Efficient computation of the extreme solutions of $X+A^*X^{-1}A=Q$ and $X-A^*X^{-1}A=Q$

DOI10.1090/S0025-5718-01-01368-0zbMath0994.65046MaRDI QIDQ4529712

Beatrice Meini

Publication date: 6 May 2002

Published in: Mathematics of Computation (Search for Journal in Brave)

zbMATH Keywords

numerical resultsNewton's methodmatrix equationcyclic reductionblock Toeplitz matrix

Mathematics Subject Classification ID

Related Items (60)

On the Matrix EquationX = Q − S∗X†S ⋮ The iterative method for solving nonlinear matrix equation \(X^{s} + A^{*}X^{-t}A = Q\) ⋮ Algorithms for hyperbolic quadratic eigenvalue problems ⋮ Necessary and sufficient conditions for the existence of a positive definite solution for the matrix equation \(X + \sum^m_{i = 1} A^T_i X^{\delta_i} A_i = I\) ⋮ The structure-preserving doubling algorithm and convergence analysis for a nonlinear matrix equation ⋮ Positive definite solutions of the matrix equations ⋮ On the matrix equation ⋮ Two structure-preserving-doubling like algorithms to solve the positive definite solution of the equation \(X-A^{\mathrm{H}}\overline{X}^{-1}A=Q\) ⋮ An accelerated technique for solving the positive definite solutions of a class of nonlinear matrix equations ⋮ Some iterative methods for the largest positive definite solution to a class of nonlinear matrix equation ⋮ Solving large-scale nonlinear matrix equations by doubling ⋮ A closed-form estimator for the multivariate GARCH(1,1) model ⋮ The iterative methods for solving nonlinear matrix equation \(X+A^\star X^{-1}A+B^{\star}X^{-1}B=Q\) ⋮ On the first degree Fejér-Riesz factorization and its applications to \(X+A^{\ast}X^{-1}A=Q\) ⋮ Fixed-point accelerated iterative method to solve nonlinear matrix equation \(X - \sum\limits_{i = 1}^m A_i^\ast X^{-1}A_i = Q^\ast\) ⋮ The inversion-free iterative methods for a system of nonlinear matrix equations ⋮ Iterative and doubling algorithms for Riccati‐type matrix equations: A comparative introduction ⋮ The extremal solution of the matrix equation \(X^s+A^\ast X^{-q}A=I\) ⋮ Solutions and perturbation analysis for the nonlinear matrix equation \(X + \sum^m_{i=1} A^*_i X^{-1} A_i = I\) ⋮ A Lipschitz constant formula for vector addition in cones with applications to Stein-like equations ⋮ An accelerated cyclic-reduction-based solvent method for solving quadratic eigenvalue problem of gyroscopic systems ⋮ Positive definite solution of the matrix equation \(X = Q - A{^*}X^{-1}A + B{^*}X^{- 1}B\) via Bhaskar-Lakshmikantham fixed point theorem ⋮ A further study on a nonlinear matrix equation ⋮ The structure-preserving doubling algorithms for positive definite solution to a system of nonlinear matrix equations ⋮ An accelerated technique for solving one type of discrete-time algebraic Riccati equations ⋮ A note on the fixed-point iteration for the matrix equations \(X \pm A^* X^{-1}A=I\) ⋮ On positive definite solution of nonlinear matrix equations ⋮ Positive definite solutions and perturbation analysis of a class of nonlinear matrix equations ⋮ On convergence of three iterative methods for solving of the matrix equation \(X+A^\ast X^{-1}A+B^\ast X^{-1}B=Q\) ⋮ On Ando-Li-Mathias geometric mean equations ⋮ Positive definite solutions of the matrix equations \(X\pm A^{\ast}X^{-q} A=Q\) ⋮ Two structure-preserving-doubling like algorithms for obtaining the positive definite solution to a class of nonlinear matrix equation ⋮ Symmetric spaces with convex metrics ⋮ On the matrix equation \(X+A^{T} \root 2^m \of {X^{-1}}A+I\) ⋮ Complex symmetric stabilizing solution of the matrix equation \(X+A^{\top}X^{-1}A=Q\) ⋮ On equations that are equivalent to the nonlinear matrix equation \(X+A^{*}X ^{-\alpha}A=Q\) ⋮ Numerical solution of a quadratic eigenvalue problem ⋮ Decoupled low-rank iterative methods for a large-scale system of nonlinear matrix equations arising from electron transport of nano materials ⋮ Solving quadratic matrix equations and factoring polynomials: new fixed point iterations based on Schur complements of Toeplitz matrices ⋮ On the Hermitian positive definite solution of the nonlinear matrix equation ⋮ Unnamed Item ⋮ Perturbation analysis for the positive definite solution of the nonlinear matrix equation \(X-\sum_{i=1}^mA_i^\ast X^{-1}A_i=Q\) ⋮ Solution of a class of cross-coupled nonlinear matrix equations ⋮ Unnamed Item ⋮ On the Hermitian positive defnite solution of the nonlinear matrix equation \(X + A^*X ^{-1} A + B^*X ^{-1} B = I\) ⋮ Unnamed Item ⋮ Solution of a class of nonlinear matrix equations ⋮ Solution of a class of nonlinear matrix equations ⋮ Necessary and sufficient conditions for the existence of positive definite solutions of the matrix equationX+A^TX⁻²A=I ⋮ On the solution of the nonlinear matrix equation \(X^n=f(X)\) ⋮ Solving two generalized nonlinear matrix equations ⋮ Newton's iterative method to solve a nonlinear matrix equation ⋮ The cyclic reduction algorithm: From Poisson equation to stochastic processes and beyond. In memoriam of Gene H. Golub ⋮ A dynamically parameterized inversion-free iteration for a system of nonlinear matrix equation ⋮ The inversion-free iterative methods for solving the nonlinear matrix equation \(X + A^H X^{- 1} A + B^H X^{- 1} B = I\) ⋮ Positive definite solution of a class of nonlinear matrix equation ⋮ The inverse mean problem of geometric mean and contraharmonic means ⋮ On the matrix equation arising in an interpolation problem ⋮ SOME ITERATIVE ALGORITHMS FOR POSITIVE DEFINITE SOLUTION TO NONLINEAR MATRIX EQUATIONS ⋮ On the tripling algorithm for large-scale nonlinear matrix equations with low rank structure

