A study of the s-step biconjugate gradient method
(2022) In Bachelor's Theses in Mathematical Sciences NUMK11 20221Mathematics (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)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9088696
- author
- Boi, Carlotta LU
- supervisor
- organization
- course
- NUMK11 20221
- year
- 2022
- 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}}, }