Accurate bidiagonal decomposition of Lagrange-Vandermonde matrices and applications
Lagrange-Vandermonde matrices are the collocation matrices corresponding to Lagrange-type bases, obtained by removing the denominators from each element of a Lagrange basis. It is proved that, provided the nodes required to create the Lagrange-type basis and the corresponding collocation matrix are...
| Autores: | , , |
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| Tipo de documento: | artigo |
| Data de publicação: | 2024 |
| País: | España |
| Recursos: | Universidad de Alcalá (UAH) |
| Repositório: | e_Buah Biblioteca Digital Universidad de Alcalá |
| Idioma: | inglês |
| OAI Identifier: | oai:ebuah.uah.es:10017/63570 |
| Acesso em linha: | http://hdl.handle.net/10017/63570 https://dx.doi.org/10.1002/nla.2527 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Bidiagonal decomposition High relative accuracy Lagrange basis Neville elimination Totallypositive matrix Matemáticas Mathematics |
| Resumo: | Lagrange-Vandermonde matrices are the collocation matrices corresponding to Lagrange-type bases, obtained by removing the denominators from each element of a Lagrange basis. It is proved that, provided the nodes required to create the Lagrange-type basis and the corresponding collocation matrix are properly ordered, such matrices are strictly totally positive. A fast algorithm to compute the bidiagonal decomposition of these matrices to high relative accuracy is presented. As an application, the problems of eigenvalue computation, linear system solving and inverse computation are solved in an efficient and accurate way for this type of matrices. Moreover, the proposed algorithms allow to solve fastly and to high relative accuracy some of the cited problems when the involved matrices are collocation matrices corresponding to the standard Lagrange basis, although such collocation matrices are not totally positive. Numerical experiments illustrating the good performance of our approach are also included. |
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