Structured Linear Algebra Problems in Digital Signal Processing

Dooren, Paul M. Van

doi:10.1007/978-3-642-75536-1_17

Cited by 15 publications

(11 citation statements)

References 22 publications

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“…Many applications lead to eigenproblems containing matrices with linear structure, and only perturbations that preserve the structure may be physically meaningful [31]. For example, the quadratic eigenvalue problem has the form (λ 2 C + λD + E)v = 0, C, D, E ∈ C n×n .…”

Section: Introductionmentioning

confidence: 99%

Structured Backward Error and Condition of Generalized Eigenvalue Problems

Higham¹,

Higham²

1998

SIAM J. Matrix Anal. & Appl.

111

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Backward errors and condition numbers are defined and evaluated for eigenvalues and eigenvectors of generalized eigenvalue problems. Both normwise and componentwise measures are used. Unstructured problems are considered first, and then the basic definitions are extended so that linear structure in the coefficient matrices (for example, Hermitian, Toeplitz, Hamiltonian, or band structure) is preserved by the perturbations.

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Section: Introductionmentioning

confidence: 99%

Structured Backward Error and Condition of Generalized Eigenvalue Problems

Higham¹,

Higham²

1998

SIAM J. Matrix Anal. & Appl.

111

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“…Applications. Matrix equations are ubiquitous in signal processing, control and system theory; see, e.g., [4], [258], [91], [70], [23], [28], [224], [63] and references therein. Most time-dependent models may be represented as linear or non-linear dynamical systems, accounting for the prediction, simulation and control of real world phenomena.…”

mentioning

confidence: 99%

Computational Methods for Linear Matrix Equations

Simoncini¹

2016

SIAM Rev.

401

434

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Given the matrices A, B, C, D and E of conforming dimensions, we consider the linear matrix equation AXE + DXB = C in the unknown matrix X. Our aim is to provide an overview of the major algorithmic developments that have taken place in the past few decades in the numerical solution of this and of related problems, which are becoming a reliable tool in the numerical formulation of advanced application models.

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“…Recent work that makes use of componentwise backward error includes [1], [2], [13], [15], [16]. [4] or Van Dooren [22] for more detailed discussion of the desirability of preserving matrix structure in definitions of backward error; Van Dooren also discusses structured condition numbers and describes various structured linear algebra problems that arise in signal processing.…”

mentioning

confidence: 99%

Backward Error and Condition of Structured Linear Systems

Higham¹,

Higham

1992

SIAM J. Matrix Anal. & Appl.

102

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Abstract. Existing definitions of backward error and condition number for linear systems do not cater to structure in the coefficient matrix, except possibly for sparsity. The definitions are extended so that when the coefficient matrix has structure the perturbed matrix has this structure too. It is shown that when the structure comprises linear dependence on a set of parameters, the structured componentwise backward error is given by the solution of minimal cx)-norm to an underdetermined linear system; an explicit expression for the condition number in this linear case is also obtained. Applications to symmetric matrices, Toeplitz matrices and the least squares problem are discussed and illustrated through numerical examples.

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Structured Linear Algebra Problems in Digital Signal Processing

Cited by 15 publications

References 22 publications

Structured Backward Error and Condition of Generalized Eigenvalue Problems

Structured Backward Error and Condition of Generalized Eigenvalue Problems

Computational Methods for Linear Matrix Equations

Backward Error and Condition of Structured Linear Systems

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