Parallel reduction of four matrices to condensed form for a generalized matrix eigenvalue algorithm

2020 ◽  
Vol 86 (1) ◽  
pp. 153-178
Author(s):  
Nela Bosner
2021 ◽  
Vol 9 (1) ◽  
pp. 240-256
Author(s):  
Quanling Deng

Abstract It is well-known that the finite difference discretization of the Laplacian eigenvalue problem −Δu = λu leads to a matrix eigenvalue problem (EVP) Ax =λx where the matrix A is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in a recent work of Strang and MacNamara. We generalize the results and develop analytical solutions to certain generalized matrix eigenvalue problems (GEVPs) Ax = λBx which arise from the finite element method (FEM) and isogeometric analysis (IGA). The FEM matrices are corner-overlapped block-diagonal while the IGA matrices are almost Toeplitz-plus-Hankel. In fact, IGA with a correction that results in Toeplitz-plus-Hankel matrices gives a better numerical method. In this paper, we focus on finding the analytical eigenpairs to the GEVPs while developing better numerical methods is our motivation. Analytical solutions are also obtained for some polynomial eigenvalue problems (PEVPs). Lastly, we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for EVPs) for GEVPs and derive some trigonometric identities.


Author(s):  
Zhechao wang ◽  
Jiafan guo ◽  
Zhejun pan ◽  
Liping qiao ◽  
Jie liu ◽  
...  

An iterative criterion for the asymptotic steadiness of a linear descriptor system is considered. The criterion is based on an iterative algorithm for computing a generalized matrix sign-function. As an example, the problem of analyzing the asymptotic steadiness of a large descriptor system is given. Keywords linear descriptor system; steadiness criterion; matrix sign-function; search algorithm


2020 ◽  
Vol 13 (1) ◽  
pp. 196
Author(s):  
Mengmeng Meng ◽  
Weiguo Fan ◽  
Jianchang Lu ◽  
Xiaobin Dong ◽  
Hejie Wei

Qinghai-Tibet Plateau is a typical resource-rich but economically backward region in western China, and it is of great urgency to improve human well-being. Combined with previous scholars’ research and the characteristics of Qinghai-Tibet Plateau, this paper constructs an index system of human well-being including four dimensions: income and consumption, means of production, means of subsistence, and resource acquisition ability. Then, it uses generalized matrix method estimations to measure the influence of energy utilization and economic development on human well-being and makes a regression analysis on the influence of energy utilization and economic development on human well-being in various provinces in this region. It is found that per capita GDP and coke utilization promote the well-being of all dimensions, while the urban registered unemployment rate only promotes the well-being of means of subsistence. The utilization of gasoline and natural gas promotes income and consumption and inhibits the means of subsistence and resource acquisition ability, but they have opposite effects on means of production. The impacts of energy utilization and economic development in different provinces on human well-being are different. This study is of great significance to the related research aiming at improving people’s livelihood and promoting regional development.


1995 ◽  
Vol 19 (3) ◽  
pp. 432-440 ◽  
Author(s):  
E. Bampis ◽  
M. Elhaddad ◽  
Y. Manoussakis ◽  
M. Santha

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