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Volume 26, Issue 2
The Restrictively Preconditioned Conjugate Gradient Methods on Normal Residual for Block Two-by-Two Linear Systems

Junfeng Yin & Zhongzhi Bai

J. Comp. Math., 26 (2008), pp. 240-249.

Published online: 2008-04

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  • Abstract

The $restrictively$ $preconditioned$ $conjugate$ $gradient$ (RPCG) method is further developed to solve large sparse system of linear equations of a block two-by-two structure. The basic idea of this new approach is that we apply the RPCG method to the normal-residual equation of the block two-by-two linear system and construct each required approximate matrix by making use of the incomplete orthogonal factorization of the involved matrix blocks. Numerical experiments show that the new method, called the $restrictively$ $preconditioned$ $conjugate$ $gradient$ $on$ $normal$ $residual$ (RPCGNR), is more robust and effective than either the known RPCG method or the standard $conjugate$ $gradient$ $on$ $normal$ $residual$ (CGNR) method when being used for solving the large sparse saddle point problems.

  • Keywords

Block two-by-two linear system, Saddle point problem, Restrictively preconditioned conjugate gradient method, Normal-residual equation, Incomplete orthogonal factorization.

  • AMS Subject Headings

65F10, 65W05.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-26-240, author = {Junfeng Yin and Zhongzhi Bai}, title = {The Restrictively Preconditioned Conjugate Gradient Methods on Normal Residual for Block Two-by-Two Linear Systems}, journal = {Journal of Computational Mathematics}, year = {2008}, volume = {26}, number = {2}, pages = {240--249}, abstract = {

The $restrictively$ $preconditioned$ $conjugate$ $gradient$ (RPCG) method is further developed to solve large sparse system of linear equations of a block two-by-two structure. The basic idea of this new approach is that we apply the RPCG method to the normal-residual equation of the block two-by-two linear system and construct each required approximate matrix by making use of the incomplete orthogonal factorization of the involved matrix blocks. Numerical experiments show that the new method, called the $restrictively$ $preconditioned$ $conjugate$ $gradient$ $on$ $normal$ $residual$ (RPCGNR), is more robust and effective than either the known RPCG method or the standard $conjugate$ $gradient$ $on$ $normal$ $residual$ (CGNR) method when being used for solving the large sparse saddle point problems.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8621.html} }
TY - JOUR T1 - The Restrictively Preconditioned Conjugate Gradient Methods on Normal Residual for Block Two-by-Two Linear Systems AU - Junfeng Yin & Zhongzhi Bai JO - Journal of Computational Mathematics VL - 2 SP - 240 EP - 249 PY - 2008 DA - 2008/04 SN - 26 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8621.html KW - Block two-by-two linear system, Saddle point problem, Restrictively preconditioned conjugate gradient method, Normal-residual equation, Incomplete orthogonal factorization. AB -

The $restrictively$ $preconditioned$ $conjugate$ $gradient$ (RPCG) method is further developed to solve large sparse system of linear equations of a block two-by-two structure. The basic idea of this new approach is that we apply the RPCG method to the normal-residual equation of the block two-by-two linear system and construct each required approximate matrix by making use of the incomplete orthogonal factorization of the involved matrix blocks. Numerical experiments show that the new method, called the $restrictively$ $preconditioned$ $conjugate$ $gradient$ $on$ $normal$ $residual$ (RPCGNR), is more robust and effective than either the known RPCG method or the standard $conjugate$ $gradient$ $on$ $normal$ $residual$ (CGNR) method when being used for solving the large sparse saddle point problems.

Junfeng Yin and Zhongzhi Bai. (2008). The Restrictively Preconditioned Conjugate Gradient Methods on Normal Residual for Block Two-by-Two Linear Systems. Journal of Computational Mathematics. 26 (2). 240-249. doi:
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