Volume 25, Issue 5
A QR Decomposition Based Solver for the Least Squares Problems from the Minimal Residual Method for the Sylvester Equation

Zhongxiao Jia & Yuquan Sun

DOI:

J. Comp. Math., 25 (2007), pp. 531-542

Published online: 2007-10

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  • Abstract

Based on the generalized minimal residual (GMRES) principle, Hu and Reichel proposed a minimal residual algorithm for the Sylvester equation. The algorithm requires the solution of a structured least squares problem. They form the normal equations of the least squares problem and then solve it by a direct solver, so it is susceptible to instability. In this paper, by exploiting the special structure of the least squares problem and working on the problem directly, a numerically stable QR decomposition based algorithm is presented for the problem. The new algorithm is more stable than the normal equations algorithm of Hu and Reichel. Numerical experiments are reported to confirm the superior stability of the new algorithm.

  • Keywords

Least-squares solution Preconditioning Generalized singular value decomposition

  • AMS Subject Headings

65F22 65K10.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-25-531, author = {}, title = {A QR Decomposition Based Solver for the Least Squares Problems from the Minimal Residual Method for the Sylvester Equation}, journal = {Journal of Computational Mathematics}, year = {2007}, volume = {25}, number = {5}, pages = {531--542}, abstract = { Based on the generalized minimal residual (GMRES) principle, Hu and Reichel proposed a minimal residual algorithm for the Sylvester equation. The algorithm requires the solution of a structured least squares problem. They form the normal equations of the least squares problem and then solve it by a direct solver, so it is susceptible to instability. In this paper, by exploiting the special structure of the least squares problem and working on the problem directly, a numerically stable QR decomposition based algorithm is presented for the problem. The new algorithm is more stable than the normal equations algorithm of Hu and Reichel. Numerical experiments are reported to confirm the superior stability of the new algorithm.}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8711.html} }
TY - JOUR T1 - A QR Decomposition Based Solver for the Least Squares Problems from the Minimal Residual Method for the Sylvester Equation JO - Journal of Computational Mathematics VL - 5 SP - 531 EP - 542 PY - 2007 DA - 2007/10 SN - 25 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8711.html KW - Least-squares solution KW - Preconditioning KW - Generalized singular value decomposition AB - Based on the generalized minimal residual (GMRES) principle, Hu and Reichel proposed a minimal residual algorithm for the Sylvester equation. The algorithm requires the solution of a structured least squares problem. They form the normal equations of the least squares problem and then solve it by a direct solver, so it is susceptible to instability. In this paper, by exploiting the special structure of the least squares problem and working on the problem directly, a numerically stable QR decomposition based algorithm is presented for the problem. The new algorithm is more stable than the normal equations algorithm of Hu and Reichel. Numerical experiments are reported to confirm the superior stability of the new algorithm.
Zhongxiao Jia & Yuquan Sun. (1970). A QR Decomposition Based Solver for the Least Squares Problems from the Minimal Residual Method for the Sylvester Equation. Journal of Computational Mathematics. 25 (5). 531-542. doi:
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