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Approximating extrema of quadratic forms using Krylov subspaces

Hallborn, Simon LU (2017) In Bachelor's Theses in Mathematical Sciences NUMK01 20171
Mathematics (Faculty of Engineering)
Abstract
In this thesis approximations to quadratic forms using Krylov subspaces are presented. These quantities are approximations of the norm and logarithmic norm of a matrix. In order to find these quantities an eigenvalue problem has to be solved, but because of limited storage and time, this is not feasible in practice with large matrices. Instead one can project this problem down to a problem of smaller size with a Krylov subspace and solve it there instead.

To test the quality of these projections, different matrices that arise in practice are tested and their norms approximated. The test matrices have known norms and behaviour so the result can be interpreted.

Overall the results show that one can obtain two digit accuracy with a low... (More)
In this thesis approximations to quadratic forms using Krylov subspaces are presented. These quantities are approximations of the norm and logarithmic norm of a matrix. In order to find these quantities an eigenvalue problem has to be solved, but because of limited storage and time, this is not feasible in practice with large matrices. Instead one can project this problem down to a problem of smaller size with a Krylov subspace and solve it there instead.

To test the quality of these projections, different matrices that arise in practice are tested and their norms approximated. The test matrices have known norms and behaviour so the result can be interpreted.

Overall the results show that one can obtain two digit accuracy with a low dimension of the subspace, even for matrices with large dimensions, which is truly promising. (Less)
Popular Abstract (Swedish)
Avhandlingen syftar till att undersöka om det är mjöligt att finna billiga approximationer till extremvärdena för kvadratiska former. Dessa extremvärden kan t.ex. användas för feluppskattningar i tidsstegningsmetoder för lösning av differentialekvationer, men då extremvärdena måste uppdateras varje steg måste metoden vara billig. Här undersöks dessutom "stora" matriser och möjligheten att finna de billiga approximationerna på Krylovrum, då denna information finns tillgänglig.
Please use this url to cite or link to this publication:
author
Hallborn, Simon LU
supervisor
organization
course
NUMK01 20171
year
type
M2 - Bachelor Degree
subject
keywords
Approximations, Quadratic forms, Krylov subspaces, Eigenvalue problems, Norms, Logarithmic norms
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFNA-4016-2017
ISSN
1654-6229
other publication id
2017:K17
language
English
id
8926601
date added to LUP
2017-12-01 15:31:35
date last changed
2017-12-01 15:41:27
@misc{8926601,
  abstract     = {In this thesis approximations to quadratic forms using Krylov subspaces are presented. These quantities are approximations of the norm and logarithmic norm of a matrix. In order to find these quantities an eigenvalue problem has to be solved, but because of limited storage and time, this is not feasible in practice with large matrices. Instead one can project this problem down to a problem of smaller size with a Krylov subspace and solve it there instead.

To test the quality of these projections, different matrices that arise in practice are tested and their norms approximated. The test matrices have known norms and behaviour so the result can be interpreted.

Overall the results show that one can obtain two digit accuracy with a low dimension of the subspace, even for matrices with large dimensions, which is truly promising.},
  author       = {Hallborn, Simon},
  issn         = {1654-6229},
  keyword      = {Approximations,Quadratic forms,Krylov subspaces,Eigenvalue problems,Norms,Logarithmic norms},
  language     = {eng},
  note         = {Student Paper},
  series       = {Bachelor's Theses in Mathematical Sciences},
  title        = {Approximating extrema of quadratic forms using Krylov subspaces},
  year         = {2017},
}