scholarly journals REGULARITY RADIUS: PROPERTIES, APPROXIMATION AND A NOT A PRIORI EXPONENTIAL ALGORITHM

2018 ◽  
Vol 33 ◽  
pp. 122-136 ◽  
Author(s):  
David Hartman ◽  
Milan Hladik
Keyword(s):  
Approximation Algorithms ◽  
A Priori ◽  
Interval Matrix ◽  
Original Matrix ◽  
Totally Positive ◽  
Classes Of Matrices ◽  
The Matrix ◽  
Rank One ◽  
Radius Of Regularity ◽  
Special Classes

The radius of regularity, sometimes spelled as the radius of nonsingularity, is a measure providing the distance of a given matrix to the nearest singular one. Despite its possible application strength this measure is still far from being handled in an efficient way also due to findings of Poljak and Rohn providing proof that checking this property is NP-hard for a general matrix. There are basically two approaches to handle this situation. Firstly, approximation algorithms are applied and secondly, tighter bounds for radius of regularity are considered. Improvements of both approaches have been recently shown by Hartman and Hlad\'{i}k (doi:10.1007/978-3-319-31769-4\_9) utilizing relaxation of the radius computation to semidefinite programming. An estimation of the regularity radius using any of the above mentioned approaches is usually applied to general matrices considering none or just weak assumptions about the original matrix. Surprisingly less explored area is represented by utilization of properties of special classes of matrices as well as utilization of classical algorithms extended to be used to compute the considered radius. This work explores a process of regularity radius analysis and identifies useful properties enabling easier estimation of the corresponding radius values. At first, checking finiteness of this characteristic is shown to be a polynomial problem along with determining a sharp upper bound on the number of nonzero elements of the matrix to obtain infinite radius. Further, relationship between maximum (Chebyshev) norm and spectral norm is used to construct new bounds for the radius of regularity. Considering situations where the known bounds are not tight enough, a new method based on Jansson-Rohn algorithm for testing regularity of an interval matrix is presented which is not a priory exponential along with numerical experiments. For a situation where an input matrix has a special form, several corresponding results are provided such as exact formulas for several special classes of matrices, e.g., for totally positive and inverse non-negative, or approximation algorithms, e.g., rank-one radius matrices. For tridiagonal matrices, an algorithm by Bar-On, Codenotti and Leoncini is utilized to design a polynomial algorithm to compute the radius of regularity.

Complexity of computing interval matrix powers for special classes of matrices

2020 ◽  
Vol 65 (5) ◽  
pp. 645-663
Author(s):  
David Hartman ◽  
Milan Hladík
Keyword(s):  
Interval Matrix ◽  
Classes Of Matrices ◽  
Special Classes

Application of cross correlation to HREM images of surfaces

1987 ◽  
Vol 45 ◽  
pp. 754-755
Author(s):  
D. E. Luzzi ◽  
L. D. Marks ◽  
M. I. Buckett
Keyword(s):  
Cross Correlation ◽  
A Priori ◽  
Matrix Material ◽  
Correlation Method ◽  
Accurate Knowledge ◽  
Complex Image ◽  
Fourier Filtering ◽  
The Matrix ◽  
Low Pass ◽  
Data Reduction Technique

As the HREM becomes increasingly used for the study of dynamic localized phenomena, the development of techniques to recover the desired information from a real image is important. Often, the important features are not strongly scattering in comparison to the matrix material in addition to being masked by statistical and amorphous noise. The desired information will usually involve the accurate knowledge of the position and intensity of the contrast. In order to decipher the desired information from a complex image, cross-correlation (xcf) techniques can be utilized. Unlike other image processing methods which rely on data massaging (e.g. high/low pass filtering or Fourier filtering), the cross-correlation method is a rigorous data reduction technique with no a priori assumptions.We have examined basic cross-correlation procedures using images of discrete gaussian peaks and have developed an iterative procedure to greatly enhance the capabilities of these techniques when the contrast from the peaks overlap.

scholarly journals Global Properties of Eigenvalues of Parametric Rank One Perturbations for Unstructured and Structured Matrices

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
André C. M. Ran ◽  
Michał Wojtylak
Keyword(s):  
Unit Circle ◽  
Structured Matrices ◽  
Additional Structure ◽  
General Properties ◽  
Global Properties ◽  
Classes Of Matrices ◽  
Rank One

