scholarly journals Weighted Block Golub-Kahan-Lanczos Algorithms for Linear Response Eigenvalue Problem

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 53 ◽  
Author(s):  
Hongxiu Zhong ◽  
Zhongming Teng ◽  
Guoliang Chen
Keyword(s):  
Eigenvalue Problem ◽  
Error Bounds ◽  
Linear Response ◽  
Numerical Instability ◽  
Lanczos Algorithm ◽  
Numerical Examples ◽  
Block Algorithm ◽  
Restart Strategy ◽  
New Algorithms

In order to solve all or some eigenvalues lied in a cluster, we propose a weighted block Golub-Kahan-Lanczos algorithm for the linear response eigenvalue problem. Error bounds of the approximations to an eigenvalue cluster, as well as their corresponding eigenspace, are established and show the advantages. A practical thick-restart strategy is applied to the block algorithm to eliminate the increasing computational and memory costs, and the numerical instability. Numerical examples illustrate the effectiveness of our new algorithms.

scholarly journals Rayleigh-Ritz Majorization Error Bounds for the Linear Response Eigenvalue Problem

2019 ◽  
Vol 17 (1) ◽  
pp. 653-667
Author(s):  
Zhongming Teng ◽  
Hong-Xiu Zhong
Keyword(s):  
Eigenvalue Problem ◽  
Quantum Chemistry ◽  
Error Bounds ◽  
Linear Response ◽  
Computational Quantum Chemistry ◽  
Matrix Anal ◽  
Deflating Subspaces ◽  
Important Notion ◽  
Theoretical Foundations ◽  
Computational Quantum

Abstract In the linear response eigenvalue problem arising from computational quantum chemistry and physics, one needs to compute a few of smallest positive eigenvalues together with the corresponding eigenvectors. For such a task, most of efficient algorithms are based on an important notion that is the so-called pair of deflating subspaces. If a pair of deflating subspaces is at hand, the computed approximated eigenvalues are partial eigenvalues of the linear response eigenvalue problem. In the case the pair of deflating subspaces is not available, only approximate one, in a recent paper [SIAM J. Matrix Anal. Appl., 35(2), pp.765-782, 2014], Zhang, Xue and Li obtained the relationships between the accuracy in eigenvalue approximations and the distances from the exact deflating subspaces to their approximate ones. In this paper, we establish majorization type results for these relationships. From our majorization results, various bounds are readily available to estimate how accurate the approximate eigenvalues based on information on the approximate accuracy of a pair of approximate deflating subspaces. These results will provide theoretical foundations for assessing the relative performance of certain iterative methods in the linear response eigenvalue problem.

scholarly journals A FEAST Algorithm for the Linear Response Eigenvalue Problem

Algorithms ◽  
2019 ◽  
Vol 12 (9) ◽  
pp. 181 ◽  
Author(s):  
Zhongming Teng ◽  
Linzhang Lu
Keyword(s):  
Eigenvalue Problem ◽  
Quantum Chemistry ◽  
Linear Response ◽  
Contour Integration ◽  
Numerical Examples ◽  
Complex Contour ◽  
Convergence Results

In the linear response eigenvalue problem arising from quantum chemistry and physics, one needs to compute several positive eigenvalues together with the corresponding eigenvectors. For such a task, in this paper, we present a FEAST algorithm based on complex contour integration for the linear response eigenvalue problem. By simply dividing the spectrum into a collection of disjoint regions, the algorithm is able to parallelize the process of solving the linear response eigenvalue problem. The associated convergence results are established to reveal the accuracy of the approximated eigenspace. Numerical examples are presented to demonstrate the effectiveness of our proposed algorithm.

scholarly journals New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices

2019 ◽  
Vol 17 (1) ◽  
pp. 1599-1614
Author(s):  
Zhiwu Hou ◽  
Xia Jing ◽  
Lei Gao
Keyword(s):  
Linear Algebra ◽  
Error Bound ◽  
Linear Complementarity Problem ◽  
Error Bounds ◽  
Complementarity Problems ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Numerical Examples ◽  
Numer Algor ◽  
Better Than

Abstract A new error bound for the linear complementarity problem (LCP) of Σ-SDD matrices is given, which depends only on the entries of the involved matrices. Numerical examples are given to show that the new bound is better than that provided by García-Esnaola and Peña [Linear Algebra Appl., 2013, 438, 1339–1446] in some cases. Based on the obtained results, we also give an error bound for the LCP of SB-matrices. It is proved that the new bound is sharper than that provided by Dai et al. [Numer. Algor., 2012, 61, 121–139] under certain assumptions.

scholarly journals Sensitivity analysis of waveguide eigenvalue problems

2011 ◽  
Vol 9 ◽  
pp. 85-89 ◽  
Author(s):  
N. Burschäpers ◽  
S. Fiege ◽  
R. Schuhmann ◽  
A. Walther
Keyword(s):  
Sensitivity Analysis ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Wave Number ◽  
Design Parameters ◽  
Dielectric Waveguides ◽  
Integration Technique ◽  
Numerical Examples ◽  
Numerical Effort ◽  
Complex Solutions

Abstract. We analyze the sensitivity of dielectric waveguides with respect to design parameters such as permittivity values or simple geometric dependencies. Based on a discretization using the Finite Integration Technique the eigenvalue problem for the wave number is shown to be non-Hermitian with possibly complex solutions even in the lossless case. Nevertheless, the sensitivity can be obtained with negligible numerical effort. Numerical examples demonstrate the validity of the approach.

scholarly journals Trace minimization method via penalty for linear response eigenvalue problems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yadan Chen ◽  
Yuan Shen ◽  
Shanshan Liu
Keyword(s):  
Eigenvalue Problem ◽  
Linear Response ◽  
Large Scale ◽  
Eigenvalue Problems ◽  
Minimum Principle ◽  
Gradient Descent Method ◽  
Step Size ◽  
Minimization Method ◽  
Energy Excitation ◽  
Orthogonality Constraint

<p style='text-indent:20px;'>In various applications, such as the computation of energy excitation states of electrons and molecules, and the analysis of interstellar clouds, the linear response eigenvalue problem, which is a special type of the Hamiltonian eigenvalue problem, is frequently encountered. However, traditional eigensolvers may not be applicable to this problem owing to its inherently large scale. In fact, we are usually more interested in computing some of the smallest positive eigenvalues. To this end, a trace minimum principle optimization model with orthogonality constraint has been proposed. On this basis, we propose an unconstrained surrogate model called trace minimization via penalty, and we establish its equivalence with the original constrained model, provided that the penalty parameter is larger than a certain threshold. By avoiding the orthogonality constraint, we can use a gradient-type method to solve this model. Specifically, we use the gradient descent method with Barzilai–Borwein step size. Moreover, we develop a restarting strategy for the proposed algorithm whereby higher accuracy and faster convergence can be achieved. This is verified by preliminary experimental results.</p>

scholarly journals Ball convergence theorem for a Steffensen-type third-order method

2017 ◽  
Vol 51 (1) ◽  
pp. 1-14
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Nonlinear Equation ◽  
Error Bounds ◽  
Order Method ◽  
Third Order ◽  
Numerical Examples ◽  
First Derivative ◽  
Fourth Derivative ◽  
Ball Convergence ◽  
Computable Error Bounds ◽  
Local Convergence Analysis

We present a local convergence analysis for a family of Steffensen-type third-order methods in order to approximate a solution of a nonlinear equation. We use hypothesis up to the first derivative in contrast to earlier studies such as [2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] using hypotheses up to the fourth derivative. This way the applicability of these methods is extended under weaker hypothesis. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.

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