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

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

65F10 65F15 65N30.

@Article{JCM-25-498, author = {}, 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 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 KW - Quadratic eigenvalue problem KW - Solvent KW - Bernoulli's iteration KW - Newton's method KW - 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.