Perturbation analysis of the Hermitian positive definite solution of the matrix equation X−A*X−2A=I

Consider the nonlinear matrix equationwhere Q is an n × n positive definite matrix, C is an mn × mn positive semidefinite matrix, I is an m × m identity matrix, and A is an arbitrary mn × n matrix. This equation is connected with a certain interpolation problem when δ = −1. Using the properties of the Kronecker product and the theory for the monotonic operator defined in a normal cone, we prove the existence and uniqueness of the positive definite solution which is contained in the set {X|I ⊗ X > C} under the condition that I ⊗ Q > C. The iterative methods to compute the unique solution is proposed. Numerical examples show that the methods are feasible and effective.

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“…We have Y 0 = I ⊗ X 0 − C > 0 from X 0 ∈ ϕ(n). Let Y k = F k (Y 0 ), k = 1, 2, · · · , the map F defined by (5). We have the following inequality…”

Section: Theorem 31 Equationmentioning

confidence: 99%

“…See [1] and [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Equation (1) has been investigated in some special cases.…”

Section: Introductionmentioning

confidence: 99%

Positive definite solution of the matrix equation $\boldsymbol {X=Q+A^{H}(I\otimes X-C)^{\delta}A}$

2010

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“…Ivanov [10] disscused two iterative methods for solving (4). Cheng [11] derived a purterbation analysis of the Hermitian solution to (4). Also, Sayed [12] and Ivanov [2] introduced iterative methods for solving slightly different equations, X − A T X −n A = Q, with n ≥ 2 being an integer, and X + A T X −2 A = Q.…”

Section: Introductionmentioning

confidence: 99%

Computing Symmetric Positive Definite Solutions of Three Types of Nonlinear Matrix Equations

Bagherpour¹,

Mahdavi–Amiri²

2014

Preprint

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Nonlinear matrix equations arise in many practical contexts related to control theory, dynamical programming and finite element methods for solving some partial differential equations. In most of these applications, it is needed to compute a symmetric and positive definite solution. Here, we propose new iterative algorithms for solving three different types of nonlinear matrix equations. We have recently proposed a new algorithm for solving positive definite total least squares problems. Making use of an iterative process for inverse of a matrix, we convert the nonlinear matrix equation to an iterative linear one, and, in every iteration, we apply our algorithm for solving a positive definite total least squares problem to solve the linear subproblem and update the newly defined variables and the matrix inverse terms using appropriate formulas. Our proposed algorithms have a number of useful features. One is that the computed unknown matrix remains symmetric and positive definite in all iterations. As the second useful feature, numerical test results show that in most cases our proposed approach turns to compute solutions with smaller errors within lower computing times. Finally, we provide some test results showing that our proposed algorithm converges to a symmetric and positive definite solution in Matlab software environment on a PC, while other methods fail to do so.

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“…In the last few years there has been a constantly increasing interest in developing the theory and numerical approaches for HPD solutions to the nonlinear matrix equations of the form (1.1) [1][2][3][4][5][7][8][9][10][11][12][13]15,16,18]. Eq.…”

Section: Introductionmentioning

confidence: 99%

On Hermitian positive definite solution of the matrix equation X−∑i=1mAi∗
Duan
¹
,
Liao
²

2009
Journal of Computational and Applied Mathematics
23
0
2
0
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Perturbation analysis of the Hermitian positive definite solution of the matrix equation X−A*X−2A=I

Cited by 18 publications

References 8 publications

Positive definite solution of the matrix equation $\boldsymbol {X=Q+A^{H}(I\otimes X-C)^{\delta}A}$

Positive definite solution of the matrix equation $\boldsymbol {X=Q+A^{H}(I\otimes X-C)^{\delta}A}$

Computing Symmetric Positive Definite Solutions of Three Types of Nonlinear Matrix Equations

On Hermitian positive definite solution of the matrix equation X−∑i=1mAi∗
Duan
¹
,
Liao
²

2009
Journal of Computational and Applied Mathematics
23
0
2
0
View full text Add to dashboard Cite

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Perturbation analysis of the Hermitian positive definite solution of the matrix equation X−A*X−2A=I

Cited by 18 publications

References 8 publications

Positive definite solution of the matrix equation $\boldsymbol {X=Q+A^{H}(I\otimes X-C)^{\delta}A}$

Positive definite solution of the matrix equation $\boldsymbol {X=Q+A^{H}(I\otimes X-C)^{\delta}A}$

Computing Symmetric Positive Definite Solutions of Three Types of Nonlinear Matrix Equations

On Hermitian positive definite solution of the matrix equation X−∑i=1mAi∗Duan1, Liao2 2009Journal of Computational and Applied Mathematics23020View full textAdd to dashboardCite

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On Hermitian positive definite solution of the matrix equation X−∑i=1mAi∗
Duan
¹
,
Liao
²

2009
Journal of Computational and Applied Mathematics
23
0
2
0
View full text Add to dashboard Cite