Chebyshev Semi−iteration in Preconditioning
It is widely believed that Krylov subspace iterative methods are better than Chebyshev semi-iterative methods. When the solution of a linear system with a symmetric and positive definite coefficient matrix is required then the Conjugate Gradient method will compute the optimal approximate solution f...
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| Format: | Report |
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Oxford University Computing Laboratory
2008
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| _version_ | 1797074051876782080 |
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| author | Wathen, T |
| author_facet | Wathen, T |
| author_sort | Wathen, T |
| collection | OXFORD |
| description | It is widely believed that Krylov subspace iterative methods are better than Chebyshev semi-iterative methods. When the solution of a linear system with a symmetric and positive definite coefficient matrix is required then the Conjugate Gradient method will compute the optimal approximate solution from the appropriate Krylov subspace, that is, it will implicitly compute the optimal polynomial. Hence a semi-iterative method, which requires eigenvalue bounds and computes an explicit polynomial, must, for just a little less computational work, give an inferior result. In this manuscript we identify a specific situation in the context of preconditioning when the Chebyshev semi-iterative method is the method of choice since it has properties which make it superior to the Conjugate Gradient method. |
| first_indexed | 2024-03-06T23:30:47Z |
| format | Report |
| id | oxford-uuid:6bf30171-dfba-4028-af2d-b21b2153857c |
| institution | University of Oxford |
| last_indexed | 2024-03-06T23:30:47Z |
| publishDate | 2008 |
| publisher | Oxford University Computing Laboratory |
| record_format | dspace |
| spelling | oxford-uuid:6bf30171-dfba-4028-af2d-b21b2153857c2022-03-26T19:07:35ZChebyshev Semi−iteration in PreconditioningReporthttp://purl.org/coar/resource_type/c_93fcuuid:6bf30171-dfba-4028-af2d-b21b2153857cDepartment of Computer ScienceOxford University Computing Laboratory2008Wathen, TIt is widely believed that Krylov subspace iterative methods are better than Chebyshev semi-iterative methods. When the solution of a linear system with a symmetric and positive definite coefficient matrix is required then the Conjugate Gradient method will compute the optimal approximate solution from the appropriate Krylov subspace, that is, it will implicitly compute the optimal polynomial. Hence a semi-iterative method, which requires eigenvalue bounds and computes an explicit polynomial, must, for just a little less computational work, give an inferior result. In this manuscript we identify a specific situation in the context of preconditioning when the Chebyshev semi-iterative method is the method of choice since it has properties which make it superior to the Conjugate Gradient method. |
| spellingShingle | Wathen, T Chebyshev Semi−iteration in Preconditioning |
| title | Chebyshev Semi−iteration in Preconditioning |
| title_full | Chebyshev Semi−iteration in Preconditioning |
| title_fullStr | Chebyshev Semi−iteration in Preconditioning |
| title_full_unstemmed | Chebyshev Semi−iteration in Preconditioning |
| title_short | Chebyshev Semi−iteration in Preconditioning |
| title_sort | chebyshev semi iteration in preconditioning |
| work_keys_str_mv | AT wathent chebyshevsemiiterationinpreconditioning |