The interval eigenvalue problem: (Review article)

Mohamed A. Shalaby

The interval eigenvalue problem: (Review article)

Mohamed A. Shalaby^*

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

2 Scopus citations

Abstract

The interval eigenvalue problem A^I x = λx is discussed with an extensive survey to methods of solution available in the literature. The solution of such a problem is to apply the techniques of interval analysis on interval matrices to answer three main questions:(i)What are the location of the eigenvalues of an interval matrix?, (ii) How does the spectrum of an interval matrix depend on the spectrum of its end matrices? And (iii) how to compute the exact lower and upper bounds for every eigenpair of an interval matrix?. The stability of interval matrices is also defined, presented, and reviewed. Some applications and related topics are also presented. These applications are: Pole Assignment Problem (PAP), Telerance Analysis Problem (TAP), atomic physics, structural analysis, vibrations and robotics. The related topics are: enclosures of eigenpairs, Singular Value Decomposition (SVD), programming languages and applications on interval computation (optimzation, identification parallelism and linear porgramming).

Original language	English (US)
Title of host publication	European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2000
State	Published - 2000
Externally published	Yes
Event	European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2000 - Barcelona, Spain Duration: Sep 11 2000 → Sep 14 2000

Other

Other	European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2000
Country	Spain
City	Barcelona
Period	09/11/00 → 09/14/00

Keywords

Eigenvalue bounds (65G99).
Eigenvalues and eigenvectors (65 F15)
Interval arithmetics (65G10)
Linear interval equations (65F10)

ASJC Scopus subject areas

Artificial Intelligence
Applied Mathematics

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title = "The interval eigenvalue problem: (Review article)",

abstract = "The interval eigenvalue problem AI x = λx is discussed with an extensive survey to methods of solution available in the literature. The solution of such a problem is to apply the techniques of interval analysis on interval matrices to answer three main questions:(i)What are the location of the eigenvalues of an interval matrix?, (ii) How does the spectrum of an interval matrix depend on the spectrum of its end matrices? And (iii) how to compute the exact lower and upper bounds for every eigenpair of an interval matrix?. The stability of interval matrices is also defined, presented, and reviewed. Some applications and related topics are also presented. These applications are: Pole Assignment Problem (PAP), Telerance Analysis Problem (TAP), atomic physics, structural analysis, vibrations and robotics. The related topics are: enclosures of eigenpairs, Singular Value Decomposition (SVD), programming languages and applications on interval computation (optimzation, identification parallelism and linear porgramming).",

keywords = "Eigenvalue bounds (65G99)., Eigenvalues and eigenvectors (65 F15), Interval arithmetics (65G10), Linear interval equations (65F10)",

author = "Shalaby, {Mohamed A.}",

year = "2000",

language = "English (US)",

isbn = "8489925704",

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note = "European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2000 ; Conference date: 11-09-2000 Through 14-09-2000",

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T1 - The interval eigenvalue problem

T2 - European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2000

AU - Shalaby, Mohamed A.

PY - 2000

Y1 - 2000

N2 - The interval eigenvalue problem AI x = λx is discussed with an extensive survey to methods of solution available in the literature. The solution of such a problem is to apply the techniques of interval analysis on interval matrices to answer three main questions:(i)What are the location of the eigenvalues of an interval matrix?, (ii) How does the spectrum of an interval matrix depend on the spectrum of its end matrices? And (iii) how to compute the exact lower and upper bounds for every eigenpair of an interval matrix?. The stability of interval matrices is also defined, presented, and reviewed. Some applications and related topics are also presented. These applications are: Pole Assignment Problem (PAP), Telerance Analysis Problem (TAP), atomic physics, structural analysis, vibrations and robotics. The related topics are: enclosures of eigenpairs, Singular Value Decomposition (SVD), programming languages and applications on interval computation (optimzation, identification parallelism and linear porgramming).

AB - The interval eigenvalue problem AI x = λx is discussed with an extensive survey to methods of solution available in the literature. The solution of such a problem is to apply the techniques of interval analysis on interval matrices to answer three main questions:(i)What are the location of the eigenvalues of an interval matrix?, (ii) How does the spectrum of an interval matrix depend on the spectrum of its end matrices? And (iii) how to compute the exact lower and upper bounds for every eigenpair of an interval matrix?. The stability of interval matrices is also defined, presented, and reviewed. Some applications and related topics are also presented. These applications are: Pole Assignment Problem (PAP), Telerance Analysis Problem (TAP), atomic physics, structural analysis, vibrations and robotics. The related topics are: enclosures of eigenpairs, Singular Value Decomposition (SVD), programming languages and applications on interval computation (optimzation, identification parallelism and linear porgramming).

KW - Eigenvalue bounds (65G99).

KW - Eigenvalues and eigenvectors (65 F15)

KW - Interval arithmetics (65G10)

KW - Linear interval equations (65F10)

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M3 - Conference contribution

AN - SCOPUS:84893344778

SN - 8489925704

SN - 9788489925700

BT - European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2000

Y2 - 11 September 2000 through 14 September 2000

ER -

The interval eigenvalue problem: (Review article)

Abstract

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Keywords

ASJC Scopus subject areas

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