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Preconditioned conjugate gradient methods for the solution of indefinite least squares problems

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Abstract

The conjugate gradient (CG) method is considered for solving the large and sparse indefinite least squares (ILS) problem min  x (b−Ax)T J(b−Ax) where J=diag (I p ,−I q ) is a signature matrix. However the rate of convergence becomes slow for ill-conditioned problems. The QR-based preconditioner is found to be effective in accelerating the convergence. Numerical results show that the sparse Householder QR-based preconditioner is superior to the CG method especially for sparse and ill-conditioned problems.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, China

    Qiaohua Liu & Xianjuan Li

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  1. Qiaohua Liu
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  2. Xianjuan Li
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Correspondence to Qiaohua Liu.

Additional information

This work was supported by the National Natural Science Foundation of China (11001167) and Shanghai Leading Academic Discipline Project (J50101).

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Liu, Q., Li, X. Preconditioned conjugate gradient methods for the solution of indefinite least squares problems. Calcolo 48, 261–271 (2011). https://doi.org/10.1007/s10092-011-0039-8

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  • DOI: https://doi.org/10.1007/s10092-011-0039-8

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