# Computational Methods for Linear Matrix Equations

@article{Simoncini2016ComputationalMF, title={Computational Methods for Linear Matrix Equations}, author={Valeria Simoncini}, journal={SIAM Rev.}, year={2016}, volume={58}, pages={377-441} }

Given the square matrices $A, B, D, E$ and the matrix $C$ of conforming dimensions, we consider the linear matrix equation $A{\mathbf X} E+D{\mathbf X} B = C$ in the unknown matrix ${\mathbf X}$. Our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems, which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and… Expand

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#### References

SHOWING 1-10 OF 393 REFERENCES

Explicit Solutions of the Matrix Equation $\sum {A^i XD_i } = C$

- Mathematics, Computer Science
- SIAM J. Matrix Anal. Appl.
- 1992

A module theoretic approach is developed to study the linear matrix equation $\sum {A^i } XD_i = C$ which involves the companion matrix of the characteristic polynomial of A and the matrices of A are determined from an auxiliary equation. Expand

On the Separation of Two Matrices

- Mathematics
- 1979

The sensitivity of the solution X to the matrix equation $AX - XB = C$ is primarily dependent on the quantity ${\operatorname{sep}}(A,B)$ introduced by Stewart (1973) in connection with the… Expand

A Computational Method for Symmetric Stein Matrix Equations

- Mathematics
- 2011

In the present paper, we propose a numerical method for solving the sparse symmetric Stein equation \(AXA^T-X+BB^T=0.\) Such problems appear in control problems, filtering and image restoration. The… Expand

Krylov Subspace Methods for Large-Scale Constrained Sylvester Equations

- Mathematics, Computer Science
- SIAM J. Matrix Anal. Appl.
- 2013

A new formulation of the problem that entails the numerical solution of an unconstrained Sylvester equation is proposed and new enriched approximation spaces are proposed and experimental evidence of their effectiveness on benchmark problems is provided. Expand

An Algorithm for Generalized Matrix Eigenvalue Problems.

- Mathematics
- 1973

A new method, called the $QZ$ algorithm, is presented for the solution of the matrix eigenvalue problem $Ax = \lambda Bx$ with general square matrices A and B. Particular attention is paid to the… Expand

A numerical algorithm for solving the matrix equation AX + XTB = C1

- Mathematics
- 2011

An algorithm of the Bartels-Stewart type for solving the matrix equation AX + XTB = C is proposed. By applying the QZ algorithm, the original equation is reduced to an equation of the same type… Expand

Error Estimates and Evaluation of Matrix Functions via the Faber Transform

- Mathematics, Computer Science
- SIAM J. Numer. Anal.
- 2009

This paper describes how the Faber transform applied to the field of values of A can be used to determine improved error bounds for popular polynomial approximation methods based on the Arnoldi process. Expand

Preconditioned Krylov Subspace Methods for Lyapunov Matrix Equations

- Mathematics, Computer Science
- SIAM J. Matrix Anal. Appl.
- 1995

It is proven that this is the case for alternating direction implicit (ADI)-type and (point) symmetric successive overrelaxation (SSOR) preconditioning in association with the quasiminimal residual (QMR) method. Expand

NUMERICAL SOLUTION OF DISCRETE STABLE LINEAR MATRIX EQUATIONS ON MULTICOMPUTERS

- Mathematics, Computer Science
- Parallel Algorithms Appl.
- 2002

This work investigates the parallel performance of numerical algorithms for solving discrete Sylvester and Stein equations as they appear in discrete-time control problems, filtering, and image restoration on distributed-memory multicomputers. Expand

A Multigrid Method to Solve Large Scale Sylvester Equations

- Mathematics, Computer Science
- SIAM J. Matrix Anal. Appl.
- 2007

A multigrid algorithm is developed that computes the solution of the Sylvester equation in a data-sparse format and is of complexity $\mathcal{O}(n+m)k^{2})$. Expand