In a wide range of applications, solving the linear
system of equations Ax = b is appeared. One of the
best
methods to solve the large sparse asymmetric linear
systems is the simplified generalized minimal resi
dual
(SGMRES(m)) method. Also, some improved versions of
SGMRES(m) exist: SGMRES-E(m, k) and
SGMRES-DR(m, k). In this paper, an intelligent heur
istic method for accelerating the convergence of th
ree
methods SGMRES(m), SGMRES-E(m, k), and SGMRES-DR(m,
k) is proposed. The numerical results
obtained from implementation of the proposed approa
ch on several University of Florida standard
matrixes confirm the efficiency of the proposed met
hod.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
SUCCESSIVE LINEARIZATION SOLUTION OF A BOUNDARY LAYER CONVECTIVE HEAT TRANSFE...ijcsa
The purpose of this paper is to discuss the flow of forced convection over a flat plate. The governing partial
differential equations are transformed into ordinary differential equations using suitable transformations.
The resulting equations were solved using a recent semi-numerical scheme known as the successive
linearization method (SLM). A comparison between the obtained results with homotopy perturbation method and numerical method (NM) has been included to test the accuracy and convergence of the method.
Regional gradient optimal control problem governed by a distributed bilinear ...TELKOMNIKA JOURNAL
This paper gives an extension of previous work on gradient optimal control of distributed parabolic systems to the case of distributed bilinear systems which are a type of nonlinear systems. We introduce the notion of flux optimal control of distributed bilinear systems. The idea is trying to achieve a neighborhood of the gradient state of the considered system by minimizing a nonlinear quadratic cost. Using optimization techniques, a method showing how to reach a desired flux at a final time, only on internal subregion of the system domain will be proposed. The proposed simulation illustrates the theoretical approach by commanding the heat bilinear equation flux to a desired profile.
Dumitru Vulcanov - Numerical simulations with Ricci flow, an overview and cos...SEENET-MTP
Lecture by prof. dr Dumitru Vulcanov (dean of the Faculty of Physics, West University of Timisoara, Romania) on October 21, 2010 at the Faculty of Science and Mathematics, Nis, Serbia.
This paper is concerned with the stability and Robust stabilization problem for 2-D continuous systems in Roesser model, based on Generalized Kalman$-$Yakubovich$-$Popov lemma in combination with frequency-partitioning approach. Sufficient conditions of stability of the systems are formulated via linear matrix inequality technique. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.
EXACT SOLUTIONS OF SCHRÖDINGER EQUATION WITH SOLVABLE POTENTIALS FOR NON PT/P...ijrap
We have obtained explicitly the exact solutions of the Schrodinger equation with Non PT/PT symmetric
Rosen Morse II, Scarf II and Coulomb potentials. Energy eigenvalues and the corresponding
unnormalized wave functions for these systems for both Non PT and PT symmetric are also obtained using
the Nikiforov-Uvarov (NU) method.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Automatic Unsupervised Data Classification Using Jaya Evolutionary Algorithmaciijournal
In this paper we attempt to solve an automatic clustering problem by optimizing multiple objectives such as
automatic k-determination and a set of cluster validity indices concurrently. The proposed automatic
clustering technique uses the most recent optimization algorithm Jaya as an underlying optimization
stratagem. This evolutionary technique always aims
to attain global best solution rather than a local best solution in larger datasets. The explorations and exploitations imposed on the proposed work results to
detect the number of automatic clusters, appropriate partitioning present in data sets and mere optimal
values towards CVIs frontiers. Twelve datasets of different intricacy are used to endorse the performance
of aimed algorithm. The experiments lay bare that the conjectural advantages of multi objective clustering optimized with evolutionary approaches decipher into realistic and scalable performance paybacks.
SUCCESSIVE LINEARIZATION SOLUTION OF A BOUNDARY LAYER CONVECTIVE HEAT TRANSFE...ijcsa
The purpose of this paper is to discuss the flow of forced convection over a flat plate. The governing partial
differential equations are transformed into ordinary differential equations using suitable transformations.
The resulting equations were solved using a recent semi-numerical scheme known as the successive
linearization method (SLM). A comparison between the obtained results with homotopy perturbation method and numerical method (NM) has been included to test the accuracy and convergence of the method.
