Is Gauss quadrature better than Clenshaw-Curtis?
We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw-Curtis. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-2 advantage of Gauss quadrature is rarely realized. Theorems are given to exp...
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| Format: | Journal article |
| Language: | English |
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2008
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| _version_ | 1826263437844414464 |
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| author | Trefethen, L |
| author_facet | Trefethen, L |
| author_sort | Trefethen, L |
| collection | OXFORD |
| description | We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw-Curtis. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following O'Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log((z + 1)/(Z - 1)) in the complex plane. Gauss quadrature corresponds to Padé approximation at z = ∞. Clenshaw-Curtis quadrature corresponds to an approximation whose order of accuracy at z = ∞ is only half as high, but which is nevertheless equally accurate near [-1, 1]. © 2008 Society for Industrial and Applied Mathematics. |
| first_indexed | 2024-03-06T19:51:45Z |
| format | Journal article |
| id | oxford-uuid:2431af96-8127-4403-840c-8f686324cb80 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T19:51:45Z |
| publishDate | 2008 |
| record_format | dspace |
| spelling | oxford-uuid:2431af96-8127-4403-840c-8f686324cb802022-03-26T11:48:40ZIs Gauss quadrature better than Clenshaw-Curtis?Journal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:2431af96-8127-4403-840c-8f686324cb80EnglishSymplectic Elements at Oxford2008Trefethen, LWe compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw-Curtis. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following O'Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log((z + 1)/(Z - 1)) in the complex plane. Gauss quadrature corresponds to Padé approximation at z = ∞. Clenshaw-Curtis quadrature corresponds to an approximation whose order of accuracy at z = ∞ is only half as high, but which is nevertheless equally accurate near [-1, 1]. © 2008 Society for Industrial and Applied Mathematics. |
| spellingShingle | Trefethen, L Is Gauss quadrature better than Clenshaw-Curtis? |
| title | Is Gauss quadrature better than Clenshaw-Curtis? |
| title_full | Is Gauss quadrature better than Clenshaw-Curtis? |
| title_fullStr | Is Gauss quadrature better than Clenshaw-Curtis? |
| title_full_unstemmed | Is Gauss quadrature better than Clenshaw-Curtis? |
| title_short | Is Gauss quadrature better than Clenshaw-Curtis? |
| title_sort | is gauss quadrature better than clenshaw curtis |
| work_keys_str_mv | AT trefethenl isgaussquadraturebetterthanclenshawcurtis |