Eigenproblem Basics and Algorithms

IF 2.2 3区 综合性期刊 Q2 MULTIDISCIPLINARY SCIENCES Symmetry-Basel Pub Date : 2023-11-10 DOI:10.3390/sym15112046
Lorentz Jäntschi
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Abstract

Some might say that the eigenproblem is one of the examples people discovered by looking at the sky and wondering. Even though it was formulated to explain the movement of the planets, today it has become the ansatz of solving many linear and nonlinear problems. Formulation in the terms of the eigenproblem is one of the key tools to solve complex problems, especially in the area of molecular geometry. However, the basic concept is difficult without proper preparation. A review paper covering basic concepts and algorithms is very useful. This review covers the basics of the topic. Definitions are provided for defective, Hermitian, Hessenberg, modal, singular, spectral, symmetric, skew-symmetric, skew-Hermitian, triangular, and Wishart matrices. Then, concepts of characteristic polynomial, eigendecomposition, eigenpair, eigenproblem, eigenspace, eigenvalue, and eigenvector are subsequently introduced. Faddeev–LeVerrier, von Mises, Gauss–Jordan, Pohlhausen, Lanczos–Arnoldi, Rayleigh–Ritz, Jacobi–Davidson, and Gauss–Seidel fundamental algorithms are given, while others (Francis–Kublanovskaya, Gram–Schmidt, Householder, Givens, Broyden–Fletcher–Goldfarb–Shanno, Davidon–Fletcher–Powell, and Saad–Schultz) are merely discussed. The eigenproblem has thus found its use in many topics. The applications discussed include solving Bessel’s, Helmholtz’s, Laplace’s, Legendre’s, Poisson’s, and Schrödinger’s equations. The algorithm extracting the first principal component is also provided.
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特征问题基础与算法
有些人可能会说,特征问题是人们通过观察天空和思考而发现的例子之一。尽管它是用来解释行星运动的,但今天它已经成为解决许多线性和非线性问题的工具。特征问题的表述是解决复杂问题的关键工具之一,特别是在分子几何领域。然而,如果没有适当的准备,基本概念是困难的。一篇涵盖基本概念和算法的综述论文是非常有用的。这篇综述涵盖了这个主题的基础知识。给出了缺陷矩阵、厄米矩阵、海森伯格矩阵、模态矩阵、奇异矩阵、谱矩阵、对称矩阵、偏对称矩阵、偏厄米矩阵、三角形矩阵和Wishart矩阵的定义。接着介绍了特征多项式、特征分解、特征对、特征问题、特征空间、特征值、特征向量等概念。给出了Faddeev-LeVerrier、von Mises、Gauss-Jordan、Pohlhausen、Lanczos-Arnoldi、rayley - ritz、Jacobi-Davidson和Gauss-Seidel基本算法,而其他基本算法(Francis-Kublanovskaya、Gram-Schmidt、Householder、Givens、Broyden-Fletcher-Goldfarb-Shanno、Davidon-Fletcher-Powell和Saad-Schultz)仅进行了讨论。因此,特征问题在许多主题中都得到了应用。讨论的应用包括求解贝塞尔方程、亥姆霍兹方程、拉普拉斯方程、勒让德方程、泊松方程和Schrödinger方程。给出了第一主成分的提取算法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Symmetry-Basel
Symmetry-Basel MULTIDISCIPLINARY SCIENCES-
CiteScore
5.40
自引率
11.10%
发文量
2276
审稿时长
14.88 days
期刊介绍: Symmetry (ISSN 2073-8994), an international and interdisciplinary scientific journal, publishes reviews, regular research papers and short notes. Our aim is to encourage scientists to publish their experimental and theoretical research in as much detail as possible. There is no restriction on the length of the papers. Full experimental and/or methodical details must be provided, so that results can be reproduced.
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