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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

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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  • TXT
@Article{JCM-25-531, author = {Zhongxiao Jia and Yuquan Sun}, 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 AU - Zhongxiao Jia & Yuquan Sun 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, Preconditioning, 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 and Yuquan Sun. (2007). 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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