The Modified Power Method for Solving the Eigenvalue Problem with the Use of Idempotent Matrix

2015 ◽  
Vol 712 ◽  
pp. 43-48
Author(s):  
Rafał Palej ◽  
Artur Krowiak ◽  
Renata Filipowska
Keyword(s):  
Eigenvalue Problem ◽  
Rate Of Convergence ◽  
Rayleigh Quotient ◽  
Frobenius Norm ◽  
Dominant Eigenvalue ◽  
Power Method ◽  
New Approach ◽  
Idempotent Matrix ◽  
The Given ◽  
Scaling Coefficient

The work presents a new approach to the power method serving the purpose of solving the eigenvalue problem of a matrix. Instead of calculating the eigenvector corresponding to the dominant eigenvalue from the formula , the idempotent matrix B associated with the given matrix A is calculated from the formula , where m stands for the method’s rate of convergence. The scaling coefficient ki is determined by the quotient of any norms of matrices Bi and or by the reciprocal of the Frobenius norm of matrix Bi. In the presented approach the condition for completing calculations has the form. Once the calculations are completed, the columns of matrix B are vectors parallel to the eigenvector corresponding to the dominant eigenvalue, which is calculated from the Rayleigh quotient. The new approach eliminates the necessity to use a starting vector, increases the rate of convergence and shortens the calculation time when compared to the classic method.

A Unified Approach to H∞ Control of Singular Markovian Jump Systems

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Guoliang Wang ◽  
Hongyi Li
Keyword(s):  
Transition Probability ◽  
Markovian Jump Systems ◽  
Linear Matrix ◽  
Markovian Jump ◽  
New Approach ◽  
Control Approach ◽  
Singular Markovian Jump Systems ◽  
Jump Systems ◽  
Bounded Uncertainties ◽  
The Given

This paper considers the H∞ control problem for a class of singular Markovian jump systems (SMJSs), where the jumping signal is not always available. The main contribution of this paper introduces a new approach to a mode-independent (MI) H∞ controller by exploiting the nonfragile method. Based on the given method, a unified control approach establishing a direct connection between mode-dependent (MD) and mode-independent controllers is presented, where both existence conditions are given in terms of linear matrix inequalities. Moreover, another three cases of transition probability rate matrix (TRPM) with elementwise bounded uncertainties, being partially unknown and to be designed are analyzed, respectively. Numerical examples are used to demonstrate the effectiveness of the proposed methods.

scholarly journals Rayleigh Quotient Methods for Estimating Common Roots of Noisy Univariate Polynomials

2019 ◽  
Vol 19 (1) ◽  
pp. 147-163 ◽  
Author(s):  
Alwin Stegeman ◽  
Lieven De Lathauwer
Keyword(s):  
Eigenvalue Problem ◽  
Simulation Study ◽  
Rayleigh Quotient ◽  
Tensor Algebra ◽  
Symbolic Algebra ◽  
Starting Values ◽  
New Methods ◽  
Sylvester Matrix ◽  
Univariate Polynomials ◽  
The Common

AbstractThe problem is considered of approximately solving a system of univariate polynomials with one or more common roots and its coefficients corrupted by noise. The goal is to estimate the underlying common roots from the noisy system. Symbolic algebra methods are not suitable for this. New Rayleigh quotient methods are proposed and evaluated for estimating the common roots. Using tensor algebra, reasonable starting values for the Rayleigh quotient methods can be computed. The new methods are compared to Gauss–Newton, solving an eigenvalue problem obtained from the generalized Sylvester matrix, and finding a cluster among the roots of all polynomials. In a simulation study it is shown that Gauss–Newton and a new Rayleigh quotient method perform best, where the latter is more accurate when other roots than the true common roots are close together.

scholarly journals Multiple crack identification in stepped beam by measurements of natural frequencies

