An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations

Feng Yin and Guang-Xin Huang

Journal of Applied Mathematics, 2012, vol. 2012, 1-28

Abstract:

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations ( ð ´ ð ‘‹ ð µ âˆ’ ð ¶ ð ‘Œ ð · , ð ¸ ð ‘‹ ð ¹ âˆ’ ð º ð ‘Œ ð » ) = ( ð ‘€ , ð ‘ ) , which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matrices ð ‘‹ and ð ‘Œ . When the matrix equations are consistent, for any initial generalized reflexive matrix pair [ ð ‘‹ 1 , ð ‘Œ 1 ] , the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair [ î î ð ‘‹ , ð ‘Œ ] to a given matrix pair [ ð ‘‹ 0 , ð ‘Œ 0 ] in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair [ î ‚ ð ‘‹ ∗ , î ‚ ð ‘Œ ∗ ] of a new corresponding generalized coupled Sylvester matrix equation pair î ‚ î ‚ î ‚ î ‚ î‚‹ î ‚ ( ð ´ ð ‘‹ ð µ âˆ’ ð ¶ ð ‘Œ ð · , ð ¸ ð ‘‹ ð ¹ âˆ’ ð º ð ‘Œ ð » ) = ( ð ‘€ , ð ‘ ) , where î‚‹ ð ‘€ = ð ‘€ − ð ´ ð ‘‹ 0 ð µ + ð ¶ ð ‘Œ 0 î ‚ ð · , ð ‘ = ð ‘ âˆ’ ð ¸ ð ‘‹ 0 ð ¹ + ð º ð ‘Œ 0 ð » . Several numerical examples are given to show the effectiveness of the presented iterative algorithm.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnljam:152805

DOI: 10.1155/2012/152805

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