Computing the Extremal Positive Definite Solutions of a Matrix Equation
SIAM Journal on Scientific Computing
Iterative solution of two matrix equations
Mathematics of Computation
Solving a Quadratic Matrix Equation by Newton's Method with Exact Line Searches
SIAM Journal on Matrix Analysis and Applications
Convergence Rate of an Iterative Method for a Nonlinear Matrix Equation
SIAM Journal on Matrix Analysis and Applications
Convergence Analysis of Structure-Preserving Doubling Algorithms for Riccati-Type Matrix Equations
SIAM Journal on Matrix Analysis and Applications
Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations
SIAM Journal on Matrix Analysis and Applications
Journal of Computational and Applied Mathematics
SIAM Journal on Matrix Analysis and Applications
Journal of Computational and Applied Mathematics
On a Nonlinear Matrix Equation Arising in Nano Research
SIAM Journal on Matrix Analysis and Applications
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The matrix equation $X+A^TX^{-1}A=Q$ has been studied extensively when $A$ and $Q$ are real square matrices and $Q$ is symmetric positive definite. The equation has positive definite solutions under suitable conditions, and in that case the solution of interest is the maximal positive definite solution. The same matrix equation plays an important role in Green's function calculations in nano research, but the matrix $Q$ there is usually indefinite (so the matrix equation has no positive definite solutions), and one is interested in the case where the matrix equation has no positive definite solutions even when $Q$ is positive definite. The solution of interest in this nano application is a special weakly stabilizing complex symmetric solution. In this paper we show how a doubling algorithm can be used to find good approximations to the desired solution efficiently and reliably.