Approximating the weighted pseudospectra of large matrices

Astudillo, R.; Castillo, Zenaida

doi:10.1016/j.mcm.2011.03.029

Cited by 1 publication

(2 citation statements)

References 13 publications

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“…Thus, pseudospectra can be readily approximated in physically relevant norms using the invariant subspace V deriving from a standard 2-norm linear stability analysis. Related ideas for approximating standard pseudospectra in weighted norms are described by Astudillo and Castillo [2]. We also note that the new reduced basis techniques for standard pseudospectra of Sirković [42] also hold great promise for estimating σ ε (A, E).…”

Section: Alternative Normsmentioning

confidence: 76%

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Pseudospectra of Matrix Pencils for Transient Analysis of Differential-Algebraic Equations

Embree¹,

Keeler²

2017

SIAM J. Matrix Anal. & Appl.

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To understand the solution of a linear, time-invariant differential-algebraic equation, one must analyze a matrix pencil (A, E) with singular E. Even when this pencil is stable (all its finite eigenvalues fall in the left-half plane), the solution can exhibit transient growth before its inevitable decay. When the equation results from the linearization of a nonlinear system, this transient growth gives a mechanism that can promote nonlinear instability. One can enrich the conventional large-scale eigenvalue calculation used for linear stability analysis to identify the potential for such transient growth. Toward this end, we introduce a new definition of the pseudospectrum of a matrix pencil, use it to bound transient growth, explain how to incorporate a physically-relevant norm, and derive approximate pseudospectra using the invariant subspace computed in conventional linear stability analysis. We apply these tools to several canonical test problems in fluid mechanics, an important source of differential-algebraic equations.

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Section: Alternative Normsmentioning

confidence: 76%

“…(The examples from fluid dynamics considered in section 6 have index 2.) Consider again the (A, E) pair in the example (2). The pencil has spectrum σ(A, E) = {−1, ∞}, where λ = −1 has algebraic multiplicity two.…”

mentioning

confidence: 99%

Pseudospectra of Matrix Pencils for Transient Analysis of Differential-Algebraic Equations

Embree¹,

Keeler²

2017

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

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Approximating the weighted pseudospectra of large matrices

Cited by 1 publication

References 13 publications

Pseudospectra of Matrix Pencils for Transient Analysis of Differential-Algebraic Equations

Pseudospectra of Matrix Pencils for Transient Analysis of Differential-Algebraic Equations

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