Regional gradient optimal control problem governed by a distributed bilinear ...TELKOMNIKA JOURNAL
This paper gives an extension of previous work on gradient optimal control of distributed parabolic systems to the case of distributed bilinear systems which are a type of nonlinear systems. We introduce the notion of flux optimal control of distributed bilinear systems. The idea is trying to achieve a neighborhood of the gradient state of the considered system by minimizing a nonlinear quadratic cost. Using optimization techniques, a method showing how to reach a desired flux at a final time, only on internal subregion of the system domain will be proposed. The proposed simulation illustrates the theoretical approach by commanding the heat bilinear equation flux to a desired profile.
Dumitru Vulcanov - Numerical simulations with Ricci flow, an overview and cos...SEENET-MTP
Lecture by prof. dr Dumitru Vulcanov (dean of the Faculty of Physics, West University of Timisoara, Romania) on October 21, 2010 at the Faculty of Science and Mathematics, Nis, Serbia.
This paper is concerned with the stability and Robust stabilization problem for 2-D continuous systems in Roesser model, based on Generalized Kalman$-$Yakubovich$-$Popov lemma in combination with frequency-partitioning approach. Sufficient conditions of stability of the systems are formulated via linear matrix inequality technique. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.
EXACT SOLUTIONS OF SCHRÖDINGER EQUATION WITH SOLVABLE POTENTIALS FOR NON PT/P...ijrap
We have obtained explicitly the exact solutions of the Schrodinger equation with Non PT/PT symmetric
Rosen Morse II, Scarf II and Coulomb potentials. Energy eigenvalues and the corresponding
unnormalized wave functions for these systems for both Non PT and PT symmetric are also obtained using
the Nikiforov-Uvarov (NU) method.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Automatic Unsupervised Data Classification Using Jaya Evolutionary Algorithmaciijournal
In this paper we attempt to solve an automatic clustering problem by optimizing multiple objectives such as
automatic k-determination and a set of cluster validity indices concurrently. The proposed automatic
clustering technique uses the most recent optimization algorithm Jaya as an underlying optimization
stratagem. This evolutionary technique always aims
to attain global best solution rather than a local best solution in larger datasets. The explorations and exploitations imposed on the proposed work results to
detect the number of automatic clusters, appropriate partitioning present in data sets and mere optimal
values towards CVIs frontiers. Twelve datasets of different intricacy are used to endorse the performance
of aimed algorithm. The experiments lay bare that the conjectural advantages of multi objective clustering optimized with evolutionary approaches decipher into realistic and scalable performance paybacks.
Audio Signal Identification and Search Approach for Minimizing the Search Tim...aciijournal
Audio or music fingerprints can be utilize to implement an economical music identification system on a
million-song library, however the system needs great deal of memory to carry the fingerprints and indexes.
Therefore, for a large-scale audio library, memory
imposes a restriction on the speed of music
identifications. So, we propose an efficient music
identification system which used a kind of space-saving
audio fingerprints. For saving space, original finger representations are sub-sample and only one quarters
of the original data is reserved. In this approach,
memory demand is far reduced and therefore the search
speed is criticalincreasing whereas the lustiness and dependability are well preserved. Mapping
audio information to time and frequency domain for
the classification, retrieval or identification tasks
presents four principal challenges. The dimension o
f the input should be considerably reduced;
the ensuing options should be strong to possible distortions of the input; the feature should be informative
for the task at hand simple. We propose distortion
free system which fulfils all four of these requirements.
Extensive study has been done to compare our system
with the already existing ones, and the results sh
ow
that our system requires less memory, provides fast
results and achieves comparable accuracy for a large-
scale database.
SURVEY SURFER: A WEB BASED DATA GATHERING AND ANALYSIS APPLICATIONaciijournal
The most important need for a web based survey technology is speedy performance and accurate results
.Though there are variety of ways a survey can be taken manually and have it assessed manually but here
we are introducing the survey site concept where the assessment of the survey responses are created
automatically by applying certain statistics. In this paper we propose the idea of automated approach to
the analysis of the survey where the results would be evident graphically to take a wise decision. Additional
to graphical view we aim on forming a platform to view complex, tedious data in simple, understandable,
interactive and visualized form.
Erca energy efficient routing and reclusteringaciijournal
The pervasive application of wireless sensor networks (WNSs) is challenged by the scarce energy constraints of sensor nodes. En-route filtering schemes, especially commutative cipher based en-route filtering (CCEF) can saves energy with better filtering capacity. However, this approach suffer from fixed paths and inefficient underlying routing designed for ad-hoc networks. Moreover, with decrease in remaining sensor nodes, the probability of network partition increases. In this paper, we propose energy-efficient routing and re-clustering algorithm (ERCA) to address these limitations. In proposed scheme with reduction in the number of sensor nodes to certain thresh-hold the cluster size and transmission range dynamically maintain cluster node-density. Performance results show that our approach demonstrate filtering-power, better energy-efficiency, and an average gain over 285% in network lifetime.
