## A new inversion free iteration for solving the equation $$X + A^{\star} X^{-1} A = Q$$.(English)Zbl 1072.65060

A new iterative method for solving the matrix equation $$X + A^\star X^{-1} A = I$$, where $$I$$ denotes the identity matrix, is proposed: $X_0 = Y_0 = I, \;Y_{n+1} = (I-X_n) Y_n + I_n, \;X_{n+1} = I-A^\star Y_{n+1} A.$ It is shown that $$X_n$$ converges to the maximal positive definite solution. Based on numerical experiments with two $$3\times 3$$ and $$4\times 4$$ examples, the authors conclude that the new method is more accurate and requires less floating point operations than some existing methods.

### MSC:

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

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