Large-scale Stein and Lyapunov equations, Smith method, and applications

Tiexiang Li Peter Chang-Yi Weng Eric King-wah Chu Wen-Wei Lin

Numerical Analysis and Scientific Computing mathscidoc:1912.43733

Numerical Algorithms, 63, (4), 727-752, 2013.8
We consider the solution of large-scale Lyapunov and Stein equations. For Stein equations, the well-known Smith method will be adapted, with A_k = A^{2^k} not explicitly computed but in the recursive form A_k = A^{2^k}, and the fast growing but diminishing components in the approximate solutions truncated. Lyapunov equations will be first treated with the Cayley transform before the Smith method is applied. For algebraic equations with numerically low-ranked solutions of dimension <i>n</i>, the resulting algorithms are of an efficient <i>O</i>(<i>n</i>) computational complexity and memory requirement per iteration and converge essentially quadratically. An application in the estimation of a lower bound of the condition number for continuous-time algebraic Riccati equations is presented, as well as some numerical results.
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@inproceedings{tiexiang2013large-scale,
  title={Large-scale Stein and Lyapunov equations, Smith method, and applications},
  author={Tiexiang Li, Peter Chang-Yi Weng, Eric King-wah Chu, and Wen-Wei Lin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205350592648297},
  booktitle={Numerical Algorithms},
  volume={63},
  number={4},
  pages={727-752},
  year={2013},
}
Tiexiang Li, Peter Chang-Yi Weng, Eric King-wah Chu, and Wen-Wei Lin. Large-scale Stein and Lyapunov equations, Smith method, and applications. 2013. Vol. 63. In Numerical Algorithms. pp.727-752. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205350592648297.
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