2014 ◽  
Vol 36 (2) ◽  
pp. 119-132
Author(s):  
Nguyen Tien Khiem ◽  
Duong The Hung ◽  
Vu Thi An Ninh
Keyword(s):  
Crack Detection ◽  
Natural Frequencies ◽  
Rayleigh Quotient ◽  
Crack Identification ◽  
Multiple Cracks ◽  
Simple Relationship ◽  
Multiple Crack ◽  
Scanning Method ◽  
New Approach ◽  
Efficient Procedure

A new approach is proposed for calculating natural frequencies and crack detection in a stepped cantilever beam with arbitrary number of cracks. This is based an explicit expression of the natural frequencies in term of crack parameter derived in the form similar to the so-called Rayleigh quotient for vibrating beam. The obtained simple relationship between natural frequencies and crack parameters enables not only accurate calculating the natural frequencies but also to develop an efficient procedure for detecting multiple cracks from given natural frequencies. The proposed technique called crack scanning method is illustrated and validated by numerical results.

Genetic Algorithms in Fuzzy Engineering Design

1999 ◽  
Author(s):  
Ralf Schleiffer ◽  
Hans-Jürgen Sebastian ◽  
Erik K. Antonsson
Keyword(s):  
Engineering Design ◽  
Clustering Algorithm ◽  
Computer Application ◽  
Test Problems ◽  
Important Class ◽  
New Approach ◽  
Design Task ◽  
Evolutionary Technique ◽  
The Given ◽  
Real World Problems

Abstract Problems in the field of engineering design represent an important class of real world problems that typically require a fuzzy and imprecise representation. This article presents and discusses a new approach to model this type of problem, by incorporating linguistic descriptions together with a variety of user-defined trade-off strategies. An interactive computer application is introduced, using stochastic optimization to solve the design task by producing a specially desired output under the given environmental conditions which are partly caused by the personal preferences of the engineer and by the expectations of the customer. It utilizes a randomized evolutionary technique, made suitable for the class of problems at hand, to generate and to optimize design solutions that are later identified by a clustering algorithm. Moreover test problems that were solved by the application are considered. In all cases the good solutions were obtained by evaluating only an extremely small fraction of all possible designs.

A simultaneous decomposition for three quaternion tensors with applications in color video signal processing

2020 ◽  
pp. 1-35
Author(s):  
Zhuo-Heng He ◽  
Chen Chen ◽  
Xiang-Xiang Wang
Keyword(s):  
Signal Processing ◽  
Video Signal ◽  
New Approach ◽  
Video Signal Processing ◽  
Encryption And Decryption ◽  
Color Video ◽  
The Given ◽  
Einstein Product

In this paper, we establish a simultaneous decomposition for three quaternion tensors via Einstein product. This simultaneous decomposition transforms the given three quaternion tensors into nice forms which have only 1 and 0. We conclude with an application in the color video signal processing. This new approach only need to store four keys to realize the simultaneous encryption and decryption of three videos.

scholarly journals A New Approach for Solving the Undamped Helmholtz Oscillator for the Given Arbitrary Initial Conditions and Its Physical Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Alvaro H. Salas S ◽  
Jairo E. Castillo H ◽  
Darin J. Mosquera P
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Analytical Solution ◽  
Initial Conditions ◽  
Negative Ions ◽  
Nonlinear Problems ◽  
New Approach ◽  
Weierstrass Elliptic Function ◽  
Electronegative Plasma ◽  
The Given

In this paper, a new analytical solution to the undamped Helmholtz oscillator equation in terms of the Weierstrass elliptic function is reported. The solution is given for any arbitrary initial conditions. A comparison between our new solution and the numerical approximate solution using the Range Kutta approach is performed. We think that the methodology employed here may be useful in the study of several nonlinear problems described by a differential equation of the form z ″ = F z in the sense that z = z t . In this context, our solutions are applied to some physical applications such as the signal that can propagate in the LC series circuits. Also, these solutions were used to describe and investigate some oscillations in plasma physics such as oscillations in electronegative plasma with Maxwellian electrons and negative ions.

