Reduction of nonselfadjoint operators to diagonal form

1965 ◽  
pp. 31-35
Author(s):  
L. A. Sahnovič
Keyword(s):  
Diagonal Form

A new version of the Fast Multipole Method for the Laplace equation in three dimensions

Acta Numerica ◽  
1997 ◽  
Vol 6 ◽  
pp. 229-269 ◽  
Author(s):  
Leslie Greengard ◽  
Vladimir Rokhlin
Keyword(s):  
Laplace Equation ◽  
Fast Multipole Method ◽  
High Accuracy ◽  
Three Dimensions ◽  
Potential Fields ◽  
Diagonal Form ◽  
Fast Multipole ◽  
Reasonable Cost ◽  
Multipole Method ◽  
Translation Operators

We introduce a new version of the Fast Multipole Method for the evaluation of potential fields in three dimensions. It is based on a new diagonal form for translation operators and yields high accuracy at a reasonable cost.

Convergence to diagonal form of block Jacobi-type methods

2014 ◽  
Vol 129 (3) ◽  
pp. 449-481 ◽  
Author(s):  
Vjeran Hari
Keyword(s):  
Diagonal Form

scholarly journals Wave Propagation Analysis in Composite Laminates Containing a Delamination Using a Three-Dimensional Spectral Element Method

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Fucai Li ◽  
Haikuo Peng ◽  
Xuewei Sun ◽  
Jinfu Wang ◽  
Guang Meng
Keyword(s):  
Wave Propagation ◽  
Lamb Wave ◽  
Composite Laminates ◽  
Composite Structures ◽  
Spectral Element Method ◽  
Three Dimensional ◽  
Spectral Element ◽  
Wave Mode ◽  
Diagonal Form ◽  
Element Method

A three-dimensional spectral element method (SEM) was developed for analysis of Lamb wave propagation in composite laminates containing a delamination. SEM is more efficient in simulating wave propagation in structures than conventional finite element method (FEM) because of its unique diagonal form of the mass matrix. Three types of composite laminates, namely, unidirectional-ply laminates, cross-ply laminates, and angle-ply laminates are modeled using three-dimensional spectral finite elements. Wave propagation characteristics in intact composite laminates are investigated, and the effectiveness of the method is validated by comparison of the simulation results with analytical solutions based on transfer matrix method. Different Lamb wave mode interactions with delamination are evaluated, and it is demonstrated that symmetric Lamb wave mode may be insensitive to delamination at certain interfaces of laminates while the antisymmetric mode is more suited for identification of delamination in composite structures.

Solution of two-qubit Rabi model with extended generalized rotating-wave approximation

2021 ◽  
pp. 2150213
Author(s):  
Zhanyuan Yan ◽  
Peihua Qu ◽  
BingBing Xu ◽  
Shihui Zhang ◽  
Jinying Ma
Keyword(s):  
Order Approximation ◽  
Photon Number ◽  
Energy Spectra ◽  
Hamiltonian Matrix ◽  
Diagonal Form ◽  
Coupling Region ◽  
Rotating Wave Approximation ◽  
Wave Approximation ◽  
Rabi Model ◽  
Rotating Wave

The generalized rotating-wave approximation (GRWA) method is extended to the two-qubit quantum Rabi model. In the first-order approximation (one photon exchange), the Hamiltonian matrix in photon number space is simplified by introducing two variational parameters. However, the Hamiltonian matrix is not a diagonalizable matrix yet. Furthermore, by presenting a constraint condition on coupling strength and atomic transition frequency, the Hamiltonian matrix is simplified and an effective solvable Hamiltonian with block diagonal form is obtained. In the even and odd parity space, the energy spectra and eigenstates of the two-qubit quantum Rabi model are achieved analytically. Most of the energy spectra, especially the lower energy levels, agree well with the numerical exact results in ultra-strong coupling region, and the ground state wave function can gives a fairly accurate result of mean photon number.

scholarly journals On the reduction of pairs of hermitian or symmetric matrices to diagonal form by congruence

