Log-lightning computation of capacity and Green's function
A basic measure of the size of a set E in the complex plane is the logarithmic capacity cap(E). Capacities are known analytically for a few simple shapes like ellipses, but in most cases they must be computed numerically. We explore their computation by the new "log-lightning'' method...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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University of Western Ontario, Western Libraries
2021
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| _version_ | 1797074958529069056 |
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| author | Baddoo, P Trefethen, LN |
| author_facet | Baddoo, P Trefethen, LN |
| author_sort | Baddoo, P |
| collection | OXFORD |
| description | A basic measure of the size of a set E in the complex plane is the logarithmic capacity cap(E). Capacities are known analytically for a few simple shapes like ellipses, but in most cases they must be computed numerically. We explore their computation by the new "log-lightning'' method based on reciprocal-log approximations in the complex plane. For a sequence of 16 examples involving both connected and disconnected sets E, we compute capacities to 8–15 digits of accuracy at great speed in MATLAB. The convergence is almost-exponential with respect to the number of reciprocal-log poles employed, so it should be possible to compute many more digits if desired in Maple or another extended-precision environment. This is the first systematic exploration of applications of the log-lightning method, which opens up the possibility of solving Laplace problems with an efficiency not achievable by previous methods. The method computes not just the capacity, but also the Green's function and its harmonic conjugate. It also extends to "domains of negative measure" and other Riemann surfaces. |
| first_indexed | 2024-03-06T23:43:41Z |
| format | Journal article |
| id | oxford-uuid:702eb8de-af6a-41f4-920a-a6c741a3f5b7 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T23:43:41Z |
| publishDate | 2021 |
| publisher | University of Western Ontario, Western Libraries |
| record_format | dspace |
| spelling | oxford-uuid:702eb8de-af6a-41f4-920a-a6c741a3f5b72022-03-26T19:35:27ZLog-lightning computation of capacity and Green's functionJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:702eb8de-af6a-41f4-920a-a6c741a3f5b7EnglishSymplectic ElementsUniversity of Western Ontario, Western Libraries2021Baddoo, PTrefethen, LNA basic measure of the size of a set E in the complex plane is the logarithmic capacity cap(E). Capacities are known analytically for a few simple shapes like ellipses, but in most cases they must be computed numerically. We explore their computation by the new "log-lightning'' method based on reciprocal-log approximations in the complex plane. For a sequence of 16 examples involving both connected and disconnected sets E, we compute capacities to 8–15 digits of accuracy at great speed in MATLAB. The convergence is almost-exponential with respect to the number of reciprocal-log poles employed, so it should be possible to compute many more digits if desired in Maple or another extended-precision environment. This is the first systematic exploration of applications of the log-lightning method, which opens up the possibility of solving Laplace problems with an efficiency not achievable by previous methods. The method computes not just the capacity, but also the Green's function and its harmonic conjugate. It also extends to "domains of negative measure" and other Riemann surfaces. |
| spellingShingle | Baddoo, P Trefethen, LN Log-lightning computation of capacity and Green's function |
| title | Log-lightning computation of capacity and Green's function |
| title_full | Log-lightning computation of capacity and Green's function |
| title_fullStr | Log-lightning computation of capacity and Green's function |
| title_full_unstemmed | Log-lightning computation of capacity and Green's function |
| title_short | Log-lightning computation of capacity and Green's function |
| title_sort | log lightning computation of capacity and green s function |
| work_keys_str_mv | AT baddoop loglightningcomputationofcapacityandgreensfunction AT trefethenln loglightningcomputationofcapacityandgreensfunction |