scholarly journals Convergence Results on Iteration Algorithms to Linear Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Zhuande Wang ◽  
Chuansheng Yang ◽  
Yubo Yuan
Keyword(s):  
Linear Systems ◽  
Spectral Radius ◽  
Positive Constant ◽  
Numerical Experiments ◽  
Large Scale ◽  
Iterative Algorithms ◽  
Convergence Results ◽  
Jacobi Iteration ◽  
The One

In order to solve the large scale linear systems, backward and Jacobi iteration algorithms are employed. The convergence is the most important issue. In this paper, a unified backward iterative matrix is proposed. It shows that some well-known iterative algorithms can be deduced with it. The most important result is that the convergence results have been proved. Firstly, the spectral radius of the Jacobi iterative matrix is positive and the one of backward iterative matrix is strongly positive (lager than a positive constant). Secondly, the mentioned two iterations have the same convergence results (convergence or divergence simultaneously). Finally, some numerical experiments show that the proposed algorithms are correct and have the merit of backward methods.

A Study of Sparse Matrix Methods on New Hardware

2015 ◽  
Vol 3 (3) ◽  
pp. 1-19
Author(s):  
Athanasios Fevgas ◽  
Konstantis Daloukas ◽  
Panagiota Tsompanopoulou ◽  
Panayiotis Bozanis
Keyword(s):  
Power Systems ◽  
Linear Systems ◽  
Large Scale ◽  
Comprehensive Evaluation ◽  
Sparse Matrix ◽  
Electrical Power ◽  
Iterative Algorithms ◽  
Direct Methods ◽  
Computer Hardware ◽  
Computing Platforms

Modeling of numerous scientific and engineering problems, such as multi-physic problems and analysis of electrical power systems, amounts to the solution of large scale linear systems. The main characteristics of such systems are the large sparsity ratio and the large number of unknowns that can reach thousands or even millions of equations. As a result, efficient solution of sparse large-scale linear systems is of great importance in order to enable analysis of such problems. Direct and iterative algorithms are the prevalent methods for solution of linear systems. Advances in computer hardware provide new challenges and capabilities for sparse solvers. The authors present a comprehensive evaluation of some, state of the art, sparse methods (direct and iterative) using modern computing platforms, aiming to determine the performance boundaries of each solver on different hardware infrastructures. By identifying the potential performance bottlenecks of out-of-core direct methods, the authors present a series of optimizations that increase their efficiency on flash-based systems.

Comparison Results between Jacobi and USSOR Iterative Methods

2014 ◽  
Vol 989-994 ◽  
pp. 1790-1793
Author(s):  
Ting Zhou ◽  
Shi Guang Zhang
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Spectral Radius ◽  
Numerical Example ◽  
Iteration Matrix ◽  
Comparison Results ◽  
Jacobi Iteration

In this paper, some comparison results between Jacobi and USSOR iteration for solving nonsingular linear systems are presented. It is showed that spectral radius of Jacobi iteration matrix B is less than that of USSOR iterative matrix under some conditions. A numerical example is also given to illustrate our results.

scholarly journals The extrapolated successive overrelaxation (ESOR) method for consistently ordered matrices

1984 ◽  
Vol 7 (2) ◽  
pp. 361-370 ◽  
Author(s):  
N. M. Missirlis ◽  
D. J. Evans
Keyword(s):  
Numerical Solution ◽  
Rate Of Convergence ◽  
Linear Systems ◽  
Spectral Radius ◽  
Sparse Linear Systems ◽  
Successive Overrelaxation ◽  
Iteration Matrix ◽  
Jacobi Iteration ◽  
Region Of Convergence

This paper develops the theory of the Extrapolated Successive Overrelaxation (ESOR) method as introduced by Sisler in [1], [2], [3] for the numerical solution of large sparse linear systems of the formAu=b, whenAis a consistently ordered2-cyclic matrix with non-vanishing diagonal elements and the Jacobi iteration matrixBpossesses only real eigenvalues. The region of convergence for the ESOR method is described and the optimum values of the involved parameters are also determined. It is shown that if the minimum of the moduli of the eigenvalues ofB,μ¯does not vanish, then ESOR attains faster rate of convergence than SOR when1−μ¯2<(1−μ¯2)12, whereμ¯denotes the spectral radius ofB.

