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A review of the Kaczmarz method

Lokrantz, Carl LU (2023) NUMK11 20212
Centre for Mathematical Sciences
Mathematics (Faculty of Sciences)
Abstract
The Kaczmarz method is an iterative method for solving linear systems of equations. The Kaczmarz method has been around since it was developed by Kaczmarz 1937. The main idea behind the original Kaczmarz method is to orthogonally project the previous x_k onto the solution space given by a row of the system. The block Kaczmarz on the other hand orthogonally projects the previous x_k onto the solution space given by a sub system of equations. Both the original Kaczmarz method and block Kaczmarz method can only solve consistent systems, however the extended Kaczmarz method is an adaptation that makes it possible to solve inconsistent systems.
We will look at both deterministic and randomized, row and block selection processes then compare... (More)
The Kaczmarz method is an iterative method for solving linear systems of equations. The Kaczmarz method has been around since it was developed by Kaczmarz 1937. The main idea behind the original Kaczmarz method is to orthogonally project the previous x_k onto the solution space given by a row of the system. The block Kaczmarz on the other hand orthogonally projects the previous x_k onto the solution space given by a sub system of equations. Both the original Kaczmarz method and block Kaczmarz method can only solve consistent systems, however the extended Kaczmarz method is an adaptation that makes it possible to solve inconsistent systems.
We will look at both deterministic and randomized, row and block selection processes then compare them on both consistent and inconsistent systems of equations. (Less)
Please use this url to cite or link to this publication:
author
Lokrantz, Carl LU
supervisor
organization
alternative title
En genomgång av Kaczmarz metoden
course
NUMK11 20212
year
type
M2 - Bachelor Degree
subject
keywords
Kaczmarz method, Least squares, Image reconstruction, Iterative method, Linear system of equations, Projection, Numerical Linear Algebra, Block Kaczmarz, Greedy Kaczmarz, Randomized Kaczmarz
report number
LUNFNA-4050-2023
ISSN
1654-6229
language
English
id
9130754
date added to LUP
2023-10-24 09:13:29
date last changed
2023-10-24 09:13:29
@misc{9130754,
  abstract     = {{The Kaczmarz method is an iterative method for solving linear systems of equations. The Kaczmarz method has been around since it was developed by Kaczmarz 1937. The main idea behind the original Kaczmarz method is to orthogonally project the previous x_k onto the solution space given by a row of the system. The block Kaczmarz on the other hand orthogonally projects the previous x_k onto the solution space given by a sub system of equations. Both the original Kaczmarz method and block Kaczmarz method can only solve consistent systems, however the extended Kaczmarz method is an adaptation that makes it possible to solve inconsistent systems.
 We will look at both deterministic and randomized, row and block selection processes then compare them on both consistent and inconsistent systems of equations.}},
  author       = {{Lokrantz, Carl}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  title        = {{A review of the Kaczmarz method}},
  year         = {{2023}},
}