## A note on the fixed-point iteration for the matrix equations $$X \pm A^* X^{-1}A=I$$.(English)Zbl 1166.65018

Consider the complex matrix equations $$X\pm A^*X^{-1} A = I$$, where $$I$$ is the identity matrix. Conditions for solvability and representation of the solutions to these equations are well known. Also, it has been observed that the rate of convergence of the simple fixed-point iteration $$X_{k+1} = I \mp A^*X_k^{-1}A$$ strongly depends on the initial state $$X_0 = \gamma I$$, where $$\gamma$$ is a parameter. The authors discuss this phenomenon and explain the fast convergence when $$A$$ is normal or nearly normal matrix.

### MSC:

 65F30 Other matrix algorithms (MSC2010) 15A24 Matrix equations and identities
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### References:

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