First grammar books in the Habsburg Monarchy: individual initiative and regulatory interference by the state (1760s–1770s)

2020 ◽  
Vol 2019 (2 (11)) ◽  
pp. 137-157
Author(s):  
Olga V. Khavanova ◽  
Keyword(s):  
Eighteenth Century ◽  
State Policy ◽  
Large Scale ◽  
Language Use ◽  
Linguistic Diversity ◽  
Mother Tongue ◽  
German Language ◽  
Habsburg Monarchy ◽  
Private Initiative ◽  
The One

The second half of the eighteenth century in the lands under the sceptre of the House of Austria was a period of development of a language policy addressing the ethno-linguistic diversity of the monarchy’s subjects. On the one hand, the sphere of use of the German language was becoming wider, embracing more and more segments of administration, education, and culture. On the other hand, the authorities were perfectly aware of the fact that communication in the languages and vernaculars of the nationalities living in the Austrian Monarchy was one of the principal instruments of spreading decrees and announcements from the central and local authorities to the less-educated strata of the population. Consequently, a large-scale reform of primary education was launched, aimed at making the whole population literate, regardless of social status, nationality (mother tongue), or confession. In parallel with the centrally coordinated state policy of education and language-use, subjects-both language experts and amateur polyglots-joined the process of writing grammar books, which were intended to ease communication between the different nationalities of the Habsburg lands. This article considers some examples of such editions with primary attention given to the correlation between private initiative and governmental policies, mechanisms of verifying the textbooks to be published, their content, and their potential readers. This paper demonstrates that for grammar-book authors, it was very important to be integrated into the patronage networks at the court and in administrative bodies and stresses that the Vienna court controlled the process of selection and financing of grammar books to be published depending on their quality and ability to satisfy the aims and goals of state policy.

The Green New Deal

2019 ◽  
Author(s):  
Robert C. Hockett
Keyword(s):  
New Deal ◽  
Large Scale ◽  
Public Investment ◽  
Climate Science ◽  
Time Frame ◽  
White Paper ◽  
Good News ◽  
Carbon Neutrality ◽  
The One ◽  
In Virtue Of

This white paper lays out the guiding vision behind the Green New Deal Resolution proposed to the U.S. Congress by Representative Alexandria Ocasio-Cortez and Senator Bill Markey in February of 2019. It explains the senses in which the Green New Deal is 'green' on the one hand, and a new 'New Deal' on the other hand. It also 'makes the case' for a shamelessly ambitious, not a low-ball or slow-walked, Green New Deal agenda. At the core of the paper's argument lies the observation that only a true national mobilization on the scale of those associated with the original New Deal and the Second World War will be up to the task of comprehensively revitalizing the nation's economy, justly growing our middle class, and expeditiously achieving carbon-neutrality within the twelve-year time-frame that climate science tells us we have before reaching an environmental 'tipping point.' But this is actually good news, the paper argues. For, paradoxically, an ambitious Green New Deal also will be the most 'affordable' Green New Deal, in virtue of the enormous productivity, widespread prosperity, and attendant public revenue benefits that large-scale public investment will bring. In effect, the Green New Deal will amount to that very transformative stimulus which the nation has awaited since the crash of 2008 and its debt-deflationary sequel.

“Community Interests” and the Role of International Law in the Creation of a Global Market for Agricultural Land

2018 ◽  
Author(s):  
Jochen von Bernstorff
Keyword(s):  
International Law ◽  
Large Scale ◽  
Agricultural Land ◽  
Food Sovereignty ◽  
Global Market ◽  
Negative Consequences ◽  
Cultural Rights ◽  
Land Grab ◽  
Large Scale Land ◽  
The One

The chapter explores the notion of “community interests” with regard to the global “land-grab” phenomenon. Over the last decade, a dramatic increase of foreign investment in agricultural land could be observed. Bilateral investment treaties protect around 75 per cent of these large-scale land acquisitions, many of which came with associated social problems, such as displaced local populations and negative consequences for food security in Third World countries receiving these large-scale foreign investments. Hence, two potentially conflicting areas of international law are relevant in this context: Economic, social, and cultural rights and the principles of permanent sovereignty over natural resources and “food sovereignty” challenging large-scale investments on the one hand, and specific norms of international economic law stabilizing them on the other. The contribution discusses the usefulness of the concept of “community interests” in cases where the two colliding sets of norms are both considered to protect such interests.

scholarly journals Convergence analysis of gradient-based iterative algorithms for a class of rectangular Sylvester matrix equations based on Banach contraction principle

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Adisorn Kittisopaporn ◽  
Pattrawut Chansangiam ◽  
Wicharn Lewkeeratiyutkul
Keyword(s):  
Spectral Radius ◽  
Matrix Equation ◽  
Iterative Algorithms ◽  
Banach Contraction Principle ◽  
Contraction Principle ◽  
Convergence Factor ◽  
Sylvester Matrix ◽  
Iteration Matrix ◽  
Banach Contraction ◽  
The Matrix

AbstractWe derive an iterative procedure for solving a generalized Sylvester matrix equation $AXB+CXD = E$ A X B + C X D = E , where $A,B,C,D,E$ A , B , C , D , E are conforming rectangular matrices. Our algorithm is based on gradients and hierarchical identification principle. We convert the matrix iteration process to a first-order linear difference vector equation with matrix coefficient. The Banach contraction principle reveals that the sequence of approximated solutions converges to the exact solution for any initial matrix if and only if the convergence factor belongs to an open interval. The contraction principle also gives the convergence rate and the error analysis, governed by the spectral radius of the associated iteration matrix. We obtain the fastest convergence factor so that the spectral radius of the iteration matrix is minimized. In particular, we obtain iterative algorithms for the matrix equation $AXB=C$ A X B = C , the Sylvester equation, and the Kalman–Yakubovich equation. We give numerical experiments of the proposed algorithm to illustrate its applicability, effectiveness, and efficiency.