1986 ◽  
Vol 73 ◽  
pp. 213-226 ◽  
Author(s):  
Yoo Pyo Hong ◽  
Roger A. Horn ◽  
Charles R. Johnson
Keyword(s):  
Diagonal Form ◽  
Symmetric Matrices

An Adaptive Orthogonal SSA Decomposition Algorithm for a Time Series

2018 ◽  
Vol 10 (01) ◽  
pp. 1850002 ◽  
Author(s):  
Kenji Kume ◽  
Naoko Nose-Togawa
Keyword(s):  
Time Series ◽  
Correlation Matrix ◽  
Singular Spectrum Analysis ◽  
Dimensional Subspace ◽  
Orthogonal Decomposition ◽  
Gram Matrix ◽  
Diagonal Form ◽  
Window Length ◽  
Original Time ◽  
Basis Vectors

Singular spectrum analysis (SSA) is a nonparametric spectral decomposition of a time series into arbitrary number of interpretable components. It involves a single parameter, window length [Formula: see text], which can be adjusted for the specific purpose of the analysis. After the decomposition of a time series, similar series are grouped to obtain the interpretable components by consulting with the [Formula: see text]-correlation matrix. To accomplish better resolution of the frequency spectrum, a larger window length [Formula: see text] is preferable and, in this case, the proper grouping is crucial for making the SSA decomposition. When the [Formula: see text]-correlation matrix does not have block-diagonal form, however, it is hard to adequately carry out the grouping. To avoid this, we propose a novel algorithm for the adaptive orthogonal decomposition of the time series based on the SSA scheme. The SSA decomposition sequences of the time series are recombined and the linear coefficients are determined so as to maximizing its squared norm. This results in an eigenvalue problem of the Gram matrix and we can obtain the orthonormal basis vectors for the [Formula: see text]-dimensional subspace. By the orthogonal projection of the original time series on these basis vectors, we can obtain adaptive orthogonal decomposition of the time series without the redundancy of the original SSA decomposition.

Controlled inertia tensor of a transformable spacecraft

2021 ◽  
Author(s):  
R.P. Simonyants ◽  
N.A. Alekhin ◽  
V.A. Tarasov
Keyword(s):  
Attitude Control ◽  
Dynamic Properties ◽  
Operating Conditions ◽  
Inertia Tensor ◽  
Simplified Model ◽  
Diagonal Form ◽  
Transformation Mechanism ◽  
External Disturbances ◽  
Type Transformation ◽  
Gravitational Moment

A simplified model of a transformable spacecraft is considered, including a rod-type transformation mechanism with movable weights. The mechanism can be used to adapt the dynamic properties of the spacecraft to the environment or the operating conditions of on-board systems, for example, to counter the moments of external disturbances during attitude control and angular stabilization. By changing the position of the transformation mechanism, the spacecraft inertia tensor can be put in diagonal form, which makes it possible to exclude the force interconnections between the channels and to eliminate the constant component of the gravitational moment. For a simplified model of the transformation mechanism, we establish the analytical dependence of the components of the inertia tensor on the parameters determining the position of the transformation mechanism. It is shown that by adjusting the moving mass, which is 0.5% of the entire spacecraft mass, we obtain the spacecraft configuration that ensures the diagonality of the inertia tensor.

Invariant Factors Under Rank One Perturbations

1980 ◽  
Vol 32 (1) ◽  
pp. 240-245 ◽  
Author(s):  
Robert C. Thompson
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Matrix Multiplication ◽  
Diagonal Form ◽  
Principal Ideal Domain ◽  
Ideal Domain ◽  
Zero Divisors ◽  
Rank One ◽  
Invariant Factors

Let R be a principal ideal domain, i.e., a commutative ring without zero divisors in which every ideal is principal. The invariant factors of a matrix A with entries in R are the diagonal elements when A is converted to a diagonal form D = UAV, where U, V have entries in R and are unimodular (invertible over R), and the diagonal entries d1 …, dn of D form a divisibility chain: d1|d2| … |dn. Very little has been proved about how invariant factors may change when matrices are added. This is in contrast to the corresponding question for matrix multiplication, where much information is now available [6].

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