Newton-type methods for solving nonlinear equations on quadratic matrix groups
Let \(H\) be a nonsingular, symmetric or skew--symmetric \(d \times d\) matrix. The quadratic group \(G\) related to \(H\) is given by \(G= \{ Y \in \mathbb{R}^{d \times d}: Y^T H Y= H \}\). Let \(g= \{ A \in\mathbb{R}^{d \times d}: A^T H+ H A= 0 \}\) be the Lie algebra associated with \(G\). The authors consider the problem of finding a solution \(Y \in G\) of equation \(F(Y)=0\), where \(F: G \to g\) is a smooth matrix function. They propose to transform the original problem into an equivalent equation \({\widetilde F} (V)=0\) with \({\widetilde F}\) acting from the algebra \(g\) into itself, and to use Newton's method \(\text{ d}{\widetilde F}_{V_n}(U_n)+ {\widetilde F}(V_n)=0, V_{n+1}=V_n+U_n\) for solving the obtained equation. The operator \({\widetilde F}\) has the form \({\widetilde F}(V)= F(Z {\Re}(V))\), where \(Z\) is an approximation of the root \(Y^*\) of \(F(Y)=0\) and \({\Re}\) denotes the Cayley transform. Let \(V^*\) be a solution of the transformed equation. Under standard conditions on \(\text{ d} F_{Y}\) and the assumption that \(\|V^*-V_0 \|\) is small, the authors prove that \(\lim_{n \to \infty} V_n=V^*, \lim_{n \to \infty} (Y^*)^{-1} Z {\Re}(V_n) =I\) and establish the estimate for the rate of convergence \(\|V^* -V_{n+1}\|\leq C \|V^* -V_n\|^2, n \geq 0\).
scientific article; zbMATH DE number 1423329