Positive definite solutions of the matrix equations X±A*X-q A=Q(q≥1)
Computers & Mathematics with Applications
The Matrix Equation $X+A^TX^{-1}A=Q$ and Its Application in Nano Research
SIAM Journal on Scientific Computing
Solving a Structured Quadratic Eigenvalue Problem by a Structure-Preserving Doubling Algorithm
SIAM Journal on Matrix Analysis and Applications
Journal of Computational and Applied Mathematics
Alternating-directional Doubling Algorithm for $M$-Matrix Algebraic Riccati Equations
SIAM Journal on Matrix Analysis and Applications
On a Nonlinear Matrix Equation Arising in Nano Research
SIAM Journal on Matrix Analysis and Applications
Monotone convergence of Newton-like methods for M-matrix algebraic Riccati equations
Numerical Algorithms
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In this paper, we review two types of doubling algorithm and some techniques for analyzing them. We then use the techniques to study the doubling algorithm for three different nonlinear matrix equations in the critical case. We show that the convergence of the doubling algorithm is at least linear with rate $1/2$. As compared to earlier work on this topic, the results we present here are more general, and the analysis here is much simpler.