On solution and perturbation estimates for the nonlinear matrix equation \(X-A^{*}e^{X}A=I\)

This work incorporates an efficient inversion free iterative scheme into Newton’s method to solve Newton’s step regardless of the singularity of the Fr $${\acute{\text {e}}}$$ chet derivative. The proposed iterative scheme is constructed by extending the idea of the foundational form of the conjugate gradient method. Moreover, the resulting scheme is refined and employed to obtain a symmetric solution of the nonlinear matrix equation $$X-A^{*}e^{X}A=I.$$ Furthermore, explicit expressions for the perturbation and residual bound estimates of the approximate positive definite solution are derived. Finally, five numerical case studies provided confirm both the preciseness of theoretical results and the effectiveness of the propounded iterative method.