LOCALIZATION OF OVERLAID TEXT BASED ON NOISE INCONSISTENCIESaciijournal
In this paper, we present a novel technique for localization of caption text in video frames based on noise inconsistencies. Text is artificially added to the video after it has been captured and as such does not form part of the original video graphics. Typically, the amount of noise level is uniform across the entire captured video frame, thus, artificially embedding or overlaying text on the video introduces yet another segment of noise level. Therefore detection of various noise levels in the video frame may signify availability of overlaid text. Hence we exploited this property by detecting regions with various noise levels to localize overlaid text in video frames. Experimental results obtained shows a great improvement in line with overlaid text localization, where we have performed metric measure based on Recall, Precision and f-measure.
Check out the exclusive range of wall wallpaper we have to offer. Browse through our wide range of wallpapers and pick according to your requirements. Contact us now!
Learning strategy with groups on page based students' profilesaciijournal
Most of students desire to know about their knowledge level to perfect their exams. In learning environment the fields of study overwhelm on page with collaboration or cooperation. Students can do their exercises either individually or collaboratively with their peers. The system provides the guidelines for students' learning system about interest fields as Java in this system. Especially the system feedbacks information about exam to know their grades without teachers. The participants who answered the exam can discuss with each others because of sharing e mail and list of them.
Web spam classification using supervised artificial neural network algorithmsaciijournal
Due to the rapid growth in technology employed by the spammers, there is a need of classifiers that are more efficient, generic and highly adaptive. Neural Network based technologies have high ability of adaption as well as generalization. As per our knowledge, very little work has been done in this field using neural network. We present this paper to fill this gap. This paper evaluates performance of three supervised learning algorithms of artificial neural network by creating classifiers for the complex problem of latest web spam pattern classification. These algorithms are Conjugate Gradient algorithm, Resilient Backpropagation learning, and Levenberg-Marquardt algorithm.
Text mining is the process of extracting interesting and non-trivial knowledge or information from unstructured text data. Text mining is the multidisciplinary field which draws on data mining, machine learning, information retrieval, computational linguistics and statistics. Important text mining processes are information extraction, information retrieval, natural language processing, text classification, content analysis and text clustering. All these processes are required to complete the preprocessing step before doing their intended task. Pre-processing significantly reduces the size of the input text documents and the actions involved in this step are sentence boundary determination, natural language specific stop-word elimination, tokenization and stemming. Among this, the most essential and important action is the tokenization. Tokenization helps to divide the textual information into individual words. For performing tokenization process, there are many open source tools are available. The main objective of this work is to analyze the performance of the seven open source tokenization tools. For this comparative analysis, we have taken Nlpdotnet Tokenizer, Mila Tokenizer, NLTK Word Tokenize, TextBlob Word Tokenize, MBSP Word Tokenize, Pattern Word Tokenize and Word Tokenization with Python NLTK. Based on the results, we observed that the Nlpdotnet Tokenizer tool performance is better than other tools.
Recommender systems have grown to be a critical research subject after the emergence of the first paper on collaborative filtering in the Nineties. Despite the fact that educational studies on recommender systems, has extended extensively over the last 10 years, there are deficiencies in the complete literature evaluation and classification of that research. Because of this, we reviewed articles on recommender structures, and then classified those based on sentiment analysis. The articles are categorized into three techniques of recommender system, i.e.; collaborative filtering (CF), content based and context based. We have tried to find out the research papers related to sentimental analysis based recommender system. To classify research done by authors in this field, we have shown different approaches of recommender system based on sentimental analysis with the help of tables. Our studies give statistics, approximately trends in recommender structures research, and gives practitioners and researchers with perception and destiny route on the recommender system using sentimental analysis. We hope that this paper enables all and sundry who is interested in recommender systems research with insight for destiny.
TRUST METRICS IN RECOMMENDER SYSTEMS: A SURVEYaciijournal
Information overload is a new challenge in e-commerce sites. The problem refers to the fast growing of
information that lead following the information flow in real world be impossible. Recommender systems, as
the most successful application of information filtering, help users to find items of their interest from huge
datasets. Collaborative filtering, as the most successful technique for recommendation, utilises social
behaviours of users to detect their interests. Traditional challenges of Collaborative filtering, such as cold
start, sparcity problem, accuracy and malicious attacks, derived researchers to use new metadata to
improve accuracy of recommenders and solve the traditional problems. Trust based recommender systems
focus on trustworthy value on relation among users to make more reliable and accurate recommends. In
this paper our focus is on trust based approach and discuss about the process of making recommendation
in these method. Furthermore, we review different proposed trust metrics, as the most important step in this
process.
