# A New Inversion-Free Iterative Scheme to Compute Maximal and Minimal Solutions of a Nonlinear Matrix Equation

## Abstract

**:**

## 1. Subject, Literature, Motivation and Progress

#### 1.1. Problem Statement

#### 1.2. Literature

#### 1.3. Our Result

#### 1.4. Organization of the Paper

## 2. Novel Iteration Method

#### 2.1. An Equivalent NME

#### 2.2. Our Method

#### 2.3. Theoretical Investigations

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

## 3. Experiments

^{®}Core™ i7-9750H. The matrix inversions whenever required were done using Mathematica 12.0 built-in commands. We found the number of iterations required to observe convergence. In the code we wrote to implement different schemes, we stopped all the applied schemes when two successive iterations in the infinity norm was less than a tolerance, as follows:

**Example**

**1.**

**Example**

**2.**

## 4. Concluding Remarks

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- El-Sayed, S.M. An algorithm for computing positive definite solutions of the nonlinear matrix equation X + A
^{*}X^{−1}A = I. Int. J. Comput. Math.**2003**, 80, 1527–1534. [Google Scholar] [CrossRef] - Engwerda, J.C.; Ran, C.M.A.; Rijkeboer, A.L. Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X + A
^{*}X^{−1}A = Q. Linear Algebra Its Appl.**1993**, 186, 255–275. [Google Scholar] [CrossRef] [Green Version] - Ivanov, I.G.; Hasanov, V.I.; Minchev, B.V. On matrix equations X ± A
^{*}X^{−2}A = I. Linear Algebra Its Appl.**2001**, 326, 27–44. [Google Scholar] [CrossRef] [Green Version] - Huang, N.; Ma, C.; Tang, J. The inversion-free iterative methods for a system of nonlinear matrix equations. Int. J. Comput. Math.
**2016**, 93, 1470–1483. [Google Scholar] [CrossRef] - Shil, S.; Nashine, H.K. Latest inversion-free iterative scheme for solving a pair of nonlinear matrix equations. J. Math.
**2021**, 2021, 2667885. [Google Scholar] [CrossRef] - Bougerol, P. Kalman filtering with random coefficients and contractions. SIAM J. Control Optim.
**1993**, 31, 942–959. [Google Scholar] [CrossRef] - Soheili, A.R.; Toutounian, F.; Soleymani, F. A fast convergent numerical method for matrix sign function with application in SDEs. J. Comput. Appl. Math.
**2015**, 282, 167–178. [Google Scholar] [CrossRef] - Tian, Y.; Xia, C. On the low-degree solution of the Sylvester matrix polynomial equation. J. Math.
**2021**, 2021, 4612177. [Google Scholar] [CrossRef] - Guo, C.H.; Lancaster, P. Iterative solution of two matrix equations. Math. Comput.
**1999**, 68, 1589–1603. [Google Scholar] [CrossRef] [Green Version] - Liu, A.; Chen, G. On the Hermitian positive definite solutions of nonlinear matrix equation X
^{s}+ A^{*}X^{−t1}A + B^{*}X^{−t2}B = Q. Math. Prob. Eng.**2011**, 2011, 163585. [Google Scholar] [CrossRef] [Green Version] - El-Sayed, S.M.; Ran, A.C.M. On an iteration method for solving a class of nonlinear matrix equations. SIAM J. Matrix Anal. Appl.
**2002**, 23, 632–645. [Google Scholar] [CrossRef] - Li, J. Solutions and improved perturbation analysis for the matrix equation X + A
^{*}X^{−p}A = Q. Abst. Appl. Anal.**2013**, 2013, 575964. [Google Scholar] - Engwerda, J.C. On the existence of a positive definite solution of the matrix equation X + A
^{*}X^{−1}A = I. Linear Algebra Its Appl.**1993**, 194, 91–108. [Google Scholar] [CrossRef] [Green Version] - Monsalve, M.; Raydan, M. A new inversion-free method for a rational matrix equation. Linear Algebra Its Appl.
**2010**, 433, 64–71. [Google Scholar] [CrossRef] [Green Version] - Zhang, L. An improved inversion-free method for solving the matrix equation X + A
^{*}X^{−α}A = Q. J. Comput. Appl. Math.**2013**, 253, 200–203. [Google Scholar] [CrossRef] - El-Sayed, S.M.; Al-Dbiban, A.M. A new inversion free iteration for solving the equation X + A
^{*}X^{−1}A = Q. J. Comput. Appl. Math.**2005**, 181, 148–156. [Google Scholar] [CrossRef] [Green Version] - Zhan, X. Computing the extremal positive definite solutions of a matrix equation. SIAM J. Sci. Comput.
**1996**, 17, 1167–1174. [Google Scholar] [CrossRef] - Soleymani, F.; Sharifi, M.; Vanani, S.K.; Haghani, F.K.; Kiliçman, A. An inversion-free method for finding positive definite solution of a rational matrix equation. Sci. World J.
**2014**, 2014, 560931. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Soleymani, F.; Sharifi, M.; Shateyi, S.; Haghani, F.K. An algorithm for computing geometric mean of two Hermitian positive definite matrices via matrix sign. Abst. Appl. Anal.
**2014**, 2014, 978629. [Google Scholar] [CrossRef] - Trott, M. The Mathematica Guide-Book for Numerics; Springer: New York, NY, USA, 2006. [Google Scholar]

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**MDPI and ACS Style**

Zaka Ullah, M.
A New Inversion-Free Iterative Scheme to Compute Maximal and Minimal Solutions of a Nonlinear Matrix Equation. *Mathematics* **2021**, *9*, 2994.
https://doi.org/10.3390/math9232994

**AMA Style**

Zaka Ullah M.
A New Inversion-Free Iterative Scheme to Compute Maximal and Minimal Solutions of a Nonlinear Matrix Equation. *Mathematics*. 2021; 9(23):2994.
https://doi.org/10.3390/math9232994

**Chicago/Turabian Style**

Zaka Ullah, Malik.
2021. "A New Inversion-Free Iterative Scheme to Compute Maximal and Minimal Solutions of a Nonlinear Matrix Equation" *Mathematics* 9, no. 23: 2994.
https://doi.org/10.3390/math9232994