A projection method for generalized eigenvalue problems using numerical integration

Sakurai, Tetsuya; Sugiura, Hiroshi

doi:10.1016/s0377-0427(03)00565-x

Cited by 304 publications

(313 citation statements)

References 15 publications

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“…Note that this procedure can take full advantage of approximated wavefunction |ψ T =1,T z =0 with the SS method, unlike the Lanczos method. Its detailed procedure has been fully discussed in Refs [9,10]. Through several applications [4][5][6], we confirmed that this method works well and quite efficiently.…”

supporting

confidence: 69%

“…In the case of large-scale shell-model calculations, this calculation becomes very difficult or impossible to carry out. For We have proposed a new shell-model method [9] based on the Sakurai-Sugiura (SS) method [10] where largescale shell-model diagonalization can be solved by the help of the Cauchy integral on the complex plane. For this formulation, a central quantity is moments μ p defined as…”

mentioning

confidence: 99%

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Isospin symmetry breaking and large-scale shell-model calculations with the Sakurai-Sugiura method

Mizusaki

Kaneko

Sun

et al. 2015

EPJ Web of Conferences

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Abstract. Recently isospin symmetry breaking for mass 60-70 region has been investigated based on large-scale shell-model calculations in terms of mirror energy differences (MED), Coulomb energy differences (CED) and triplet energy differences (TED). Behind these investigations, we have encountered a subtle problem in numerical calculations for odd-odd N = Z nuclei with large-scale shell-model calculations. Here we focus on how to solve this subtle problem by the Sakurai-Sugiura (SS) method, which has been recently proposed as a new diagonalization method and has been successfully applied to nuclear shell-model calculations.The isospin symmetry breaking is one of the current topics in nuclear structure physics. An asymmetry between spectra of isospin analogue states with mirror pair nuclei has been extensively investigated in the upper sd− and the lower f p−shell regions [1], and in mass 60-70 region [2][3][4][5][6]. This asymmetry, that is, isospin symmetry breaking arises partly due to the Coulomb force and partly due to the strong nucleon-nucleon interaction.To analyze this isospin symmetry breaking, mirror energy differences (MED) [7], Coulomb energy differences (CED)[8] and triplet energy differences(TED) [7] have been discussed. The MED is a measure of charge symmetry breaking in an effective interaction and is defined aswhere E x J,T T z is the excitation energies of analog states with spin J and isospin T, T z . The TED of T = 1 states in triplet nuclei is a measure of the charge-independence breaking and is defined asFor the TED, we have to evaluate the energy of the T = 1 and T z = 0 state for odd-odd N = Z nuclei precisely, employing shell-model calculations with isospin non-conserving interactions [5,6]. This calculation involves a rather subtle problem in numerical calculations, especially for large-scale shell-model calculation with the widely used Lanczos method.For the p f -shell or larger shells, we need largescale shell-model calculations where the M-scheme (shellmodel space with definite total magnetic quantum number) is often used. Third component of isospin, T z , is also given, choosing proton and neutron numbers. As the shellmodel interaction is rotationally invariant, the obtained eigenstates naturally have a good total angular momentum. For isospin-conserving interactions, they also have a good total isospin. This property is often utilized by taking a state with good total angular momentum and isospin as an initial one of the Lanczos method.Here we consider isospin symmetry breaking with large-scale shell-model calculations. Once the shell-model interaction contains several isospin symmetry breaking terms, total isospin is no longer a conserved quantity. This non-conservation is not, however, quantitatively significant [5] but it brings numerical difficulty for Lanczos diagonalizations, especially for odd-odd N = Z nuclei with large-scale shell-model calculations.Without isospin symmetry breaking, the low-lying states of odd-odd N = Z nuclei have T = 0 or T = 1. As the ground state or l...