HYBRID DATA CLUSTERING APPROACH USING K-MEANS AND FLOWER POLLINATION ALGORITHMaciijournal
Data clustering is a technique for clustering set of objects into known number of groups. Several approaches are widely applied to data clustering so that objects within the clusters are similar and objects in different clusters are far away from each other. K-Means, is one of the familiar center based clustering algorithms since implementation is very easy and fast convergence. However, K-Means algorithm suffers from initialization, hence trapped in local optima. Flower Pollination Algorithm (FPA) is the global optimization technique, which avoids trapping in local optimum solution. In this paper, a novel hybrid data clustering approach using Flower Pollination Algorithm and K-Means (FPAKM) is proposed. The proposed algorithm results are compared with K-Means and FPA on eight datasets. From the experimental results, FPAKM is better than FPA and K-Means.
Blood groups matcher frame work based on three level rules in myanmaraciijournal
Today Blood donation is a global interest for world to be survival lives when people are in trouble because of natural disaster. The system provides the ability how to decide to donate the blood according to the rules for blood donation not to meet the physicians. In this system, there are three main parts to accept blood from donors when they want to donate according to the features like personal health. The application facilitates to negotiate between blood donors and patients who need to get blood seriously on page without going to Blood Banks and waiting time in queue there.
Prediction of lung cancer is most challenging problem due to structure of cancer cell, where most of the cells are overlapped each other. The image processing techniques are mostly used for prediction of lung cancer and also for early detection and treatment to prevent the lung cancer. To predict the lung cancer various features are extracted from the images therefore, pattern recognition based approaches are useful to predict the lung cancer. Here, a comprehensive review for the prediction of lung cancer by previous researcher using image processing techniques is presented. The summary for the prediction of lung cancer by previous researcher using image processing techniques is also presented.
An Intelligent Method for Accelerating the Convergence of Different Versions ...aciijournal
In a wide range of applications, solving the linear system of equations Ax = b is appeared. One of the best
methods to solve the large sparse asymmetric linear systems is the simplified generalized minimal residual
(SGMRES(m)) method. Also, some improved versions of SGMRES(m) exist: SGMRES-E(m, k) and
SGMRES-DR(m, k). In this paper, an intelligent heuristic method for accelerating the convergence of three
methods SGMRES(m), SGMRES-E(m, k), and SGMRES-DR(m, k) is proposed. The numerical results
obtained from implementation of the proposed approach on several University of Florida standard
matrixes confirm the efficiency of the proposed method.
AN IMPROVED ITERATIVE METHOD FOR SOLVING GENERAL SYSTEM OF EQUATIONS VIA GENE...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
An Improved Iterative Method for Solving General System of Equations via Gene...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
An Improved Iterative Method for Solving General System of Equations via Gene...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
Computer Science
Active and Programmable Networks
Active safety systems
Ad Hoc & Sensor Network
Ad hoc networks for pervasive communications
Adaptive, autonomic and context-aware computing
Advance Computing technology and their application
Advanced Computing Architectures and New Programming Models
Advanced control and measurement
Aeronautical Engineering,
Agent-based middleware
Alert applications
Automotive, marine and aero-space control and all other control applications
Autonomic and self-managing middleware
Autonomous vehicle
Biochemistry
Bioinformatics
BioTechnology(Chemistry, Mathematics, Statistics, Geology)
Broadband and intelligent networks
Broadband wireless technologies
CAD/CAM/CAT/CIM
Call admission and flow/congestion control
Capacity planning and dimensioning
Changing Access to Patient Information
Channel capacity modelling and analysis
Civil Engineering,
Cloud Computing and Applications
Collaborative applications
Communication application
Communication architectures for pervasive computing
Communication systems
Computational intelligence
Computer and microprocessor-based control
Computer Architecture and Embedded Systems
Computer Business
Computer Sciences and Applications
Computer Vision
Computer-based information systems in health care
Computing Ethics
Computing Practices & Applications
Congestion and/or Flow Control
Content Distribution
Context-awareness and middleware
Creativity in Internet management and retailing
Cross-layer design and Physical layer based issue
Cryptography
Data Base Management
Data fusion
Data Mining
Data retrieval
Data Storage Management
Decision analysis methods
Decision making
Digital Economy and Digital Divide
Digital signal processing theory
Distributed Sensor Networks
Drives automation
Drug Design,
Drug Development
DSP implementation
E-Business
E-Commerce
E-Government
Electronic transceiver device for Retail Marketing Industries
Electronics Engineering,
Embeded Computer System
Emerging advances in business and its applications
Emerging signal processing areas
Enabling technologies for pervasive systems
Energy-efficient and green pervasive computing
Environmental Engineering,
Estimation and identification techniques
Evaluation techniques for middleware solutions
Event-based, publish/subscribe, and message-oriented middleware
Evolutionary computing and intelligent systems
Expert approaches
Facilities planning and management
Flexible manufacturing systems
Formal methods and tools for designing
Fuzzy algorithms
Fuzzy logics
GPS and location-based app
Numerical Solution of Third Order Time-Invariant Linear Differential Equation...theijes
In this paper, we state the Adomian Decomposition Method (ADM) for third order time-invariant linear homogeneous differential equations. And we applied it to find solutions to the same class of equations. Three test problems were used as concrete examples to validate the reliability of the method, and the result shows remarkable solutions as those that are obtained by any knows analytical method(s).
