Characterization and construction of the nearest defective matrix via coalescence of pseudospectral components

Alam, Rafikul; Bora, Shreemayee; Byers, Ralph; Overton, Michael L.

doi:10.1016/j.laa.2010.09.022

Cited by 18 publications

(26 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If A is normal with distinct eigenvalues, then Λ ε (A) is the union of n disks with radius ε centered at the eigenvalues of A, and it is easy to see that the perpendicular bisector of the line segment joining the two eigenvalues closest to each other contains an interval of points z for which σ n (A−zI) is a double singular value, but these points are not maximizers of (1.5). See [ABBO11] for some examples of nonnormal matrices for which double singular values of A − zI occur, but again such points z are not local maximizers. See also [LP08] for an extensive study of mathematical properties of the pseudospectrum.…”

Section: Basic Results Definitions and An Assumptionmentioning

confidence: 99%

Fast Algorithms for the Approximation of the Pseudospectral Abscissa and Pseudospectral Radius of a Matrix

Guglielmi¹,

Overton²

2011

SIAM J. Matrix Anal. & Appl.

Self Cite

116

View full text Add to dashboard Cite

The ε-pseudospectral abscissa and radius of an n × n matrix are, respectively, the maximal real part and the maximal modulus of points in its ε-pseudospectrum, defined using the spectral norm. Existing techniques compute these quantities accurately, but the cost is multiple singular value decompositions and eigenvalue decompositions of order n, making them impractical when n is large. We present new algorithms based on computing only the spectral abscissa or radius of a sequence of matrices, generating a sequence of lower bounds for the pseudospectral abscissa or radius. We characterize fixed points of the iterations, and we discuss conditions under which the sequence of lower bounds converges to local maximizers of the real part or modulus over the pseudospectrum, proving a locally linear rate of convergence for ε sufficiently small. The convergence results depend on a perturbation theorem for the normalized eigenprojection of a matrix as well as a characterization of the group inverse (reduced resolvent) of a singular matrix defined by a rankone perturbation. The total cost of the algorithms is typically only a constant times the cost of computing the spectral abscissa or radius, where the value of this constant usually increases with ε, and may be less than 10 in many practical cases of interest.

show abstract

Section: Basic Results Definitions and An Assumptionmentioning

confidence: 99%

Fast Algorithms for the Approximation of the Pseudospectral Abscissa and Pseudospectral Radius of a Matrix

Guglielmi¹,

Overton²

2011

SIAM J. Matrix Anal. & Appl.

Self Cite

116

View full text Add to dashboard Cite

show abstract

“…Let bd Λ ε (A) denote the boundary of the pseudospectrum of A. We now state our main result on the resolvent-critical points of bd Λ ε (A), which is based on a result in [ABBO10]. Proof.…”

Section: Previous Results and Notationmentioning

confidence: 99%

Some Regularity Results for the Pseudospectral Abscissa and Pseudospectral Radius of a Matrix

Gürbüzbalaban¹,

Overton²

2012

SIAM J. Optim.

Self Cite

View full text Add to dashboard Cite

Abstract. The ε-pseudospectral abscissa αε and radius ρε of an n × n matrix are, respectively, the maximal real part and the maximal modulus of points in its ε-pseudospectrum, defined using the spectral norm. It was proved in [A.S. Lewis and C.H.J. Pang, SIAM J. Optim., 19 (2008), pp. 1048-1072] that for fixed ε > 0, αε and ρε are Lipschitz continuous at a matrix A except when αε and ρε are attained at a critical point of the norm of the resolvent (in the nonsmooth sense), and it was conjectured that the points where αε and ρε are attained are not resolvent-critical. We prove this conjecture, which leads to the new result that αε and ρε are Lipschitz continuous, and also establishes the Aubin property with respect to both ε and A of the ε-pseudospectrum for the points z ∈ C where αε and ρε are attained. Finally, we give a proof showing that the pseudospectrum can never be "pointed outwards."

show abstract

“…By using the procedure described in [ABBO11] we identify as initial eigenvalues, candidate to coalesce, λ 1 = 1.416177710 + 1.260523165 i , λ 2 = 0.338991381 + 0.455810180 i . Compute γ k and ε 0, * k by (7.19)…”

Section: Example 1 Consider the Complex Matrixmentioning

confidence: 99%

“…In the recent and very interesting paper by Alam, Bora, Byers and Overton [ABBO11], which provides an extensive historical analysis of the problem, the authors have shown that when K = C the infimum in (1.1) is actually a minimum. Furthermore, in the same paper, the authors have proposed a computational approach to approximate the nearest defective matrix by a variant of Newton's method.…”

Section: Introductionmentioning

confidence: 99%

Differential Equations for Real-Structured Defectivity Measures

Buttà¹,

Guglielmi²,

Manetta³

et al. 2015

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

Let A be a real matrix with all distinct eigenvalues. We propose a new method for the computation of the distance w_R(A) of the matrix A from the set of real defective matrices, i.e., the set of those real matrices with at least one multiple eigenvalue with algebraic multiplicity larger than its geometric multiplicity. For 0 < ε ≤ w_R(A), this problem is closely related to the computation of the most ill-conditioned ε-pseudoeigenvalues of A, that is, points in the ε-pseudospectrum of A characterized by the highest condition number. The method we propose couples a system of differential equations on a low-rank manifold which determines the ε-pseudoeigenvalue closest to coalescence, with a fast Newton-like iteration aiming to determine the minimal value ε such that an ε- pseudoeigenvalue becomes defective. The method has a local behavior; this means that, in general, we find upper bounds for w_R(A). However, these bounds usually provide good approximations, in those (simple) cases where we can check this. The methodology can be extended to a structured matrix, where it is required that the distance be computed within some manifold defining the structure of the matrix. In this paper we extensively examine the case of real matrices. As far as we know, there do not exist methods in the literature able to compute such distance

show abstract

Characterization and construction of the nearest defective matrix via coalescence of pseudospectral components

Cited by 18 publications

References 14 publications

Fast Algorithms for the Approximation of the Pseudospectral Abscissa and Pseudospectral Radius of a Matrix

Fast Algorithms for the Approximation of the Pseudospectral Abscissa and Pseudospectral Radius of a Matrix

Some Regularity Results for the Pseudospectral Abscissa and Pseudospectral Radius of a Matrix

Differential Equations for Real-Structured Defectivity Measures

Contact Info

Product

Resources

About