Numerical algorithms based on analytic function values at roots of unity
Let $f(z)$ be an analytic or meromorphic function in the closed unit disk sampled at the $n$th roots of unity. Based on these data, how can we approximately evaluate $f(z)$ or $f^{(m)}(z)$ at a point $z$ in the disk? How can we calculate the zeros or poles of $f$ in the disk? These questions exhibit...
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| Format: | Report |
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SINUM
2013
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| _version_ | 1797104140178948096 |
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| author | Austin, A Kravanja, P Trefethen, L |
| author_facet | Austin, A Kravanja, P Trefethen, L |
| author_sort | Austin, A |
| collection | OXFORD |
| description | Let $f(z)$ be an analytic or meromorphic function in the closed unit disk sampled at the $n$th roots of unity. Based on these data, how can we approximately evaluate $f(z)$ or $f^{(m)}(z)$ at a point $z$ in the disk? How can we calculate the zeros or poles of $f$ in the disk? These questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations, partial differential equations, and large-scale linear algebra. We analyze the possibilities and emphasize a basic distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of Cauchy integrals. We then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension. |
| first_indexed | 2024-03-07T06:29:38Z |
| format | Report |
| id | oxford-uuid:f58b4eb8-94f3-4afd-bc32-8edf44835927 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T06:29:38Z |
| publishDate | 2013 |
| publisher | SINUM |
| record_format | dspace |
| spelling | oxford-uuid:f58b4eb8-94f3-4afd-bc32-8edf448359272022-03-27T12:28:07ZNumerical algorithms based on analytic function values at roots of unityReporthttp://purl.org/coar/resource_type/c_93fcuuid:f58b4eb8-94f3-4afd-bc32-8edf44835927Mathematical Institute - ePrintsSINUM2013Austin, AKravanja, PTrefethen, LLet $f(z)$ be an analytic or meromorphic function in the closed unit disk sampled at the $n$th roots of unity. Based on these data, how can we approximately evaluate $f(z)$ or $f^{(m)}(z)$ at a point $z$ in the disk? How can we calculate the zeros or poles of $f$ in the disk? These questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations, partial differential equations, and large-scale linear algebra. We analyze the possibilities and emphasize a basic distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of Cauchy integrals. We then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension. |
| spellingShingle | Austin, A Kravanja, P Trefethen, L Numerical algorithms based on analytic function values at roots of unity |
| title | Numerical algorithms based on analytic function values at roots of unity |
| title_full | Numerical algorithms based on analytic function values at roots of unity |
| title_fullStr | Numerical algorithms based on analytic function values at roots of unity |
| title_full_unstemmed | Numerical algorithms based on analytic function values at roots of unity |
| title_short | Numerical algorithms based on analytic function values at roots of unity |
| title_sort | numerical algorithms based on analytic function values at roots of unity |
| work_keys_str_mv | AT austina numericalalgorithmsbasedonanalyticfunctionvaluesatrootsofunity AT kravanjap numericalalgorithmsbasedonanalyticfunctionvaluesatrootsofunity AT trefethenl numericalalgorithmsbasedonanalyticfunctionvaluesatrootsofunity |