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Nonlinear Viscoelastic Analysis of Laminated Composite Plates – A Multi Scale...rtme
Laminated composite plates are widely used in modern structures. Resins of composites are almost made of
polymers which show time dependent and and in some cases stress dependent behaviour. In this paper, a
laminated composite plate is analysed using a multiscale method. At first, material properties of a lamina is
obtained using an analytical micromechanical approach called simplified unit cell method (SUCM) and
then in macromechanical level, Generalized Differential Quadrature Method (GDQM) is used to analyse
laminated composite plate. Schapery's integral is used to model nonlinear viscoelastic behaviour of the
matrix. Prony series is considered to define the compliance of matrix. Micromechanical process includes
obtaining overall properties of the composite by SUCM. Both geometrical and material nonlinearity are
taken into account in order to multiscale analysis of laminated composite plate.
This poster was created in LaTeX on a Dell Inspiron laptop with a Linux Fedora Core 4 operating system. The background image and the animation snapshots are dxf meshes of elastic waveform solutions, rendered on a Windows machine using 3D Studio Max.
LOGNORMAL ORDINARY KRIGING METAMODEL IN SIMULATION OPTIMIZATIONorajjournal
This paper presents a lognormal ordinary kriging (LOK) metamodel algorithm and its application to
optimize a stochastic simulation problem. Kriging models have been developed as an interpolation method
in geology. They have been successfully used for the deterministic simulation optimization (SO) problem. In
recent years, kriging metamodeling has attracted a growing interest with stochastic problems. SO
researchers have begun using ordinary kriging through global optimization in stochastic systems. The
goals of this study are to present LOK metamodel algorithm and to analyze the result of the application
step-by-step. The results show that LOK is a powerful alternative metamodel in simulation optimization
when the data are too skewed.
MARGINAL PERCEPTRON FOR NON-LINEAR AND MULTI CLASS CLASSIFICATION ijscai
Generalization error of classifier can be reduced by larger margin of separating hyperplane. The proposed classification algorithm implements margin in classical perceptron algorithm, to reduce generalized errors by maximizing margin of separating hyperplane. Algorithm uses the same updation rule with the perceptron, to converge in a finite number of updates to solutions, possessing any desirable fraction of the margin. This solution is again optimized to get maximum possible margin. The algorithm can process linear, non-linear and multi class problems. Experimental results place the proposed classifier equivalent to the support vector machine and even better in some cases. Some preliminary experimental results are briefly discussed.
Differential evolution (DE) algorithm has been applied as a powerful tool to find optimum switching angles for selective harmonic elimination pulse width modulation (SHEPWM) inverters. However, the DE’s performace is very dependent on its control parameters. Conventional DE generally uses either trial and error mechanism or tuning technique to determine appropriate values of the control paramaters. The disadvantage of this process is that it is very time comsuming. In this paper, an adaptive control parameter is proposed in order to speed up the DE algorithm in optimizing SHEPWM switching angles precisely. The proposed adaptive control parameter is proven to enhance the convergence process of the DE algorithm without requiring initial guesses. The results for both negative and positive modulation index (M) also indicate that the proposed adaptive DE is superior to the conventional DE in generating SHEPWM switching patterns.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Similar to An Intelligent Method for Accelerating the Convergence of Different Versions of SGMRES Algorithm (20)
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
Forklift Classes Overview by Intella PartsIntella Parts
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An Intelligent Method for Accelerating the Convergence of Different Versions of SGMRES Algorithm
1. Advanced Computational Intelligence: An International Journal (ACII), Vol.3, No.2, April 2016
DOI:10.5121/acii.2016.3201 1
AN INTELLIGENT METHOD FOR ACCELERATING
THE CONVERGENCE OF DIFFERENT VERSIONS OF
SGMRES ALGORITHM
MohadesehEntezari Zarch1,2
, *
SeyedAbolfazlShahzadeh Fazeli1,2
, and Mohammad
Bagher Dowlatshahi1,3
1
Parallel Processing Laboratory, Yazd University,Yazd,Iran.
