An inversion algorithm for a banded matrix

Ran, Ruisheng; Huang, Ting‐Zhu

doi:10.1016/j.camwa.2009.07.069

Cited by 11 publications

(11 citation statements)

References 12 publications

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“…For tridiagonal and banded matrices, many efficient direct algorithms and explicit expressions of the entries of B have been found in the last decades, see the recent paper of Ran and Huang 17 and the references therein. In this paper, an inversion algorithm for a banded matrix is proposed by employing so‐called twisted decompositions of matrices.…”

Section: Literature Overviewmentioning

confidence: 99%

A probing method for computing the diagonal of a matrix inverse

Tang

Saad

2011

Numerical Linear Algebra App

120

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The computation of some entries of a matrix inverse arises in several important applications in practice. This paper presents a probing method for determining the diagonal of the inverse of a sparse matrix in the common situation when its inverse exhibits a decay property, i.e., when many of the entries of the inverse are small. A few simple properties of the inverse suggest a way to determine effective probing vectors based on standard graph theory results. An iterative method is then applied to solve the resulting sequence of linear systems, from which the diagonal of the matrix inverse is extracted. Results of numerical experiments are provided to demonstrate the effectiveness of the probing method.

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Section: Literature Overviewmentioning

confidence: 99%

A probing method for computing the diagonal of a matrix inverse

Tang

Saad

2011

Numerical Linear Algebra App

120

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“…Some approaches allow the inverse to be calculated in an efficient way for some special cases. For banded matrices, for instance, Ran and Huang 34 developed an algorithm that is about twice as fast as the standard method based on the LU decomposition. To determine the diagonal entries of the inverse, Tang and Saad 35 developed a probing method for situations where the inverse exhibits decay properties.…”

Section: Introductionmentioning

confidence: 99%

Efficient Computation of Sparse Matrix Functions for Large-Scale Electronic Structure Calculations: The CheSS Library

Mohr

Dawson

Wagner

et al. 2017

J. Chem. Theory Comput.

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We present CheSS, the "Chebyshev Sparse Solvers" library, which has been designed to solve typical problems arising in large-scale electronic structure calculations using localized basis sets. The library is based on a flexible and efficient expansion in terms of Chebyshev polynomials and presently features the calculation of the density matrix, the calculation of matrix powers for arbitrary powers, and the extraction of eigenvalues in a selected interval. CheSS is able to exploit the sparsity of the matrices and scales linearly with respect to the number of nonzero entries, making it well-suited for large-scale calculations. The approach is particularly adapted for setups leading to small spectral widths of the involved matrices and outperforms alternative methods in this regime. By coupling CheSS to the DFT code BigDFT, we show that such a favorable setup is indeed possible in practice. In addition, the approach based on Chebyshev polynomials can be massively parallelized, and CheSS exhibits excellent scaling up to thousands of cores even for relatively small matrix sizes.

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“…Since the equations set has a banded coefficients matrix in which all elements are nested sub-matrixes, we use a banded matrix decomposition principle [21] to solve the equation set indirectly. Our method only requires the coefficients matrix to be assembled and decomposed once.…”

Section: Introductionmentioning

confidence: 99%

A New Unconditionally Stable Scheme for FDTD Method Using Associated Hermite Orthogonal Functions

Huang

Shi

Chen

et al. 2014

IEEE Trans. Antennas Propagat.

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An unconditionally stable solution using associated Hermite (AH) functions is proposed for the finite-difference time-domain (FDTD) method. The electromagnetic fields and their time derivatives in time-domain Maxwell's equations are expanded by these orthonormal basis functions. By applying Galerkin temporal testing procedure to these expanded equations the time variable can be eliminated from the calculations. A set of implicit equations is derived to calculate the magnetic filed expansion coefficients of all orders of AH functions for the temporal variable. And the electrical field coefficients can be obtained respectively. With the appropriate translation and scale parameters, we can find a minimum-order basis functions subspace to approach a particular electromagnetic field. The numerical results have shown that the proposed method can reduce the CPU time to 0.59% of the traditional FDTD method while maintaining good accuracy.

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An inversion algorithm for a banded matrix

Cited by 11 publications

References 12 publications

A probing method for computing the diagonal of a matrix inverse

A probing method for computing the diagonal of a matrix inverse

Efficient Computation of Sparse Matrix Functions for Large-Scale Electronic Structure Calculations: The CheSS Library

A New Unconditionally Stable Scheme for FDTD Method Using Associated Hermite Orthogonal Functions

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