increase numerical stability, thus at least partly curing a problem previously observed by several authors. Moreover, we develop modifications of the "global" variants which almost halve the number of matrix-vector multiplications. We present a discussion as well as numerical evidence which both indicate that the additional work present in the block methods can be substantial, and that the new "economic" versions of the "global" BiCG and QMR method can be considered as good alternatives to the BiCGStab variants.

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Section: Discussion Of the Literaturementioning

confidence: 99%

On short recurrence Krylov type methods for linear systems with many right-hand sides

Rashedi

Ebadi

Birk

et al. 2016

Journal of Computational and Applied Mathematics

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“…It is known that the CGS method [19] for the general linear systems may exhibit erratic convergence behavior. Unfortunately, the disadvantage can also be carried over to the Gl-CGS method [20]. In this section, we give a global transpose-free quasi-minimal residual method (Gl-TFQMR) for the Sylvester equation (1).…”

Section: A Global Transpose-free Quasi-minimal Residual Methodsmentioning

confidence: 99%

“…Further, Zhang etc. have extended the CGS algorithm to solve linear systems with multiple right-hand sides, i.e., MX = N, which yields the global CGS algorithm (Gl-CGS) [20]. To clarify the derivation of our new algorithm in Section 3, we give the Gl-CGS algorithm below.…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…To clarify the derivation of our new algorithm in Section 3, we give the Gl-CGS algorithm below. For more detailed description of Gl-CGS, we refer to [20].…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…presented a global Arnoldi method and corresponding theory for solving multi-input Sylvesterobserver equations from the construction of the Luenberger observer in control theory [5]. Using the global techniques, some new methods for matrix equations, for instance, the global conjugate squared method (Gl-CGS) and its generalization [20], and the global bi-conjugate residual method (Gl-BCR) [21], have been proposed following [12]. Though some of these methods were originally contrived for linear systems with multiple right-hand sides, they can also be applied for the Sylvester equation in a similar way.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A global transpose-free method with quasi-minimal residual strategy for the Sylvester equations

Zhang¹,

Gu²

2013

Filomat

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The Sylvester equation has numerous applications in control and system theory, power system, linear algebra, model reduction and signal processing. In the present paper, a global transpose-free quasiminimal residual method for the Sylvester equations is proposed. The resulting method, with shortterm recurrences, does not involve the multiplication with the transpose of the coefficient matrices, and often exhibits smoother convergence behavior than some other existing global methods. Finally, several numerical examples are reported by comparing the proposed method with some global methods.

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Global GPBiCGstab(L) method for solving linear matrix equations

Horiuchi¹,

Aihara

Suzuki

et al. 2022

Numer Algor

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Global Krylov subspace methods are effective iterative solvers for large linear matrix equations. Several Lanczos-type product methods (LTPMs) for solving standard linear systems of equations have been extended to their global versions. However, the GPBiCGstab(L) method, which unifies two well-known LTPMs (i.e., BiCGstab(L) and GPBiCG methods), has been developed recently, and it has been shown that this novel method has superior convergence when compared to the conventional LTPMs. In the present study, we therefore extend the GPBiCGstab(L) method to its global version. Herein, we present not only a naive extension of the original GPBiCGstab(L) algorithm but also its alternative implementation. This variant enables the preconditioning technique to be applied stably and efficiently. Numerical experiments were performed, and the results demonstrate the effectiveness of the proposed global GPBiCGstab(L) method.

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Generalized global conjugate gradient squared algorithm

Cited by 9 publications

References 25 publications

On short recurrence Krylov type methods for linear systems with many right-hand sides

On short recurrence Krylov type methods for linear systems with many right-hand sides

A global transpose-free method with quasi-minimal residual strategy for the Sylvester equations

Global GPBiCGstab(L) method for solving linear matrix equations

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