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A study of the s-step biconjugate gradient method

Boi, Carlotta LU (2022) In Bachelor's Theses in Mathematical Sciences NUMK11 20221
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
Abstract
In this thesis we will examine how to solve linear systems using the s-step biconjugate gradient algorithm, which is an iterative method based on the Krylov subspaces. It is useful especially when we have a large and sparse matrix. We begin looking over the biconjugate gradient algorithm (BiCG), in order to understand how to construct the s-step BiCG algorithm. We will go through some numerical examples to see which method can give a better numerical solution and which one is able to converge. At the end we will talk about finite precision arithmetic and study roundoff errors of the s-step BiCG method.
Popular Abstract
Iterative algorithms are important methods to make of solutions for systems of linear equations. They do it by creating a succession of approximate solutions which can drive the user to a solution that can be closer to the exact one. These methods are valuable in different fields of science, for instance materials science and statistics. One of the most known iterative techniques are the Krylov subspace methods (KSMs). This thesis focuses on an algorithm based on the KSMs, named s-step biconjugate method, which is very useful especially for decreasing the communication costs caused by exchanging information among different levels of computer storage and among different devices. But this comes with a price: as we increment the s number for... (More)
Iterative algorithms are important methods to make of solutions for systems of linear equations. They do it by creating a succession of approximate solutions which can drive the user to a solution that can be closer to the exact one. These methods are valuable in different fields of science, for instance materials science and statistics. One of the most known iterative techniques are the Krylov subspace methods (KSMs). This thesis focuses on an algorithm based on the KSMs, named s-step biconjugate method, which is very useful especially for decreasing the communication costs caused by exchanging information among different levels of computer storage and among different devices. But this comes with a price: as we increment the s number for minimizing the price of transferring information, we can experience side effects like the decrease of precision of the solution computed by the algorithm, or the increase of the number of iterations for arriving at a solution. In this thesis we will explore these side effects and compare our results to another iterative technique named biconjugate gradient method, which is the technique used for building the s-step method. (Less)
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author
Boi, Carlotta LU
supervisor
organization
course
NUMK11 20221
year
type
M2 - Bachelor Degree
subject
keywords
biconjugate gradient methods, s-step biconjugate gradient methods, nonsymmetric linear systems, sparse matrices.
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFNA-4041-2022
ISSN
1654-6229
other publication id
2022:K14
language
English
id
9088696
date added to LUP
2022-08-25 13:57:51
date last changed
2022-08-25 13:57:51
@misc{9088696,
  abstract     = {{In this thesis we will examine how to solve linear systems using the s-step biconjugate gradient algorithm, which is an iterative method based on the Krylov subspaces. It is useful especially when we have a large and sparse matrix. We begin looking over the biconjugate gradient algorithm (BiCG), in order to understand how to construct the s-step BiCG algorithm. We will go through some numerical examples to see which method can give a better numerical solution and which one is able to converge. At the end we will talk about finite precision arithmetic and study roundoff errors of the s-step BiCG method.}},
  author       = {{Boi, Carlotta}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor's Theses in Mathematical Sciences}},
  title        = {{A study of the s-step biconjugate gradient method}},
  year         = {{2022}},
}