Consider the nonlinear matrix equation X + A * X −q A = Q where 0 < q ≤ 1. A new sufficient condition for this equation to have positive definite solution is provided and two iterative methods for the maximal positive definite solution are proposed. Applying the theory of condition number developed by Rice, an explicit expression of the condition number of the maximal positive definite solution is obtained. The theoretical results are illustrated by numerical examples.

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In this paper, we present an efficient ΓQR algorithm for solving the linear response eigenvalue problem H x x x = λx x x, where H is Π Π Π −-symmetric with respect to Γ 0 = diag(I n , −I n). Based on newly introduced Γ-orthogonal transformations, the ΓQR algorithm preserves the Π Π Π −-symmetric structure of H throughout the whole process, and thus guarantees the computed eigenvalues to appear pairwise (λ, −λ) as they should. With the help of a newly established implicit Γ-orthogonality theorem, we incorporate the implicit multi-shift technique to accelerate the convergence of the ΓQR algorithm. Numerical experiments are given to show the effectiveness of the algorithm.