Accurate eigenvalue decomposition of real symmetric arrowhead matrices and applications

Stor, Nevena Jakovcevic; Slapničar, Ivan; Barlow, Jesse L.

doi:10.1016/j.laa.2013.10.007

Cited by 27 publications

(30 citation statements)

References 13 publications

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“…The form of a matrix A (q) n is particularly special; this is obtained by a permutation of the row (or column) 1 with the row (or column) q of a symmetric upward arrowhead matrix. On the other hand, it coincides with the structure of the inverse of a downward arrowhead matrix, whose element in the position (q, q) (1 < q < n), is equal to zero (see [7]). An analogous situation happens, in both cases, if we have a downward or upward arrowhead matrix, respectively.…”

Section: Introductionmentioning

confidence: 73%

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Inverse eigenproblems of real symmetric doubly arrowhead matrices

Pickmann-Soto

Arela-Pérez

Arancibia

et al. 2020

Proyecciones (Antofagasta)

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Section: Introductionmentioning

confidence: 73%

“…. , 7 and eigenpair ³ λ (7,4) , x (4)´. Again, the results show that the reconstructed matrix A is as expected.…”

Section: Firstmentioning

confidence: 99%

Inverse eigenproblems of real symmetric doubly arrowhead matrices

Pickmann-Soto

Arela-Pérez

Arancibia

et al. 2020

Proyecciones (Antofagasta)

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“…The middle matrix is a k + 1 × k + 1 half-arrowhead matrix, and we can compute its SVD USV * in O k 2 time [66]. The rank k + 1 TSVD of B + is then given by B + = U + S + V + * , with U + = U 0 0 1 U, S + = S, and V + = V q V .…”

Section: Building the Compressed Representation On The Flymentioning

confidence: 99%

Low rank compression in the numerical solution of the nonequilibrium Dyson equation

Kaye

Golež

2021

SciPost Phys.

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We propose a method to improve the computational and memory efficiency of numerical solvers for the nonequilibrium Dyson equation in the Keldysh formalism. It is based on the empirical observation that the nonequilibrium Green's functions and self energies arising in many problems of physical interest, discretized as matrices, have low rank off-diagonal blocks, and can therefore be compressed using a hierarchical low rank data structure. We describe an efficient algorithm to build this compressed representation on the fly during the course of time stepping, and use the representation to reduce the cost of computing history integrals, which is the main computational bottleneck. For systems with the hierarchical low rank property, our method reduces the computational complexity of solving the nonequilibrium Dyson equation from cubic to near quadratic, and the memory complexity from quadratic to near linear. We demonstrate the full solver for the Falicov-Kimball model exposed to a rapid ramp and Floquet driving of system parameters, and are able to increase feasible propagation times substantially. We present examples with 262 144 time steps, which would require approximately five months of computing time and 2.2 TB of memory using the direct time stepping method, but can be completed in just over a day on a laptop with less than 4 GB of memory using our method. We also confirm the hierarchical low rank property for the driven Hubbard model in the weak coupling regime within the GW approximation, and in the strong coupling regime within dynamical mean-field theory.

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“…We recall some basic features regarding arrowhead matrices , hub matrices , and directed multigraphs . In this work, only the finite case will be considered.…”

Section: Preliminariesmentioning

confidence: 99%

Associating hub‐directed multigraphs to arrowhead matrices

Marrero

Valdés

Villar

2017

Math Methods in App Sciences

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In this paper, we use the arrowhead matrices as a tool to study graph theory. More precisely, we deal with an interesting class of directed multigraphs, the hub‐directed multigraphs. We associate the arrowhead matrices with the adjacency matrices of a class of directed multigraph, and we obtain new properties of the second objects by using properties of the first ones. The hub‐directed multigraphs with potential use in applications are also defined. As main result, we show that a hub‐directed multigraph G(H) with adjacency matrix H∗ is a dominant hub‐directed multigraph if and only if H∗=CE, where C is the adjacency matrix of another directed multigraph and E is the adjacency matrix of a particular elementary dominant hub‐directed pseudo‐graph. Another decomposition of its Gram (arrowhead) matrix A=()H∗TH∗ is also given. Copyright © 2017 John Wiley & Sons, Ltd.

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Accurate eigenvalue decomposition of real symmetric arrowhead matrices and applications

Cited by 27 publications

References 13 publications

Inverse eigenproblems of real symmetric doubly arrowhead matrices

Inverse eigenproblems of real symmetric doubly arrowhead matrices

Low rank compression in the numerical solution of the nonequilibrium Dyson equation

Associating hub‐directed multigraphs to arrowhead matrices

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