A Relaxed Gradient Based Algorithm for Solving Extended S ylvester‐Conjugate Matrix Equations

Self Cite

In this paper, we present a modified gradient-based algorithm for solving extended Sylvester-conjugate matrix equations. The idea is from the gradient-based method introduced in [14] and the relaxed gradient-based algorithm proposed in [16]. The convergence analysis of the algorithm is investigated. We show that the iterative solution converges to the exact solution for any initial value based on some appropriate assumptions. A numerical example is given to illustrate the effectiveness of the proposed method and to test its efficiency and accuracy compared with those presented in [14] and [16].

“…Ramadan et al presented the following algorithm for solving equation .

X_{1} ((), k) = X_{1} ((), k - 1) + μ ((), 1 - ω) A^{H} ((), F - {AX}_{1} ((), k - 1) B - C \overset{X_{1} ((), k - 1)}{true} D) B^{H}

X_{2} ((), k) = X_{2} ((), k - 1) + italicμω {\overset{true¯}{C}}^{H} ((), \overset{F}{true} - \overset{{AX}_{1} ((), k - 1) B}{true} - \overset{C}{true} X ((), k - 1) \overset{D}{true}) {\overset{true¯}{D}}^{H}

as the approximate solution X ( k ) is wanted rather than X 1 ( k ) and X 2 ( k ), so we propose the following balanced strategy to form the k − th approximate solution

X ((), k) = ω X_{1} ((), k) + ((), 1 - ω) X_{2} ((), k)

where ω is a relaxation parameter satisfying 0 < ω < 1, it controls the relative importance of two residual matrices.…”

Section: Preliminariesmentioning

confidence: 99%

“…

X_{1} ((), k) = X_{1} ((), k - 1) + μ ((), 1 - ω) A^{H} ((), F - {AX}_{1} ((), k - 1) B - C \overset{X_{1} ((), k - 1)}{true} D) B^{H}

X_{2} ((), k) = X_{2} ((), k - 1) + italicμω {\overset{true¯}{C}}^{H} ((), \overset{F}{true} - \overset{{AX}_{1} ((), k - 1) B}{true} - \overset{C}{true} X ((), k - 1) \overset{D}{true}) {\overset{true¯}{D}}^{H}

as the approximate solution X ( k ) is wanted rather than X 1 ( k ) and X 2 ( k ), so we propose the following balanced strategy to form the k − th approximate solution

X ((), k) = ω X_{1} ((), k) + ((), 1 - ω) X_{2} ((), k)

0 < μ < \frac{2}{ω ((), 1 - ω) ((), λ_{\max} ((), {italicAA}^{H}) λ_{\max} ((), B^{H} B) + λ_{\max} ((), \overset{C}{true} {\overset{true¯}{C}}^{H}) λ_{\max} ((), {\overset{true¯}{D}}^{H} \overset{D}{true}) + 2 σ_{\max} ((), \overset{D}{true}))}

…”

Section: Preliminariesmentioning

confidence: 99%

italicAXB + C \overset{X}{true} D = F

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A modified gradient‐based algorithm for solving extended Sylvester‐conjugate matrix equations

Ramadan

Bayoumi

2017

Self Cite

“…Such problems arise in the solution of large eigenvalue problems and in the boundary value problems . They play an important role in linear control and filtering theory for continuous or discrete‐time large‐scale dynamical systems, image restoration and processing, and other applications such as model reduction, numerical solution of matrix differential Riccati equations, and many more; see .…”

Section: Introductionmentioning

confidence: 99%

The Unified Frame of Alternating Direction Method of Multipliers for Three Classes of Matrix Equations Arising in Control Theory

2017

In this paper, the unified frame of alternating direction method of multipliers (ADMM) is proposed for solving three classes of matrix equations arising in control theory including the linear matrix equation, the generalized Sylvester matrix equation and the quadratic matrix equation. The convergence properties of ADMM and numerical results are presented. The numerical results show that ADMM tends to deliver higher quality solutions with less computing time on the tested problems.

Explicit and Iterative Methods for Solving the Matrix Equation AV + BW = EVF + C

Ramadan

Bayoumi

2014

Self Cite

In this paper, we consider explicit and iterative methods for solving the Generalized Sylvester matrix equation AV + BW = EVF + C. Based on the use of Kronecker map and Sylvester sum some lemmas and theorems are stated and proved where explicit and iterative solutions are obtained. The proposed methods are illustrated by numerical example. The obtained results show that the methods are very neat and efficient.