55 research outputs found
Verified partial eigenvalue computations using contour integrals for Hermitian generalized eigenproblems
We propose a verified computation method for partial eigenvalues of a
Hermitian generalized eigenproblem. The block Sakurai-Sugiura Hankel method, a
contour integral-type eigensolver, can reduce a given eigenproblem into a
generalized eigenproblem of block Hankel matrices whose entries consist of
complex moments. In this study, we evaluate all errors in computing the complex
moments. We derive a truncation error bound of the quadrature. Then, we take
numerical errors of the quadrature into account and rigorously enclose the
entries of the block Hankel matrices. Each quadrature point gives rise to a
linear system, and its structure enables us to develop an efficient technique
to verify the approximate solution. Numerical experiments show that the
proposed method outperforms a standard method and infer that the proposed
method is potentially efficient in parallel.Comment: 15 pages, 4 figures, 1 tabl
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Complex moment-based methods for differential eigenvalue problems
This paper considers computing partial eigenpairs of differential eigenvalue
problems (DEPs) such that eigenvalues are in a certain region on the complex
plane. Recently, based on a "solve-then-discretize" paradigm, an operator
analogue of the FEAST method has been proposed for DEPs without discretization
of the coefficient operators. Compared to conventional "discretize-then-solve"
approaches that discretize the operators and solve the resulting matrix
problem, the operator analogue of FEAST exhibits much higher accuracy; however,
it involves solving a large number of ordinary differential equations (ODEs).
In this paper, to reduce the computational costs, we propose operation
analogues of Sakurai-Sugiura-type complex moment-based eigensolvers for DEPs
using higher-order complex moments and analyze the error bound of the proposed
methods. We show that the number of ODEs to be solved can be reduced by a
factor of the degree of complex moments without degrading accuracy, which is
verified by numerical results. Numerical results demonstrate that the proposed
methods are over five times faster compared with the operator analogue of FEAST
for several DEPs while maintaining almost the same high accuracy. This study is
expected to promote the "solve-then-discretize" paradigm for solving DEPs and
contribute to faster and more accurate solutions in real-world applications.Comment: 26 pages, 9 figure
Contour integral method for obtaining the self-energy matrices of electrodes in electron transport calculations
We propose an efficient computational method for evaluating the self-energy
matrices of electrodes to study ballistic electron transport properties in
nanoscale systems. To reduce the high computational cost incurred in large
systems, a contour integral eigensolver based on the Sakurai-Sugiura method
combined with the shifted biconjugate gradient method is developed to solve
exponential-type eigenvalue problem for complex wave vectors. A remarkable
feature of the proposed algorithm is that the numerical procedure is very
similar to that of conventional band structure calculations. We implement the
developed method in the framework of the real-space higher-order finite
difference scheme with nonlocal pseudopotentials. Numerical tests for a wide
variety of materials validate the robustness, accuracy, and efficiency of the
proposed method. As an illustration of the method, we present the electron
transport property of the free-standing silicene with the line defect
originating from the reversed buckled phases.Comment: 36 pages, 13 figures, 2 table
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