Normal form for GLT sequences, functions of normal GLT sequences, and spectral distribution of perturbed normal matrices
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866909063495811072 |
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| author | Barbarino, Giovanni Garoni, Carlo |
| author_facet | Barbarino, Giovanni Garoni, Carlo |
| contents | The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices $A_n$ arising from numerical discretizations of differential equations. Indeed, when the mesh fineness parameter $n$ tends to infinity, these matrices $A_n$ give rise to a sequence $\{A_n\}_n$, which often turns out to be a GLT sequence. In this paper, we extend the theory of GLT sequences in several directions: we show that every GLT sequence enjoys a normal form, we identify the spectral symbol of every GLT sequence formed by normal matrices, and we prove that, for every GLT sequence $\{A_n\}_n$ formed by normal matrices and every continuous function $f:\mathbb C\to\mathbb C$, the sequence $\{f(A_n)\}_n$ is again a GLT sequence whose spectral symbol is $f(κ)$, where $κ$ is the spectral symbol of $\{A_n\}_n$. In addition, using the theory of GLT sequences, we prove a spectral distribution result for perturbed normal matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1805_08708 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Normal form for GLT sequences, functions of normal GLT sequences, and spectral distribution of perturbed normal matrices Barbarino, Giovanni Garoni, Carlo Numerical Analysis 15B05, 15A18, 47B06, 47B15, 15A16 The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices $A_n$ arising from numerical discretizations of differential equations. Indeed, when the mesh fineness parameter $n$ tends to infinity, these matrices $A_n$ give rise to a sequence $\{A_n\}_n$, which often turns out to be a GLT sequence. In this paper, we extend the theory of GLT sequences in several directions: we show that every GLT sequence enjoys a normal form, we identify the spectral symbol of every GLT sequence formed by normal matrices, and we prove that, for every GLT sequence $\{A_n\}_n$ formed by normal matrices and every continuous function $f:\mathbb C\to\mathbb C$, the sequence $\{f(A_n)\}_n$ is again a GLT sequence whose spectral symbol is $f(κ)$, where $κ$ is the spectral symbol of $\{A_n\}_n$. In addition, using the theory of GLT sequences, we prove a spectral distribution result for perturbed normal matrices. |
| title | Normal form for GLT sequences, functions of normal GLT sequences, and spectral distribution of perturbed normal matrices |
| topic | Numerical Analysis 15B05, 15A18, 47B06, 47B15, 15A16 |
| url | https://arxiv.org/abs/1805.08708 |