AbstractGeneral properties of eigenvalues of $$A+\tau uv^*$$ A + τ u v ∗ as functions of $$\tau \in {\mathbb {C} }$$ τ ∈ C or $$\tau \in {\mathbb {R} }$$ τ ∈ R or $$\tau ={{\,\mathrm{{e}}\,}}^{{{\,\mathrm{{i}}\,}}\theta }$$ τ = e i θ on the unit circle are considered. In particular, the problem of existence of global analytic formulas for eigenvalues is addressed. Furthermore, the limits of eigenvalues with $$\tau \rightarrow \infty $$ τ → ∞ are discussed in detail. The following classes of matrices are considered: complex (without additional structure), real (without additional structure), complex H-selfadjoint and real J-Hamiltonian.

scholarly journals MC2: a two-phase algorithm for leveraged matrix completion

2018 ◽  
Vol 7 (3) ◽  
pp. 581-604 ◽  
Author(s):  
Armin Eftekhari ◽  
Michael B Wakin ◽  
Rachel A Ward
Keyword(s):  
Broad Class ◽  
Matrix Completion ◽  
A Priori ◽  
Total Sample ◽  
Second Phase ◽  
Sample Complexity ◽  
Two Phase ◽  
The Matrix ◽  
Leverage Scores ◽  
Uniform Random Sampling

Abstract Leverage scores, loosely speaking, reflect the importance of the rows and columns of a matrix. Ideally, given the leverage scores of a rank-r matrix $M\in \mathbb{R}^{n\times n}$, that matrix can be reliably completed from just $O (rn\log ^{2}n )$ samples if the samples are chosen randomly from a non-uniform distribution induced by the leverage scores. In practice, however, the leverage scores are often unknown a priori. As such, the sample complexity in uniform matrix completion—using uniform random sampling—increases to $O(\eta (M)\cdot rn\log ^{2}n)$, where η(M) is the largest leverage score of M. In this paper, we propose a two-phase algorithm called MC2 for matrix completion: in the first phase, the leverage scores are estimated based on uniform random samples, and then in the second phase the matrix is resampled non-uniformly based on the estimated leverage scores and then completed. For well-conditioned matrices, the total sample complexity of MC2 is no worse than uniform matrix completion, and for certain classes of well-conditioned matrices—namely, reasonably coherent matrices whose leverage scores exhibit mild decay—MC2 requires substantially fewer samples. Numerical simulations suggest that the algorithm outperforms uniform matrix completion in a broad class of matrices and, in particular, is much less sensitive to the condition number than our theory currently requires.

scholarly journals Generalization of real interval matrices to other fields

2019 ◽  
Vol 35 ◽  
pp. 285-296
Author(s):  
Elena Rubei
Keyword(s):  
Full Rank ◽  
Maximal Rank ◽  
Interval Matrix ◽  
Real Interval ◽  
Rational Matrix ◽  
Rank One ◽  
Rational Rank ◽  
Q Matrix ◽  
Rational Interval ◽  
Rank Of A Matrix

An interval matrix is a matrix whose entries are intervals in $\R$. This concept, which has been broadly studied, is generalized to other fields. Precisely, a rational interval matrix is defined to be a matrix whose entries are intervals in $\Q$. It is proved that a (real) interval $p \times q$ matrix with the endpoints of all its entries in $\Q$ contains a rank-one matrix if and only if it contains a rational rank-one matrix, and contains a matrix with rank smaller than $\min\{p,q\}$ if and only if it contains a rational matrix with rank smaller than $\min\{p,q\}$; from these results and from the analogous criterions for (real) inerval matrices, a criterion to see when a rational interval matrix contains a rank-one matrix and a criterion to see when it is full-rank, that is, all the matrices it contains are full-rank, are deduced immediately. Moreover, given a field $K$ and a matrix $\al$ whose entries are subsets of $K$, a criterion to find the maximal rank of a matrix contained in $\al$ is described.

scholarly journals Institutional Traps of Digitalization of Russian Higher Education

2021 ◽  
Vol 30 (3) ◽  
pp. 59-75
Author(s):  
M. A. Golovchin
Keyword(s):  
Higher Education ◽  
Large Scale ◽  
New Technologies ◽  
Teaching Staff ◽  
Original Matrix ◽  
Russian Higher Education ◽  
The Matrix ◽  
Institutional Traps ◽  
The Impact ◽  
Russian Universities

