Criteria for positive definiteness of some band matrices

Allgower, Eugene L.

doi:10.1007/bf02308868

Cited by 9 publications

(3 citation statements)

References 7 publications

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“…[Trench, 1974] Trench presents a method for inverting {p, q}-banded Toeplitz matrices by exploiting the banded structure. [Bevilacqua and Capovani, 1976] Bevilacqua and Capovani extend the results of the papers [Greenberg and Sarhan, 1959] and [Allgower, 1970[Allgower, , 1973 to band matrices and to block band matrices (not necessarily symmetric). Formulas are presented for inverting band matrices whose elements on the extreme diagonals are different from zero.…”

Section: The Overviewmentioning

confidence: 75%

A bibliography on semiseparable matrices*

Vandebril

Barel

Golub

et al. 2005

Calcolo

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Currently there is a growing interest in semiseparable matrices and generalized semiseparable matrices. To gain an appreciation of the historical evolution of this concept, we present in this paper an extensive list of publications related to the field of semiseparable matrices. It is interesting to see that semiseparable matrices were investigated in different fields, e.g., integral equations, statistics, vibrational analysis, independently of-each other. Also notable is the fact that leading statisticians at that time Used semiseparable matrices without knowing their inverses to be tridiagonal. During this historical evolution the definition of semiseparable matrices has always been a difficult point leading to misunderstandings; sometimes they were defined as the inverses of irreducible tridiagonal matrices leading to generator representable matrices, while in other cases they were defined as matrices having low rank blocks below the diagonal.

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Section: The Overviewmentioning

confidence: 75%

A bibliography on semiseparable matrices*

Vandebril

Barel

Golub

et al. 2005

Calcolo

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“…Both of such approaches have been utilized in numerical analysis. The former is used, for example, in [3,5,13,14]. The latter has been used in the past, for example by Dahlquist, in studying the stability problem for numerical methods [8].…”

Section: A Characterization Of Positive Definite Band Symmetric Toeplmentioning

confidence: 99%

“…Such new criterium seems to be more useful with respect to the classical ones based on the study of the spectrum (see, for example, [3,5]) or on the positivity of the real part of the generating polynomial [10]. In fact, it does not depends on the dimension of the matrix and it only requires information on some real roots of the generating polynomial.…”

Section: Introductionmentioning

confidence: 99%

On the properties of matrices defining some classes of BVMs

Aceto

Trigiante

2003

Numerische Mathematik

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It is known that the matrices defining the discrete problem generated by a k-step Boundary Value Method (BVM) have a quasi-Toeplitz band structure. In particular, when the boundary conditions are skipped, they become Toeplitz matrices. In this paper, by introducing a characterization of positive definiteness for such matrices, we shall prove that the Toeplitz matrices which arise when using the methods in the classes of BVMs known as Generalized BDF and Top Order Methods have such property

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Exact inverses of certain band matrices

Allgower

1973

Numer. Math.

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Criteria for positive definiteness of some band matrices

Cited by 9 publications

References 7 publications

A bibliography on semiseparable matrices*

A bibliography on semiseparable matrices*

On the properties of matrices defining some classes of BVMs

Exact inverses of certain band matrices

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