Computing the common zeros of two bivariate functions via Bezout resultants
The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and Bezout matrices with polynomial ent...
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| Format: | Report |
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Unspecified
2013
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| _version_ | 1826274254709063680 |
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| author | Nakatsukasa, Y Noferini, V Townsend, A |
| author_facet | Nakatsukasa, Y Noferini, V Townsend, A |
| author_sort | Nakatsukasa, Y |
| collection | OXFORD |
| description | The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and Bezout matrices with polynomial entries. Using techniques including domain subdivision, Bezoutian regularization and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree ($\ge$ 1000). We analyze the resultant method and its conditioning by noting that the Bezout matrices are matrix polynomials. Our robust algorithm is implemented in the roots command in Chebfun2, a software package written in object-oriented MATLAB for computing with bivariate functions. |
| first_indexed | 2024-03-06T22:40:40Z |
| format | Report |
| id | oxford-uuid:5b6f5fad-c3e6-4485-866b-ec13c2960353 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T22:40:40Z |
| publishDate | 2013 |
| publisher | Unspecified |
| record_format | dspace |
| spelling | oxford-uuid:5b6f5fad-c3e6-4485-866b-ec13c29603532022-03-26T17:22:02ZComputing the common zeros of two bivariate functions via Bezout resultants Reporthttp://purl.org/coar/resource_type/c_93fcuuid:5b6f5fad-c3e6-4485-866b-ec13c2960353Mathematical Institute - ePrintsUnspecified2013Nakatsukasa, YNoferini, VTownsend, AThe common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and Bezout matrices with polynomial entries. Using techniques including domain subdivision, Bezoutian regularization and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree ($\ge$ 1000). We analyze the resultant method and its conditioning by noting that the Bezout matrices are matrix polynomials. Our robust algorithm is implemented in the roots command in Chebfun2, a software package written in object-oriented MATLAB for computing with bivariate functions. |
| spellingShingle | Nakatsukasa, Y Noferini, V Townsend, A Computing the common zeros of two bivariate functions via Bezout resultants |
| title | Computing the common zeros of two bivariate functions via Bezout resultants
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| title_full | Computing the common zeros of two bivariate functions via Bezout resultants
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| title_fullStr | Computing the common zeros of two bivariate functions via Bezout resultants
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| title_full_unstemmed | Computing the common zeros of two bivariate functions via Bezout resultants
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| title_short | Computing the common zeros of two bivariate functions via Bezout resultants
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| title_sort | computing the common zeros of two bivariate functions via bezout resultants |
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