scholarly journals BiCGCR2: A new extension of conjugate residual method for solving non-Hermitian linear systems

2016 ◽  
Vol 305 ◽  
pp. 115-128
Author(s):  
Xian-Ming Gu ◽  
Ting-Zhu Huang ◽  
Bruno Carpentieri
Keyword(s):  
Linear Systems ◽  
Residual Method ◽  
Conjugate Residual

scholarly journals A Breakdown-Free Block COCG Method for Complex Symmetric Linear Systems with Multiple Right-Hand Sides

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1302
Author(s):  
Hong-Xiu Zhong ◽  
Xian-Ming Gu ◽  
Shao-Liang Zhang
Keyword(s):  
Linear Systems ◽  
Spectral Properties ◽  
Coefficient Matrix ◽  
Complex Symmetric Linear Systems ◽  
Right Hand ◽  
Orthogonality Conditions ◽  
Free Block ◽  
Complex Symmetric ◽  
The Right ◽  
Conjugate Residual

The block conjugate orthogonal conjugate gradient method (BCOCG) is recognized as a common method to solve complex symmetric linear systems with multiple right-hand sides. However, breakdown always occurs if the right-hand sides are rank deficient. In this paper, based on the orthogonality conditions, we present a breakdown-free BCOCG algorithm with new parameter matrices to handle rank deficiency. To improve the spectral properties of coefficient matrix A, a precondition version of the breakdown-free BCOCG is proposed in detail. We also give the relative algorithms for the block conjugate A-orthogonal conjugate residual method. Numerical results illustrate that when breakdown occurs, the breakdown-free algorithms yield faster convergence than the non-breakdown-free algorithms.

An Extension of the COCR Method to Solving Shifted Linear Systems with Complex Symmetric Matrices

2011 ◽  
Vol 1 (2) ◽  
pp. 97-107 ◽  
Author(s):  
Tomohiro Sogabe ◽  
Shao-Liang Zhang
Keyword(s):  
Linear Systems ◽  
Large Scale ◽  
Electronic Structure Calculations ◽  
Symmetric Matrices ◽  
Complex Symmetric Linear Systems ◽  
Shifted Linear Systems ◽  
Complex Symmetric ◽  
Complex Symmetric Matrices ◽  
Structure Calculations ◽  
Conjugate Residual

AbstractThe Conjugate Orthogonal Conjugate Residual (COCR) method [T. Sogabe and S.-L. Zhang, JCAM, 199 (2007), pp. 297-303.] has recently been proposed for solving complex symmetric linear systems. In the present paper, we develop a variant of the COCR method that allows the efficient solution of complex symmetric shifted linear systems. Some numerical examples arising from large-scale electronic structure calculations are presented to illustrate the performance of the variant.

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