Developing HSS iteration schemes for solving the quadratic matrix equation AX2+BX+C=0$AX^{2}+BX+C=0$

Abstract The quadratic matrix equation (QME) occurs in the branches such as the quadratic eigenvalue problems and quasi‐birth‐death processes. Also, the numerical solution of QMEs is an essential step in many computational methods for linear‐quadratic and robust control, filtering, controller order reduction, inner‐outer factorization, spectral factorization, and other applications. In this study, schemes are presented to solve the QME based on the Hermitian and skew‐Hermitian splitting (HSS). It is shown that the proposed schemes converge to the solutions of the QME. Finally, some examples are solved to discover the application of the schemes in comparison with Newton's method.