2
Department of Computer Science, Faculty of Mathematics, Yazd
University,Yazd,Iran
3
Department of Computer Engineering, Faculty of Electronic and Computer, Yazd
University,Yazd,Iran.
ABSTRACT
In a wide range of applications, solving the linear system of equations Ax = b is appeared. One of the best
methods to solve the large sparse asymmetric linear systems is the simplified generalized minimal residual
(SGMRES(m)) method. Also, some improved versions of SGMRES(m) exist: SGMRES-E(m, k) and
SGMRES-DR(m, k). In this paper, an intelligent heuristic method for accelerating the convergence of three
methods SGMRES(m), SGMRES-E(m, k), and SGMRES-DR(m, k) is proposed. The numerical results
obtained from implementation of the proposed approach on several University of Florida standard
matrixes confirm the efficiency of the proposed method.
Keywords
Artificial Intelligence, Heuristic Algorithms, Linear systems of equations, SGMRES.
1. INTRODUCTION
In a wide range of applied sciences, solution of problems is obtained from solving a linear system
of equations Ax = b. In numerical analysis, the algorithms for solving linear systems commonly
use one of the two following methods: Direct methods, Iterative methods.
According to the coefficient of linear devices and matrix A, we can use one of these methods. If
the matrix dimension is too small, direct methods such as Gaussian elimination method, Gauss-
Jordan and LU decomposition is preferred. These methods are composed of a finite number of
steps. On the other hand, iterative methods are based on calculating a sequence of approximations
to find the solution. In these methods, to stop the algorithm either an accurate solution is found or
a certain number of iterations is performed. If the matrix A is relatively large, triangular-based
direct methods are not recommended, because this method requires plenty of time and storage
space. In addition, in many cases the coefficient matrix is sparse and triangular process destroys
the sparseness of matrix, such we are faced with a large dense matrix. To solve such problems, it
is recommended to use iterative methods which do not change the sparseness nature of matrix A.
2. Advanced Computational Intelligence: An International Journal (ACII), Vol.3, No.2, April 2016
2
One of the main iterative methods, which nowadays has found many applications, is the
generalized minimal residual method (GMRES) proposed by Saad and Schultz [1]. Simpler
GMRES (SGMRES) method is a new version of GMRES, which was proposed by Walker and
Zhou [2] and analyzed by Jiránek et al. [4]. SGMRES(m)) is a restarted version of SGMRES
which restarts at the iteration when the Krylov subspace reaches dimension m, then the current
solution is used as the new initial guess for the next m iterations.
Despite the good performance of these methods, it seems that we can make changes in their
structure to accelerate their convergence. In this study, an intelligent heuristic method to change
the update equations of three methods SGMRES(m), SGMRES-E(m, k), and SGMRES-DR(m, k)
is proposed. The experimental results on several University of Florida standard matrixes confirm
the efficiency of the proposed method.
2. THE SGMRES AND ITS VARIANTS
The generalized minimal residual (GMRES) method developed by Saad and Schultz [1] is one of
the most popular iterative methods for solving the large sparse nonsymmetrical system of linear
equations Ax=b, in which A∈Rn×n
is nonsingular, b∈Rn
is a right-hand side vector, and x∈Rn
is the
sought-after solution. Let x0∈Rn
be the initial guess, and r0 = b−Ax0 the initial residual vector. At
the mth
iteration step, GMRES finds the approximate solution xmin the affine subspace x0+Km(A,
r0), such that xm minimizes the Euclidean norm of the residual, i.e.:
‖ݎ‖ = ‖ܾ − ݔܣ‖ = ‖ܾ − ݔ(ܣ + ݖ)‖
= min
௭∈(,బ)
‖ܾ − ݔ(ܣ + ‖)ݖ = min
௭∈(,బ)
‖ݎ − )1( ‖)ݖܣ
Note that ܭ(,ܣ ݎ) = ݊ܽݏሼݎ, ݎܣ, … , ܣିଵ
ݎሽis the mth
Krylov subspace constructed by the
matrix A and the initial residual vector r0. It is obvious that (1) is corresponding to the
orthogonality condition, i.e.:
ݎ ⊥ ܭܣ(,ܣ ݎ) (2)
where the orthogonality relation is established upon the Euclidean inner product.