Uses Software

• na12

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Positive solutions to \(X=A-BX^{-1}B^*\)
• Block Toeplitz matrices and preconditioning
• Some interlacing properties of the Schur complement of a Hermitian matrix
• Spectral and computational analysis of block Toeplitz matrices having nonnegative definite matrix-valued generating functions
• On the existence of a positive definite solution of the matrix equation \(X+A^ T X^{-1} A=I\)
• Computations with infinite Toeplitz matrices and polynomials
• Improved cyclic reduction for solving queueing problems
• New convergence results on functional iteration techniques for the numerical solution of M/G/1 type Markov chains
• Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation \(X+A^*X^{-1}A=Q\)
• On the matrix equation \(X+A^ TX^{-1}A=I\)
• Hermitian solutions of the equation \(X=Q+NX^{-1}N^*\)
• A note on two matrices occurring in the solution of quasi-birth-and-death processes
• Introduction to Matrix Analytic Methods in Stochastic Modeling
• Asymptotic Results on the Spectra of Block Toeplitz Preconditioned Matrices
• Newton's Method for Discrete Algebraic Riccati Equations when the Closed-Loop Matrix Has Eigenvalues on the Unit Circle
• Effective Methods for Solving Banded Toeplitz Systems
• Iterative solution of two matrix equations
• A logarithmic reduction algorithm for quasi-birth-death processes
• An invariant subspace approach in m/g/l and g/m/l type markov chains
• Solution of the Sylvester matrix equation AXB ^T + CXD ^T = E
• On the Solution of a Nonlinear Matrix Equation Arising in Queueing Problems
• Computing the Extremal Positive Definite Solutions of a Matrix Equation
• Asymptotic Spectra of Hermitian Block Toeplitz Matrices and Preconditioning Results
• Factorization of analytic functions by means of Koenig's theorem and Toeplitz computations

This page was built for publication: Efficient computation of the extreme solutions of $X+A^*X^{-1}A=Q$ and $X-A^*X^{-1}A=Q$