Parallel solution of block tridiagonal linear systems

Bevilacqua, Roberto; Codenotti, Bruno; Romani, Francesco

doi:10.1016/0024-3795(88)90305-9

Cited by 23 publications

(14 citation statements)

References 12 publications

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“…In 1986, Romani [30] studied the additive structure of the inverses of banded matrices, namely, that the inverse of a 2k + 1 diagonal symmetric banded matrix can be expressed as a sum of k symmetric matrices belonging to the class of inverses of symmetric irreducible tridiagonal matrices. In 1988, Bevilacqua, Codenotti, and Romani [9] gave formulas for block Hessenberg and block tridiagonal matrices with nonsingular outer blocks.…”

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confidence: 99%

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A Review on the Inverse of Symmetric Tridiagonal and Block Tridiagonal Matrices

Meurant¹

1992

SIAM J. Matrix Anal. & Appl.

322

190

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Abstract. In this paper some results are reviewed concerning the characterization of inverses of symmetric tridiagonal and block tridiagonal matrices as well as results concerning the decay of the elements of the inverses. These results are obtained by relating the elements of inverses to elements of the Cholesky decompositions of these matrices. This gives explicit formulas for the elements of the inverse and gives rise to stable algorithms to compute them. These expressions also lead to bounds for the decay of the elements of the inverse for problems arising from discretization schemes. Basically there are two kinds of papers: the first gives analytic formulas for special cases; the second gives characterizations of matrices whose inverse has certain properties, e.g., being tridiagonal or banded.Historically, the oldest paper we found considering the explicit inverse of matrices

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confidence: 99%

“…When no explicit solutions for the elements of the inverse can be found, they are usually given in terms of solutions of second-order linear recurrences [5], [9], [14]. However, as it was shown in Concus and Meurant [14] [6]).…”

mentioning

confidence: 99%

A Review on the Inverse of Symmetric Tridiagonal and Block Tridiagonal Matrices

Meurant¹

1992

SIAM J. Matrix Anal. & Appl.

322

190

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“…When K is proper, i.e., when B i are nonsingular [2], there exists two (non-unique) sequences of matrices {U i } and…”

Section: B Inverses Of Block-tridiagonal Matricesmentioning

confidence: 99%

A fast band matching technique for impedance extraction

Jain

Koh

et al. 2008

2008 IEEE International Symposium on Circuits and Systems (ISCAS)

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We present an efficient technique for the fast and accurate extraction of inductance of large-scale on-chip interconnects. Several simulation techniques exploit the sparsity of L −1 for reducing storage and computational costs during computation. Recently, band matching technique was introduced for sparsification of L −1 . We extend the band-matching technique for inductance extraction problems. We introduce an efficient technique to invert the approximate band-matched impedance matrix R f + jωL f . The new technique relies on a numerically stable and efficient representation for inverses of banded matrices. Numerical results show that our approach is both accurate as well as computationally efficient, with orders-of-magnitude improvement over traditional techniques.

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“…The factors u i and v j can easily be calculated using a simple recurrence relation (Brevilacqua et al, 1988).…”

Section: A2 Inverse Of a Symmetrical Jacobian Matrixmentioning

confidence: 99%

The Darwin procedure in optics of layered media and the matrix theory

Dub

Litzman

1999

Acta Cryst Sect A

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The Darwin dynamical theory of diffraction for two beams yields a nonhomogeneous system of linear algebraic equations with a tridiagonal matrix. It is shown that different formulae of the two-beam Darwin theory can be obtained by a uniform view of the basic properties of tridiagonal matrices, their determinants (continuants) and their close relationship to continued fractions and difference equations. Some remarks concerning the relation of the Darwin theory in the three-beam case to tridiagonal block matrices are also presented.

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Parallel solution of block tridiagonal linear systems

Cited by 23 publications

References 12 publications

A Review on the Inverse of Symmetric Tridiagonal and Block Tridiagonal Matrices

A Review on the Inverse of Symmetric Tridiagonal and Block Tridiagonal Matrices

A fast band matching technique for impedance extraction

The Darwin procedure in optics of layered media and the matrix theory

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