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Arnoldi iteration and Chebyshev polynomials

Ufymtseva, Kateryna LU (2021) In Bachelor's Theses in Mathematical Sciences NUMK11 20211
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
Abstract
In this thesis, we examine the Arnoldi iteration - an iterative algorithm used for finding a Hessenberg form of a matrix as well as approximating its eigenvalues by forming an orthonormal basis of a Krylov subspace. We explore the mechanism behind the work of the algorithm and how the values it finds approximate the eigenvalues.
In the process of answering these questions we consider the concepts of the ideal Arnoldi approximation problem, ideal Arnoldi polynomial and Chebyshev polynomial of a matrix, and how they are related to the original Arnoldi approximation problem, which is solved by the iteration. We also look into the pseudospectra of a matrix, its connection to eigenvalue estimates, also known as Ritz values, and Chebyshev... (More)
In this thesis, we examine the Arnoldi iteration - an iterative algorithm used for finding a Hessenberg form of a matrix as well as approximating its eigenvalues by forming an orthonormal basis of a Krylov subspace. We explore the mechanism behind the work of the algorithm and how the values it finds approximate the eigenvalues.
In the process of answering these questions we consider the concepts of the ideal Arnoldi approximation problem, ideal Arnoldi polynomial and Chebyshev polynomial of a matrix, and how they are related to the original Arnoldi approximation problem, which is solved by the iteration. We also look into the pseudospectra of a matrix, its connection to eigenvalue estimates, also known as Ritz values, and Chebyshev polynomials. (Less)
Please use this url to cite or link to this publication:
author
Ufymtseva, Kateryna LU
supervisor
organization
course
NUMK11 20211
year
type
M2 - Bachelor Degree
subject
keywords
Arnoldi iteration, Chebyshev polynomials, Ritz values, eigenvalues, Krylov subspaces, pseudospectra
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFNA-4055-2021
ISSN
1654-6229
other publication id
2021:K46
language
English
id
9066866
date added to LUP
2024-04-15 17:09:17
date last changed
2024-04-15 17:09:17
@misc{9066866,
  abstract     = {{In this thesis, we examine the Arnoldi iteration - an iterative algorithm used for finding a Hessenberg form of a matrix as well as approximating its eigenvalues by forming an orthonormal basis of a Krylov subspace. We explore the mechanism behind the work of the algorithm and how the values it finds approximate the eigenvalues.
In the process of answering these questions we consider the concepts of the ideal Arnoldi approximation problem, ideal Arnoldi polynomial and Chebyshev polynomial of a matrix, and how they are related to the original Arnoldi approximation problem, which is solved by the iteration. We also look into the pseudospectra of a matrix, its connection to eigenvalue estimates, also known as Ritz values, and Chebyshev polynomials.}},
  author       = {{Ufymtseva, Kateryna}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor's Theses in Mathematical Sciences}},
  title        = {{Arnoldi iteration and Chebyshev polynomials}},
  year         = {{2021}},
}