Which constraints of a numerical problem cause ill-conditioning?
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | |
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| _version_ | 1866917221810307072 |
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| author | Dewaele, Nick Vannieuwenhoven, Nick |
| author_facet | Dewaele, Nick Vannieuwenhoven, Nick |
| contents | Many numerical problems with input $x$ and output $y$ can be formulated as a system of equations $F(x, y) = 0$ where the goal is to solve for $y$. The condition number measures the change of $y$ for small perturbations to $x$. From this numerical problem, one can derive a (typically underdetermined) relaxation by omitting any number of equations from $F$. We propose a condition number for underdetermined systems that relates the condition number of a numerical problem to those of its relaxations, thereby detecting the ill-conditioned constraints. We illustrate the use of our technique by computing the condition of two problems that do not have a finite condition number in the classic sense: two-factor matrix decompositions and Tucker decompositions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_11547 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Which constraints of a numerical problem cause ill-conditioning? Dewaele, Nick Vannieuwenhoven, Nick Numerical Analysis 15A12, 15A23, 49Q12, 53B20, 15A69, 65F35 Many numerical problems with input $x$ and output $y$ can be formulated as a system of equations $F(x, y) = 0$ where the goal is to solve for $y$. The condition number measures the change of $y$ for small perturbations to $x$. From this numerical problem, one can derive a (typically underdetermined) relaxation by omitting any number of equations from $F$. We propose a condition number for underdetermined systems that relates the condition number of a numerical problem to those of its relaxations, thereby detecting the ill-conditioned constraints. We illustrate the use of our technique by computing the condition of two problems that do not have a finite condition number in the classic sense: two-factor matrix decompositions and Tucker decompositions. |
| title | Which constraints of a numerical problem cause ill-conditioning? |
| topic | Numerical Analysis 15A12, 15A23, 49Q12, 53B20, 15A69, 65F35 |
| url | https://arxiv.org/abs/2305.11547 |