The traditional implementation of GMRES is inspired from the Arnoldi process [3]. Application
of m steps of the Arnoldi process to the matrix A with the nonzero residual vector r0 yields the
Arnoldi factorization
ܸܣ = ܸାଵܪ
where the columns of the matrix Vm form the orthonormal basis of the Krylov subspace Km(A, r0)
and ܪ∈R(m+1)×m
is an upper Hessenberg matrix. We can reformulate the least-squares problem as
the reduced minimization problem
‖ܾ − ݔܣ‖ = ‖ܾ − ݔ(ܣ + ܸݕ)‖ = min
௬∈ோ
ฮߚ݁ଵ − ܪݕฮ
whereߚ = ‖ݎ‖and e1 is the first column of an identity matrix of order m + 1. The QR
factorization of ܪwith Givens rotations can be used to solve the reduced minimization problem.
3. Advanced Computational Intelligence: An International Journal (ACII), Vol.3, No.2, April 2016
3
Simpler GMRES (SGMRES) method is a new version of GMRES, which was proposed by
Walker and Zhou [2] and analyzed by Jiránek et al. [4]. In their method, instead of building an
orthonormal basis of Km(A, r0), an orthonormal basis Vmof AKm(A, r0) is established and then
carries out the orthogonality relation (2). In a specific manner, suppose Zm= [z1,z2, . . ., zm] be a
basis of Km(A, r0). Then, the QR factorization of AZmcan be used to acquire the orthonormal basis
VmofAKm(A, r0), i.e.
AZ୫ = V୫R୫
whereVm=[v1, v2, . . . , vm] has orthonormal columns and Rmis an m×m upper triangular and
nonsingular matrix. By accomplishing the orthogonality relation (2), we can compute the mth
residual vector rmrecursively as
r୫ = r − V୫൫V୫
r൯ = r୫ିଵ − α୫v୫ m ≥ 1
where αm=vT
mr0. The corresponding approximate solution is
x୫ = x + Z୫t୫
where tm is the solution of the upper triangular system
R୫t୫ = ሾαଵ, αଶ, … , α୫ሿ
.
By lapse of iterations, the amount of required computational time and space for GMRES or
simpler GMRES increases meaningfully; and this subject makes GMRES and simpler GMRES
impractical. In order to overcome this shortcoming, the restarted version of these algorithms is
proposed. In restarted GMRES (GMRES(m)) or restarted simpler GMRES (SGMRES(m)),
GMRES or SGMRES is restarted at the moment that the Krylov subspace reaches dimension m,
then the current solution is used as the new initial guess for the next m iterations. Some details of
SGMRES(m) is presented in Algorithm 1[13].
Algorithm 1 (SGMRES(m)).
Input: A : the coefficient matrix; b : the right-hand side; x0 : an initial approximation; m : the
maximal dimension of the Krylov subspace; ε : a tolerance.
Output: An approximate solution x.