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confidence: 69%

mentioning

confidence: 99%

Isospin symmetry breaking and large-scale shell-model calculations with the Sakurai-Sugiura method

Mizusaki

Kaneko

Sun

et al. 2015

EPJ Web of Conferences

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“…Later on Tetsuya Sakurai joined us in our study and co-authored some papers [13,17,12]. To solve eigenvalue problems, Sakurai and his co-authors applied the idea of the generalized eigenvalue problem involving the Hankel and shifted Hankel matrix using moments based on the resolvent function [18,10,15,9,19,25,1,2]. Eric Polizzi and co-authors also used contour integrals based on the resolvent function resulting in the FEAST algorithm [16,22,7].…”

Section: Introductionmentioning

confidence: 99%

Designing rational filter functions for solving eigenvalue problems by contour integration

Barel

2016

Linear Algebra and its Applications

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Solving (nonlinear) eigenvalue problems by contour integration, requires an e↵ective discretization for the corresponding contour integrals. In this paper it is shown that good rational filter functions can be computed using (nonlinear least squares) optimization techniques as opposed to designing those functions based on a thorough understanding of complex analysis. The conditions that such an effective filter function should satisfy, are derived and translated in a nonlinear least squares optimization problem solved by optimization algorithms from Tensorlab. Numerical experiments illustrate the validity of this approach.Keywords : linear, polynomial, nonlinear eigenvalue problems; contour integration; filter function; rational approximation; nonlinear least squares; resolvent. MSC : Primary : 65-F15, Secondary : 47-J10, 30-E05. Designing rational filter functions for solving eigenvalue problems by contour integrationMarc Van Barel ⇤ January 20, 2015Abstract Solving (nonlinear) eigenvalue problems by contour integration, requires an e↵ective discretization for the corresponding contour integrals. In this paper it is shown that good rational filter functions can be computed using (nonlinear least squares) optimization techniques as opposed to designing those functions based on a thorough understanding of complex analysis. The conditions that such an e↵ective filter function should satisfy, are derived and translated in a nonlinear least squares optimization problem solved by optimization algorithms from Tensorlab. Numerical experiments illustrate the validity of this approach.

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“…The spectrum sweeping method is not to be confused with another set of methods under the name of "spectrum slicing" methods [22,23,24,25,26,27]. The idea of the spectrum slicing methods is still to obtain a partial diagonalization of the matrix A.…”

Section: Introductionmentioning

confidence: 99%

Randomized estimation of spectral densities of large matrices made accurate

Lin

2016

Numer. Math.

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Abstract. For a large Hermitian matrix A ∈ C N ×N , it is often the case that the only affordable operation is matrix-vector multiplication. In such case, randomized method is a powerful way to estimate the spectral density (or density of states) of A. However, randomized methods developed so far for estimating spectral densities only extract information from different random vectors independently, and the accuracy is therefore inherently limitedwhere Nv is the number of random vectors.In this paper we demonstrate that the "O(1/ √ Nv) barrier" can be overcome by taking advantage of the correlated information of random vectors when properly filtered by polynomials of A. Our method uses the fact that the estimation of the spectral density essentially requires the computation of the trace of a series of matrix functions that are numerically low rank. By repeatedly applying A to the same set of random vectors and taking different linear combination of the results, we can sweep through the entire spectrum of A by building such low rank decomposition at different parts of the spectrum. Under some assumptions, we demonstrate that a robust and efficient implementation of such spectrum sweeping method can compute the spectral density accurately with O(N 2 ) computational cost and O(N ) memory cost. Numerical results indicate that the new method can significantly outperform existing randomized methods in terms of accuracy. As an application, we demonstrate a way to accurately compute a trace of a smooth matrix function, by carefully balancing the smoothness of the integrand and the regularized density of states using a deconvolution procedure.

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A projection method for generalized eigenvalue problems using numerical integration

Cited by 304 publications

References 15 publications

Isospin symmetry breaking and large-scale shell-model calculations with the Sakurai-Sugiura method

Isospin symmetry breaking and large-scale shell-model calculations with the Sakurai-Sugiura method

Designing rational filter functions for solving eigenvalue problems by contour integration

Randomized estimation of spectral densities of large matrices made accurate

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