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Volume 25, Issue 5
Modified Bernoulli Iteration Methods for Quadratic Matrix Equation

Zhong-Zhi Bai & Yong-Hua Gao

J. Comp. Math., 25 (2007), pp. 498-511.

Published online: 2007-10

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  • Abstract

We construct a modified Bernoulli iteration method for solving the quadratic matrix equation $AX^{2} + BX + C = 0$, where $A$, $B$ and $C$ are square matrices. This method is motivated from the Gauss-Seidel iteration for solving linear systems and the Sherman-Morrison-Woodbury formula for updating matrices. Under suitable conditions, we prove the local linear convergence of the new method. An algorithm is presented to find the solution of the quadratic matrix equation and some numerical results are given to show the feasibility and the effectiveness of the algorithm. In addition, we also describe and analyze the block version of the modified Bernoulli iteration method.

  • Keywords

Quadratic matrix equation, Quadratic eigenvalue problem, Solvent, Bernoulli's iteration, Newton's method, Local convergence.

  • AMS Subject Headings

65F10, 65F15, 65N30.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-25-498, author = {Zhong-Zhi Bai and Yong-Hua Gao}, title = {Modified Bernoulli Iteration Methods for Quadratic Matrix Equation}, journal = {Journal of Computational Mathematics}, year = {2007}, volume = {25}, number = {5}, pages = {498--511}, abstract = {

We construct a modified Bernoulli iteration method for solving the quadratic matrix equation $AX^{2} + BX + C = 0$, where $A$, $B$ and $C$ are square matrices. This method is motivated from the Gauss-Seidel iteration for solving linear systems and the Sherman-Morrison-Woodbury formula for updating matrices. Under suitable conditions, we prove the local linear convergence of the new method. An algorithm is presented to find the solution of the quadratic matrix equation and some numerical results are given to show the feasibility and the effectiveness of the algorithm. In addition, we also describe and analyze the block version of the modified Bernoulli iteration method.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8708.html} }
TY - JOUR T1 - Modified Bernoulli Iteration Methods for Quadratic Matrix Equation AU - Zhong-Zhi Bai & Yong-Hua Gao JO - Journal of Computational Mathematics VL - 5 SP - 498 EP - 511 PY - 2007 DA - 2007/10 SN - 25 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8708.html KW - Quadratic matrix equation, Quadratic eigenvalue problem, Solvent, Bernoulli's iteration, Newton's method, Local convergence. AB -

We construct a modified Bernoulli iteration method for solving the quadratic matrix equation $AX^{2} + BX + C = 0$, where $A$, $B$ and $C$ are square matrices. This method is motivated from the Gauss-Seidel iteration for solving linear systems and the Sherman-Morrison-Woodbury formula for updating matrices. Under suitable conditions, we prove the local linear convergence of the new method. An algorithm is presented to find the solution of the quadratic matrix equation and some numerical results are given to show the feasibility and the effectiveness of the algorithm. In addition, we also describe and analyze the block version of the modified Bernoulli iteration method.

Zhong-Zhi Bai and Yong-Hua Gao. (2007). Modified Bernoulli Iteration Methods for Quadratic Matrix Equation. Journal of Computational Mathematics. 25 (5). 498-511. doi:
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