scholarly journals State Management for Cloud-Native Applications

Electronics ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 423
Author(s):  
Márk Szalay ◽  
Péter Mátray ◽  
László Toka
Keyword(s):  
Large Scale ◽  
Distributed Databases ◽  
Access Time ◽  
Replica Placement ◽  
State Management ◽  
Placement Decisions ◽  
Dynamic Replication ◽  
Cloud Databases ◽  
Placement Algorithm ◽  
The One

The stateless cloud-native design improves the elasticity and reliability of applications running in the cloud. The design decouples the life-cycle of application states from that of application instances; states are written to and read from cloud databases, and deployed close to the application code to ensure low latency bounds on state access. However, the scalability of applications brings the well-known limitations of distributed databases, in which the states are stored. In this paper, we propose a full-fledged state layer that supports the stateless cloud application design. In order to minimize the inter-host communication due to state externalization, we propose, on the one hand, a system design jointly with a data placement algorithm that places functions’ states across the hosts of a data center. On the other hand, we design a dynamic replication module that decides the proper number of copies for each state to ensure a sweet spot in short state-access time and low network traffic. We evaluate the proposed methods across realistic scenarios. We show that our solution yields state-access delays close to the optimal, and ensures fast replica placement decisions in large-scale settings.

scholarly journals A Neural Network-Based Approach for Approximating Arbitrary Roots of Polynomials

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 317
Author(s):  
Diogo Freitas ◽  
Luiz Guerreiro Lopes ◽  
Fernando Morgado-Dias
Keyword(s):  
Iterative Methods ◽  
Fundamental Problem ◽  
Iterative Algorithms ◽  
Rounding Errors ◽  
Suggested Approach ◽  
Simultaneous Iterative Methods ◽  
Complex Roots ◽  
Roots Of Polynomials ◽  
Comparative Results ◽  
The One

Finding arbitrary roots of polynomials is a fundamental problem in various areas of science and engineering. A myriad of methods was suggested to address this problem, such as the sequential Newton’s method and the Durand–Kerner (D–K) simultaneous iterative method. The sequential iterative methods, on the one hand, need to use a deflation procedure in order to compute approximations to all the roots of a given polynomial, which can produce inaccurate results due to the accumulation of rounding errors. On the other hand, the simultaneous iterative methods require good initial guesses to converge. However, Artificial Neural Networks (ANNs) are widely known by their capacity to find complex mappings between the dependent and independent variables. In view of this, this paper aims to determine, based on comparative results, whether ANNs can be used to compute approximations to the real and complex roots of a given polynomial, as an alternative to simultaneous iterative algorithms like the D–K method. Although the results are very encouraging and demonstrate the viability and potentiality of the suggested approach, the ANNs were not able to surpass the accuracy of the D–K method. The results indicated, however, that the use of the approximations computed by the ANNs as the initial guesses for the D–K method can be beneficial to the accuracy of this method.

scholarly journals Constructing Large-Scale Genetic Maps Using an Evolutionary Strategy Algorithm

Genetics ◽  
2003 ◽  
Vol 165 (4) ◽  
pp. 2269-2282
Author(s):  
D Mester ◽  
Y Ronin ◽  
D Minkov ◽  
E Nevo ◽  
A Korol
Keyword(s):  
Discrete Optimization ◽  
High Performance ◽  
Large Scale ◽  
Simulated Data ◽  
Real Data ◽  
Genetic Maps ◽  
Chromosome 1 ◽  
Evolutionary Strategy ◽  
Group A ◽  
The One

Abstract This article is devoted to the problem of ordering in linkage groups with many dozens or even hundreds of markers. The ordering problem belongs to the field of discrete optimization on a set of all possible orders, amounting to n!/2 for n loci; hence it is considered an NP-hard problem. Several authors attempted to employ the methods developed in the well-known traveling salesman problem (TSP) for multilocus ordering, using the assumption that for a set of linked loci the true order will be the one that minimizes the total length of the linkage group. A novel, fast, and reliable algorithm developed for the TSP and based on evolution-strategy discrete optimization was applied in this study for multilocus ordering on the basis of pairwise recombination frequencies. The quality of derived maps under various complications (dominant vs. codominant markers, marker misclassification, negative and positive interference, and missing data) was analyzed using simulated data with ∼50-400 markers. High performance of the employed algorithm allows systematic treatment of the problem of verification of the obtained multilocus orders on the basis of computing-intensive bootstrap and/or jackknife approaches for detecting and removing questionable marker scores, thereby stabilizing the resulting maps. Parallel calculation technology can easily be adopted for further acceleration of the proposed algorithm. Real data analysis (on maize chromosome 1 with 230 markers) is provided to illustrate the proposed methodology.

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