Showing papers on "Divide-and-conquer eigenvalue algorithm published in 2018"

Frequency response as a surrogate eigenvalue problem in topology optimization

Abstract: Summary This article discusses the use of frequency response surrogates for eigenvalue optimization problems in topology optimization that may be used to avoid solving the eigenvalue problem. The motivation is to avoid complications that arise from multiple eigenvalues and the computational complexity associated with computation of eigenvalues in very large problems. Copyright © 2017 John Wiley & Sons, Ltd.

Approximating Test Statistics Using Eigenvalue Block Averaging

Abstract: We introduce and evaluate a new class of approximations to common test statistics in structural equation modeling. Such test statistics asymptotically follow the distribution of a weighted sum of i.i.d. chi-square variates, where the weights are eigenvalues of a certain matrix. The proposed eigenvalue block averaging (EBA) method involves creating blocks of these eigenvalues and replacing them within each block with the block average. The Satorra–Bentler scaling procedure is a special case of this framework, using one single block. The proposed procedure applies also to difference testing among nested models. We investigate the EBA procedure both theoretically in the asymptotic case, and with simulation studies for the finite-sample case, under both maximum likelihood and diagonally weighted least squares estimation. Comparison is made with 3 established approximations: Satorra–Bentler, the scaled and shifted, and the scaled F tests.

C 0 IPG adaptive algorithms for the biharmonic eigenvalue problem

Abstract: This paper focuses on C 0IPG adaptive algorithms for the biharmonic eigenvalue problem with the clamped boundary condition. We prove the reliability and efficiency of the a posteriori error indicator of the approximating eigenfunctions and analyze the reliability of the a posteriori error indicator of the approximating eigenvalues. We present two adaptive algorithms, and numerical experiments indicate that both algorithms are efficient.

The Eigenvalue Problem in Phase-Space

Abstract: We formulate the standard quantum mechanical eigenvalue problem in quantum phase space. The equation obtained involves the c-function that corresponds to the quantum operator. We use the Wigner distribution for the phase space function. We argue that the phase space eigenvalue equation obtained has, in addition to the proper solutions, improper solutions. That is, solutions for which no wave function exists which could generate the distribution. We discuss the conditions for ascertaining whether a position momentum function is a proper phase space distribution. We call these conditions psi-representability conditions, and show that if these conditions are imposed, one extracts the correct phase space eigenfunctions. We also derive the phase space eigenvalue equation for arbitrary phase space distributions functions. © 2017 Wiley Periodicals, Inc.

On one class eigenvalue problem with eigenvalue parameter in the boundary condition at one end-point

Abstract: In the paper we investigate the spectrum of operator corresponding to eigenvalue problm with parameter dependent boundary condition. Trace formula for that operator is also established

Fast and Stable Low-Rank Symmetric Eigen-Update

Abstract: Author(s): Liang, Ruochen | Advisor(s): Gu, Ming | Abstract: Updating the eigensystem of modified symmetric matrices is an important task arising from certain fields of applications The core of the problem is computing the eigenvalues and orthogonal eigenvectors of a diagonal matrix with symmetric low rank modifications, {\em ie} $D + UHU^T$ The eigenproblem of this type of matrix has long been studied since Golub etal~\cite{pa} proposed theoretical considerations about it Currently, there exist methods to compute eigenvalues accurately, but the difficulty remains in numerical stability and orthogonality of computed eigenvectors The main contribution of this thesis is a new method to compute all the eigenvalues and eigenvectors of a real diagonal matrix with a symmetric low rank perturbation The algorithm computes an orthogonal matrix $Q = [q_1, q_2, \ldots, q_n]$ and a diagonal matrix $\Lambda = diag\{\lambda_1, \lambda_2, \ldots, \lambda_n\}$ such that $AQ = Q\Lambda$ Here the matrix $A = D + UHU^T$ has the special structure that $D \in\mathbb{R}^{n\times n}$ is a diagonal matrix, $U\in\mathbb{R}^{n\times r}$ is a column orthorgonal matrix and $H\in\mathbb{R}^{r\times r}$ is a symmetric matrix $n$ is the dimension of $A$ and $rlAside from solving the eigensystem update problem mentioned above, our proposed method can also be used in the divide and conquer eigenvalue algorithm Cuppen's divide and conquer algorithm~\cite{cuppen} solves a rank-one update of eigensystem in its merge step for a symmetric tri-diagonal matrix A symmetric banded matrix will require solving the eigensystem of a low-rank perturbed diagonal matrix, {\em ie}, $D+UHU^T$ Efficient solution to this problem in the merge step can potentially enable application of divide and conquer algorithm directly on symmetric banded matrixIn our proposed algorithm, eigenpairs are mostly computed by Rayleigh Quotient Iteration safe-guarded with bisection, with each eigenpair requiring $O(nr^2)$ flops to compute Hence the overall computational complexity for our algorithm is $O(n^2r^2)$ This is an appealing quadratic algorithm since $r$ is usually considered a small constant relative to $n$ To ensure numerical stability, eigenvectors corresponding to eigenvalue clusters are computed through a special \textit{orthogonal deflation} method that completely avoids re-orthogonalization In case of tight clusters, extended precision arithmetic is used for eigenvectors corresponding to close eigenvaluesWe present both theoretical analysis and numerical results to support our claim that the proposed algorithm is numerically stable