Finding eigenvalues for heptadiagonal symmetric Toeplitz matrices

Solary, Maryam Shams

doi:10.1016/j.jmaa.2013.02.008

Cited by 21 publications

(8 citation statements)

References 1 publication

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“…Pada penelitian ini matriksnya juga berbeda dari matriks-matriks yang dipakai oleh peneliti-peneliti sebelumnya. Bentuk umum matriks yang digunakan ada pada [9].…”

Section: +1 (𝑎) 𝑛 Untuk N Genapunclassified

Trace Matriks Toeplitz Heptadiagonal Simetris Berpangkat Bilangan Bulat Positif

Aryani,

Ramadhani,

Muda

et al. 2022

JFOURIER

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Penelitian ini bertujuan untuk mendapatkan bentuk umum trace matriks Toeplitz heptadiagonal simetris berpangkat dua sampai empat. Untuk mendapatkan bentuk umum trace matriks tersebut dimulai dengan menentukan bentuk umum perpangkatan matriks Toeplitz heptadiagonal simetris berpangkat dua. Sedangkan perpangkatan tiga dan empat cukup dengan menentukan entri-entri diagonal utama saja dari perpangkatan matriks Toeplitz heptadiagonal simetris tersebut. Selanjutnya, diperoleh bentuk umum trace matriks Toeplitz heptadiagonal simetris berpangkat dua sampai empat. Kedua bentuk umum tersebut dibuktikan dengan cara pembuktian langsung. Diberikan juga contoh aplikasi dari trace matriks Toeplitz heptadiagonal simetris berpangkat dua sampai empat.

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“…Pada penelitian ini matriksnya juga berbeda dari matriks-matriks yang dipakai oleh peneliti-peneliti sebelumnya. Bentuk umum matriks yang digunakan ada pada [9].…”

Section: +1 (𝑎) 𝑛 Untuk N Genapunclassified

Trace Matriks Toeplitz Heptadiagonal Simetris Berpangkat Bilangan Bulat Positif

Aryani,

Ramadhani,

Muda

et al. 2022

JFOURIER

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“…A partial solution is provided by linking the eigenvalues-eigenvectors as a function of the frequency for symmetric positive definite Toeplitz matrices [10]. However, the analytic form of the eigenvalues for this type of matrices is only known for heptadiagonal matrices [39]. We generalize the link between the eigenstructure of a matrix and the associated frequencies by the use of circulant matrices allowing non-periodic signals.…”

Section: Rd Step: Groupingmentioning

confidence: 99%

Circulant Singular Spectrum Analysis: A new automated procedure for signal extraction

Bógalo,

Poncela,

Senra

2020

Preprint

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Sometimes, it is of interest to single out the fluctuations associated to a given frequency. We propose a new variant of SSA, Circulant SSA (CiSSA), that allows to extract the signal associated to any frequency specified beforehand. This is a novelty when compared with other procedures that need to identify ex-post the frequencies associated to extracted signals. We prove that CiSSA is asymptotically equivalent to these alternative procedures although with the advantage of avoiding the need of the subsequent frequency identification. We check its good performance and compare it to alternative SSA methods through several simulations for linear and nonlinear time series. We also prove its validity in the nonstationary case. To show how it works with real data, we apply CiSSA to extract the business cycle and deseasonalize the Industrial Production Index of six countries. Economists follow this indicator in order to assess the state of the economy in real time. We find that the estimated cycles match the dated recessions from the OECD showing its reliability for business cycle analysis. Finally, we analyze the strong separability of the estimated components. In particular, we check that the deseasonalized time series do not show any evidence of residual seasonality.

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“…The numerical algorithms for computing determinants or eigenvalues of heptadiagonal Toeplitz matrices are proposed for example in [1][2][3]. In the present paper we are to show that the determinant of heptadiagonal symmetric Toeplitz matrix can be described by a system of three linear recurrence equations.…”

Section: Introductionmentioning

confidence: 96%