Eigenpairs of a family of tridiagonal matrices: three decades later

Fonseca, Carlos M. da; Kowalenko, Victor

doi:10.1007/s10474-019-00970-1

Cited by 41 publications

(30 citation statements)

References 33 publications

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“…Other possible approximations of the eigenpairs of B may be obtained from those of the Toeplitz-type matrices T + or T # described in Sect. 3.2, which are explicitly known (Fonseca and Kowalenko 2020;Losonczi 1992). More precisely, the eigenvalues of T + = T (n; x| − x, −x − y, −y|y) are 0 and…”

Section: Approximation Of the Eigenvalues And Eigenvectors Of A Bd-mamentioning

confidence: 97%

“…It is natural that such approximant matrices are chosen among those with tridiagonal structure. Explicit formulas for the eigenvalues and eigenvectors of tridiagonal Toeplitz matrices and some top and bottom corner perturbations of them, called in the sequel Toeplitz-type matrices, are known (Fonseca and Kowalenko 2020;Losonczi 1992;Noschese and Reichel 2019). Our purpose is to investigate the possibility of approximating a BD-matrix by a Toeplitz-type one.…”

Section: Introductionmentioning

confidence: 99%

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Approximating the eigenvalues and eigenvectors of birth and death matrices

Bebiano

Furtado

2020

Comp. Appl. Math.

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Section: Approximation Of the Eigenvalues And Eigenvectors Of A Bd-mamentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Approximating the eigenvalues and eigenvectors of birth and death matrices

Bebiano

Furtado

2020

Comp. Appl. Math.

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“…In addition, Eigenpairs, determinants and inverses of the tridiagonal matrices, anti-tridiagonal Hankel matrices, pentadiagonal matrices and cyclic pentadiagonal matrices with Toeplitz structure have been studied in [10][11][12][13][14][15][16][17][18][19][20][21][22][23]. The determinant and inversion of A(c , an; d k , a k , c k ; a , cn) have been studied extensively and found with simple and analytic expression (see [24][25][26]).…”

Section: Introductionmentioning

confidence: 99%

Analytic determinants and inverses of Toeplitz and Hankel tridiagonal matrices with perturbed columns

Jiang

et al. 2020

Special Matrices

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In this paper, our main attention is paid to calculate the determinants and inverses of two types Toeplitz and Hankel tridiagonal matrices with perturbed columns. Specifically, the determinants of the n × n Toeplitz tridiagonal matrices with perturbed columns (type I, II) can be expressed by using the famous Fibonacci numbers, the inverses of Toeplitz tridiagonal matrices with perturbed columns can also be expressed by using the well-known Lucas numbers and four entries in matrix 𝔸. And the determinants of the n×n Hankel tridiagonal matrices with perturbed columns (type I, II) are (−1]) {\left( { - 1} \right)^{{{n\left( {n - 1} \right)} \over 2}}} times of the determinant of the Toeplitz tridiagonal matrix with perturbed columns type I, the entries of the inverses of the Hankel tridiagonal matrices with perturbed columns (type I, II) are the same as that of the inverse of Toeplitz tridiagonal matrix with perturbed columns type I, except the position. In addition, we present some algorithms based on the main theoretical results. Comparison of our new algorithms and some recent works is given. The numerical result confirms our new theoretical results. And we show the superiority of our method by comparing the CPU time of some existing algorithms studied recently.

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“…Perhaps the most important non-trivial case is due to Losonczi [3]. A very recent and important survey in this topic can be found in da Fonseca and Kowalenko [4].…”

Section: Introductionmentioning

confidence: 99%

“…They denoted determinants of matrices A n , A n , and A n by W n , W n , and W n , respectively, and derived the following system of linear recurrence relations (see formulae (4) and (5) in [14], where all the needed initial conditions can be found too)…”

Section: Introductionmentioning

confidence: 99%