TY - JOUR

T1 - The shift techniques for a nonsymmetric algebraic Riccati equation

AU - Lin, Matthew M.

AU - Chiang, Chun Yueh

N1 - Funding Information:
We gratefully thank the editor and anonymous referees for their helpful comments that substantially improved this article. This work is partially supported by the National Science Council and the National Center for Theoretical Sciences in Taiwan . The first author (Matthew M. Lin) like to thank the support from National Science Council under the grant number NSC101-2115-M-194-007-MY3 , and the second author (Chun-Yueh Chiang) like to thank the support from National Science Council under the grant number NSC101-2115-M-150-002 .

PY - 2013

Y1 - 2013

N2 - In this paper, we want to analyze a special instance of a nonsymmetric algebraic matrix Riccati equation arising from transport theory. Traditional approaches for finding its minimal nonnegative solution are based on fixed point iterations and the speed of the convergence is linear. Recently, iterative methods such as Newton method and the structure-preserving doubling algorithm with quadratic convergence are designed for improving the speed of convergence. But, in some case, the speed of convergence will significantly decrease so that linear convergence becomes sublinear convergence and quadratic convergence becomes linear convergence. Our contribution in this work is to provide a thorough analysis to show that after the shift techniques, the speed of linear or quadratic convergence is preserved. Finally, we apply the shift procedures to the discussion of the simple iteration algorithm, improve its speed of convergence, and reduce its total elapsed CPU time.

AB - In this paper, we want to analyze a special instance of a nonsymmetric algebraic matrix Riccati equation arising from transport theory. Traditional approaches for finding its minimal nonnegative solution are based on fixed point iterations and the speed of the convergence is linear. Recently, iterative methods such as Newton method and the structure-preserving doubling algorithm with quadratic convergence are designed for improving the speed of convergence. But, in some case, the speed of convergence will significantly decrease so that linear convergence becomes sublinear convergence and quadratic convergence becomes linear convergence. Our contribution in this work is to provide a thorough analysis to show that after the shift techniques, the speed of linear or quadratic convergence is preserved. Finally, we apply the shift procedures to the discussion of the simple iteration algorithm, improve its speed of convergence, and reduce its total elapsed CPU time.

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U2 - 10.1016/j.amc.2012.11.018

DO - 10.1016/j.amc.2012.11.018

M3 - Article

AN - SCOPUS:84872122880

VL - 219

SP - 5083

EP - 5095

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

IS - 10

ER -