Newton's method for the quadratic matrix equation
Quadratic matrix equations (QME) arise in many areas of scientific computing. In this paper, the author studies the quadratic matrix equation \({\mathcal L}(X)=AX^2+BX+C=0\), where \(A, B, C \in {\mathbb R}^{n\times n}\), and \((x^T B x)^2>4(x^T A x)(x^T C x),\,\forall x\not=0.\) These equations are connected with the overdamped quadratic eigenvalue problem in the analysis of damped structural systems and vibration problem. Some techniques for analysis and solution to the equation \({\mathcal L}(X)=0\), and other nonlinear problems have been addressed by many authors. In this paper, the author studies another different technique to solve the equation \({\mathcal L}(X)=0\), with the condition \((x^T B x)^2>4(x^T A x)(x^T C x),\,\forall x\not=0.\) It is readily seen that if the quadratic matrix equation \({\mathcal L}(X)=0\) has a nonsingular solution \(S\), \(S\) is also the solution of the nonlinear matrix equation (NME) \({\mathcal F}(X)=AX+CX^{-1}+B=0.\) The author applies Newton's method to the nonlinear matrix equation \({\mathcal F}(X)=0\) for computing the \textit{dominant solvent} and the \textit{minimal solvent} of the quadratic matrix equation \({\mathcal L}(X)=0\), because the \textit{minimal solvent} of the quadratic matrix equation \({\mathcal L}(X)=0\) is also the dominant solution of the nonlinear matrix equation \({\mathcal F}(X)=0\) (provided that the minimal solvent is nonsingular). The author derives a local convergence theorem and a semilocal convergence theorem for Newton's method. Also, numerical results are presented to show the feasibility and effectiveness of the method developed.
scientific article; zbMATH DE number 5115448