In 2016-2018 the state in Russia adopted a package of program documents, which implies the transfer of education to the large-scale introduction of digital technologies. This phenomenon has been called “digitalization of education”. In scientific literature, electronization and digitalization are increasingly called one of the institutional traps for the development of Russian universities, since the corresponding institutional environment has not yet been formed due to the forced nature of innovations. As a result, the processes of introducing new technologies into education are still not regulated. Within the framework of the purpose of the study, the manifestations of the trap of electronization and digitalization of Russian higher education were analyzed on the basis of sociological data, and the theoretical modeling of the process of adaptation of educational agents to the institution of digitalization was carried out.In the course of the study, the approaches were summarized that have been developed in discussions on educational digitalization. The article presents the author’s vision of the studied phenomenon as an institutional trap; as well as understanding of the institutional features and characteristics of electronization and digitalization in education.The research method is the analysis of estimates obtained in the course of an expert survey which was conducted by the Vologda Scientific Center of the Russian Academy of Sciences among the representatives of the teaching staff of state universities in the Vologda region. In the course of this analysis, the indicators of educational digitalization as an effective innovation were clarified such as an increased accessibility of educational resources; simplification of communication and the process of transferring knowledge from teacher to student; increased opportunities for training specialists for the new (digital) economy; improving the quality of education in universities, etc. Based on the results of the empirical study, it has been determined that the conditions for the development of digitalization in Russian universities are currently ambiguous, which is closely related to the level of competitiveness of the educational organization.The scientific novelty of the research consists in the presentation of an original matrix describing the process of university employees adaptation to the conditions of digital transformation of education. The matrix is proposed on the basis of a sociological analysis of the impact of the trap of electronization and digitalization on the activities of educational agents. The matrix can be taken into account in the practice of higher education management.

scholarly journals On the C-determinantal range for special classes of matrices

2016 ◽  
Vol 275 ◽  
pp. 86-94
Author(s):  
Alexander Guterman ◽  
Rute Lemos ◽  
Graça Soares
Keyword(s):  
Classes Of Matrices ◽  
Special Classes

Solving Systems of Linear Algebraic Equations with the Use of an Selective Regularization

2021 ◽  
pp. 11-15
Author(s):  
Vladimir N. Lutay
Keyword(s):  
Diagonal Element ◽  
Decomposition Process ◽  
Algebraic Equations ◽  
Original Matrix ◽  
Triangular Matrices ◽  
First Case ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Gauss Method ◽  
Scalar Products

The solution of systems of linear algebraic equations, which matrices can be poorly conditioned or singular is considered. As a solution method, the original matrix is decomposed into triangular components by Gauss or Chole-sky with an additional operation, which consists in increasing the small or zero diagonal terms of triangular matrices during the decomposition process. In the first case, the scalar products calculated during decomposition are divided into two positive numbers such that the first is greater than the second, and their sum is equal to the original one. In further operations, the first number replaces the scalar product, as a result of which the value of the diagonal term increases, and the second number is stored and used after the decomposition process is completed to correct the result of calculations. This operation increases the diagonal elements of triangular matrices and prevents the appearance of very small numbers in the Gauss method and a negative root expression in the Cholesky method. If the matrix is singular, then the calculated diagonal element is zero, and an arbitrary positive number is added to it. This allows you to complete the decomposition process and calculate the pseudo-inverse matrix using the Greville method. The results of computational experiments are presented.

Using a Priori Information in Energy-Dispersive X-Ray Fluorescence Analysis of Complex Samples

1991 ◽  
Vol 35 (B) ◽  
pp. 1205-1209
Author(s):  
I. A. Kondurov ◽  
P. A. Sushkov ◽  
T. M. Tjukavina ◽  
G. I. Shulyak
Keyword(s):  
High Temperature Superconductors ◽  
A Priori ◽  
Fluorescence Analysis ◽  
A Priori Information ◽  
Complex Samples ◽  
X Ray ◽  
Edxrf Analysis ◽  
The Matrix ◽  
Priori Information

In multielement EDXRF analysis of very complex unknowns, some problems in data evaluation may be simplified if one can take into account a priori information on the properties of the incident and detected radiations, and also available data on the matrix of the sample. The number of variables can be drastically shortened in the LSM procedures in this case. One of the best examples of complex unknowns is the determination of the rare earth element content of ores, and most recently in samples of high temperature superconductors (HiTc).

Calibrating compartmental models of neurons

1978 ◽  
Vol 235 (1) ◽  
pp. R93-R98 ◽  
Author(s):  
D. H. Perkel ◽  
B. Mulloney
Keyword(s):  
Steady State ◽  
Electrical Properties ◽  
Differential Equations ◽  
Compartmental Model ◽  
Compartmental Models ◽  
Electrical Measurements ◽  
Original Matrix ◽  
Systematic Procedure ◽  
The Matrix ◽  
Voltage Attenuation

Numerical parameters for a compartmental model of a neuron can be chosen to conform both to the neuron's structure and to its measured steady-state electrical properties. A systematic procedure for assigning parameters is described that makes use of the matrix of coefficients of the set of differential equations that embodies the compartmental model. The inverse of this matrix furnishes input resistances and voltage attenuation factors for the model, and an interactive modification of the original matrix and its inverse may be used to fit the model to anatomic and electrical measurements.

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