scholarly journals A New Approach to Newton-Type Polynomial Interpolation with Parameters

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Le Zou ◽  
Liangtu Song ◽  
Xiaofeng Wang ◽  
Thomas Weise ◽  
Yanping Chen ◽  
...  
Keyword(s):  
Polynomial Interpolation ◽  
Information Matrix ◽  
Divided Differences ◽  
Mathematical Representation ◽  
Interpolation Algorithm ◽  
New Approach ◽  
Multiple Parameters ◽  
Parameter Values ◽  
The Given ◽  
Interpolation Functions

Newton’s interpolation is a classical polynomial interpolation approach and plays a significant role in numerical analysis and image processing. The interpolation function of most classical approaches is unique to the given data. In this paper, univariate and bivariate parameterized Newton-type polynomial interpolation methods are introduced. In order to express the divided differences tables neatly, the multiplicity of the points can be adjusted by introducing new parameters. Our new polynomial interpolation can be constructed only based on divided differences with one or multiple parameters which satisfy the interpolation conditions. We discuss the interpolation algorithm, theorem, dual interpolation, and information matrix algorithm. Since the proposed novel interpolation functions are parametric, they are not unique to the interpolation data. Therefore, its value in the interpolant region can be adjusted under unaltered interpolant data through the parameter values. Our parameterized Newton-type polynomial interpolating functions have a simple and explicit mathematical representation, and the proposed algorithms are simple and easy to calculate. Various numerical examples are given to demonstrate the efficiency of our method.

A New Approach for Supervised Dimensionality Reduction

2018 ◽  
Vol 14 (4) ◽  
pp. 20-37 ◽  
Author(s):  
Yinglei Song ◽  
Yongzhong Li ◽  
Junfeng Qu
Keyword(s):  
Eigenvalue Problem ◽  
Dimensionality Reduction ◽  
Image Databases ◽  
Data Sets ◽  
Data Set ◽  
New Approach ◽  
Local Structures ◽  
Benchmark Data ◽  
Global And Local

This article develops a new approach for supervised dimensionality reduction. This approach considers both global and local structures of a labelled data set and maximizes a new objective that includes the effects from both of them. The objective can be approximately optimized by solving an eigenvalue problem. The approach is evaluated based on a few benchmark data sets and image databases. Its performance is also compared with a few other existing approaches for dimensionality reduction. Testing results show that, on average, this new approach can achieve more accurate results for dimensionality reduction than existing approaches.

A New Approach to the Classification and Measurement of Stuttering

1972 ◽  
Vol 37 (2) ◽  
pp. 242-251 ◽  
Author(s):  
Eleanor S. Wertheim
Keyword(s):  
Quantitative Relationship ◽  
Management Plan ◽  
Clinical Validity ◽  
New Approach ◽  
Quantitative Estimates ◽  
The Social ◽  
Method Of Measurement ◽  
Qualitative Pattern ◽  
The Stability ◽  
The Given

A quantitative, multidimensional approach to clinical diagnosis and management of stuttering is proposed. This approach would involve (1) the measurement of stuttering behavior under a number of contrived social conditions relevant to the given population of stutterers; (2) a method of measurement which provides separate quantitative estimates of the qualitative aspects of the stuttering pattern, for example, total frequency of disfluencies as well as incidence of hard blocking; (3) an analysis of the quantitative relationship between the social context and the qualitative pattern of stuttering and its severity; (4) a diagnostic formulation of “stutterogenic” situations for a given individual and of the stability and severity of the stuttering pattern across the total range of situations; and (5) a management plan geared to these diagnostic considerations. The approach is illustrated, using the findings of an experimental study primarily designed to test the author’s recently postulated bioadaptive theory of stuttering. Further research is needed to refine the proposed diagnostic approach and to establish its clinical validity and usefulness.

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