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A matrix-less method to approximate the spectrum and the spectral function of Toeplitz matrices with real eigenvalues
- Source :
- Numerical Algorithms. 89:701-720
- Publication Year :
- 2021
- Publisher :
- Springer Science and Business Media LLC, 2021.
-
Abstract
- It is known that the generating function f of a sequence of Toeplitz matrices {Tn(f)}n may not describe the asymptotic distribution of the eigenvalues of Tn(f) if f is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of Tn(f) are real for all n, then they admit an asymptotic expansion of the same type as considered in previous works, where the first function, called the eigenvalue symbol $\mathfrak {f}$ f , appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of Tn(f). This eigenvalue symbol $\mathfrak {f}$ f is in general not known in closed form. After validating this working hypothesis through a number of numerical experiments, we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function $\mathfrak {f}$ f . The proposed algorithm, which opposed to previous versions, does not need any information about neither f nor $\mathfrak {f}$ f is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of $\mathfrak {f}$ f . Future research directions are outlined at the end of the paper.
- Subjects :
- Beräkningsmatematik
Applied Mathematics
010102 general mathematics
Generating function
Order (ring theory)
Asymptotic expansion
Spectral analysis
Numerical Analysis (math.NA)
010103 numerical & computational mathematics
Function (mathematics)
Type (model theory)
01 natural sciences
Toeplitz matrix
Combinatorics
Computational Mathematics
Matrix (mathematics)
Toeplitz matrices
FOS: Mathematics
Mathematics - Numerical Analysis
0101 mathematics
Structured matrices
Eigenvalues and eigenvectors
Mathematics
Subjects
Details
- ISSN :
- 15729265 and 10171398
- Volume :
- 89
- Database :
- OpenAIRE
- Journal :
- Numerical Algorithms
- Accession number :
- edsair.doi.dedup.....bc73c8362a437f12a3ec8e8b8cd67ee4