# The Algebraic Eigenvalue Problem

About: This article is published in Mathematics of Computation.The article was published on 1966-10-01. It has received 2342 citation(s) till now. The article focuses on the topic(s): Algebraic number & Eigenvalues and eigenvectors.

Topics: Algebraic number (56%), Eigenvalues and eigenvectors (52%)

##### Citations

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TL;DR: Computationally efficient methods for random eigenvalue problems arising in the dynamics of multi-degree-of-freedom systems are developed using a Galerkin formulation to obtain the unknown coefficients.

Abstract: Uncertainties need to be taken into account in the dynamic analysis of complex structures. This is because in some cases uncertainties can have a significant impact on the dynamic response and ignoring it can lead to unsafe design. For complex systems with uncertainties, the dynamic response is characterised by the eigenvalues and eigenvectors of the underlying generalised matrix eigenvalue problem. This paper aims at developing computationally efficient methods for random eigenvalue problems arising in the dynamics of multi-degree-of-freedom systems. There are efficient methods available in the literature for obtaining eigenvalues of random dynamical systems. However, the computation of eigenvectors remains challenging due to the presence of a large number of random variables within a single eigenvector. To address this problem, we project the random eigenvectors on the basis spanned by the underlying deterministic eigenvectors and apply a Galerkin formulation to obtain the unknown coefficients. The overall approach is simplified using an iterative technique. Two numerical examples are provided to illustrate the proposed method. Full-scale Monte Carlo simulations are used to validate the new results.

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TL;DR: The results of numerical examples, including the analysis of real seismic data and financial data, show the promising empirical properties of the proposed principal component analysis method, which is useful for dimension reduction and feature extraction of multiple non-stationary time series.

Abstract: This paper proposes a new principal component analysis method in the wavelet domain, which is useful for dimension reduction and feature extraction of multiple non-stationary time series. The proposed method is constructed using a novel combination of eigenanalysis and the local wavelet spectrum defined in the locally stationary wavelet process. Therefore, we can expect the proposed method to reflect a more generalized non-stationary time series beyond some limited types of signals that existing methods have performed. We investigate the theoretical results of estimated principal components and their loadings. The results of numerical examples, including the analysis of real seismic data and financial data, show the promising empirical properties of the proposed approach.

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Abstract: We provide a modification to the quantum phase estimation algorithm (QPEA) inspired on classical windowing methods for spectral density estimation. From this modification we obtain an upper bound in the cost that implies a cubic improvement with respect to the algorithm's error rate. Numerical evaluation of the costs also demonstrates an improvement. Moreover, with similar techniques, we detail an iterative projective measurement method for ground state preparation that gives an exponential improvement over previous bounds using QPEA. Numerical tests that confirm the expected scaling behavior are also obtained. For these numerical tests we have used a Lattice Thirring model as testing ground. Using well-known perturbation theory results, we also show how to more appropriately estimate the cost scaling with respect to state error instead of evolution operator error.

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Abstract: We propose a learning-based approach for the sparse Gaussian Elimination. There are many hard combinatorial optimization problems in modern sparse solver. These NP-hard problems could be handled in the framework of Markov Decision Process, especially the Q-Learning technique. We proposed some Q-Learning algorithms for the main modules of sparse solver: minimum degree ordering, task scheduling and adaptive pivoting. Finally, we recast the sparse solver into the framework of Q-Learning.
Our study is the first step to connect these two classical mathematical models: Gaussian Elimination and Markov Decision Process. Our learning-based algorithm could help improve the performance of sparse solver, which has been verified in some numerical experiments.

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Abstract: This paper proposes a new algorithm, named Householder Dice (HD), for simulating dynamics on dense random matrix ensembles with translation-invariant properties. Examples include the Gaussian ensemble, the Haar-distributed random orthogonal ensemble, and their complex-valued counterparts. A "direct" approach to the simulation, where one first generates a dense $n \times n$ matrix from the ensemble, requires at least $\mathcal{O}(n^2)$ resource in space and time. The HD algorithm overcomes this $\mathcal{O}(n^2)$ bottleneck by using the principle of deferred decisions: rather than fixing the entire random matrix in advance, it lets the randomness unfold with the dynamics. At the heart of this matrix-free algorithm is an adaptive and recursive construction of (random) Householder reflectors. These orthogonal transformations exploit the group symmetry of the matrix ensembles, while simultaneously maintaining the statistical correlations induced by the dynamics. The memory and computation costs of the HD algorithm are $\mathcal{O}(nT)$ and $\mathcal{O}(nT^2)$, respectively, with $T$ being the number of iterations. When $T \ll n$, which is nearly always the case in practice, the new algorithm leads to significant reductions in runtime and memory footprint. Numerical results demonstrate the promise of the HD algorithm as a new computational tool in the study of high-dimensional random systems.