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Iterative positive definite solutions of the two nonlinear matrix equations \(X \pm A^{T}X^{-2} A =I \). (English) Zbl 1072.65065

Consider the matrix equation \(X + \varepsilon A^\top X^{-2}A= I\) in \(\mathbb{R}^{n\times n}\), where \(\varepsilon= \pm1\). The iteration scheme \(X_{k+1}= I -\varepsilon A^\top X^{-2}_k A\), \(X_0= \alpha I\), \(\alpha> 0\), for finding the positive definite solution is justified under certain restrictions on the coefficient matrix \(A\).

MSC:

65F30 Other matrix algorithms (MSC2010)
15A24 Matrix equations and identities
65F10 Iterative numerical methods for linear systems
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