Arnoldi iteration and Chebyshev polynomials
(2021) In Bachelor's Theses in Mathematical Sciences NUMK11 20211Mathematics (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:
http://lup.lub.lu.se/student-papers/record/9066866
- author
- Ufymtseva, Kateryna LU
- supervisor
-
- Claus Führer LU
- organization
- course
- NUMK11 20211
- year
- 2021
- 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}}, }