1. Compute r = b − Ax , zଵ = r ‖r‖⁄ , zොଵ = Azଵ , rଵଵ = ‖zොଵ‖ , vଵ = zොଵ/rଵଵ
2. Compute αଵ = vଵ
r , rଵ = r − αଵvଵ
3. For j = 2,3, … , m
3.1. z୨ = v୨ିଵ
3.2.zො୨ = Az୨
3.3. For i = 1,2, … , j − 1
r୧୨ = v୧
zො୨
zො୨ = zො୨ − r୧୨v୧
End For
3.4.r୧୨ = ฮzො୨ฮ , v୨ = zො୨ r୨୨⁄ , α୨ = v୨
r୨ିଵ
4. Advanced Computational Intelligence: An International Journal (ACII), Vol.3, No.2, April 2016
4
3.5.r୨ = r୨ିଵ − α୨v୨
3.6.If ฮr୨ฮ < ߝthen solve the linear triangular system
R୨t = α
with α = ൣαଵ, αଶ, … , α୨൧
، and form the approximate solutionx୫ = x + Z୫t and stop
4. Solve the linear triangular system
R୨t = α
With α = ൣαଵ, αଶ, … , α୨൧
and form the approximate solution x୫ = x + Z୫t
5. Set x0 = xm and go to Step 1.
This implementation of SGMRES(m) was proposed by Walker and Zhou [2]. In their
implementation, they used ܼ = ሾr ‖r‖⁄ , V୫ିଵሿ. Note that different restarted simpler GMRES,
called the residual-based restarted simpler GMRES (RB-SGMRES(m)), is proposed in [4]. RB-
SGMRES(m) uses
Z୫ = ሾr ‖r‖⁄ , rଵ ‖rଵ‖⁄ , … , r୫ିଵ ‖r୫ିଵ‖⁄ ሿ
and is equivalent to SGMRES(m). The implementation of RB-SGMRES(m) is closely similar to
that of SGMRES(m) except that ݖ = ݎିଵ/ฮݎିଵฮ in Step 3.1 in Algorithm 1. In [4], it has been
shown that SGMRES(m) is essentially unstable, but RB-SGMRES(m) is conditionally stable
provided that we have some reasonable residual decrease.
In general, because the Krylov subspace dimension is restricted at each cycle for the restarted
methods, often, the convergence for the matrix with Ahaving small eigenvalues slows down. The
main reason of this is that the Krylov subspace used at each cycle does not include good
approximations to the eigenvectors corresponding to small eigenvalues. To accelerate the
convergence of restarted GMRES, it is proposed to compute the approximate eigenvectors
corresponding to small eigenvalues in modulus at each restart, and then augment the Krylov
subspace by these approximate eigenvectors to improve the convergence of restarted GMRES.
This kind of methods includes GMRES-E [5], GMRES-IR [6], and GMRES-DR [7]. These
methods are equivalent at the end of each cycle. Although, GMRES-IR needs less matrix-vector
products than GMRES-E at each cycle, it is slightly more complicated to implement. Moreover,
GMRES-IR suffers from stability problems because it uses the implicitly restarted Arnoldi
process [8], which may suffer from stability when a shift used in the implicitly restarted Arnoldi
process is close to a true eigenvalue of the problem. GMRES-DR which is a good improvement
of GMRES-IR has the efficiency of GMRES-IR, but is simpler to use and does not have the
numerical stability problems [7].
Boojhawon and Bhuruth[9] by using augmentation technique applied the idea in GMRES-E to
SGMRES, and proposed the simpler GMRES method augmented with approximate eigenvectors
(SGMRES-E). The main advantage of SGMRES-E over GMRES-E is SGMRES-E requires a less
computational time than GMRES-E.
Ref [13] improves the SGMRES-E by following the idea behind GMRES-DR and proposed the
simpler GMRES method with deflated restarting (SGMRES-DR). SGMRES-DR includes the
harmonic Ritz vectors corresponding to the smallest eigenvalues of the coefficient matrix A at the
5. Advanced Computational Intelligence: An International Journal (ACII), Vol.3, No.2, April 2016
5
beginning of each cycle. SGMRES-DR requires less matrix-vector products than SGMRES-E,
and thus is more efficient.
2.1 SGMRES with deflated restarting
Suppose W ⊂Rnbe a subspace with dimension m, the columns of Zm = [z1, z2, . . .,zm] be a basis of
W, x0∈Rn be the initial guess, and r0 = b−Ax0 the initial residual vector. The purpose is to find the
approximate solution xmin the affine subspace x0+W by imposing the minimization of the residual
norm. It is equivalent to finding xm such that
r୫ ⊥ AW (3)
whererm= b−Axm, and xm= x0 + Zmymfor some ym∈Rm.
By applying the QR decomposition of AZm we have:
AZ୫ = V୫R୫(4)
and applying the orthogonality relation (3), the expression of the mth
residual vector is obtained:
r୫ = r − V୫V୫
r = r୫ିଵ − v୫൫v୫
r൯
The harmonic Ritz vectors corresponding to the smallest eigenvalues of A is used in SGMRES-
DR because they are better approximate eigenvectors for eigenvalues with small modulus [10].
Note that the term “the smallest eigenvalues” means the smallest eigenvalues in modulus. The
harmonic Ritz vector u = Zmy satisfies the following orthogonality condition
(Au − λu) ⊥ AZ୫
which is equivalent to
V୫
(AZ୫y − λZ୫y) = 0
i.e.,
R୫y = λV୫
Z୫y (5)
Suppose y1, y2, . . .,ykare k eigenvectors corresponding to k smallest eigenvalues of the reduced
generalized eigenvalue problem (5).
In the SGMRES-E, proposed by Boojhawon and Bhuruth in [9], approximate eigenvectors used
to augment the Krylov subspaces are harmonic Ritz vectorsxො୨ = Z୫y୨, j = 1,2, … , k. The
implementation of SGMRES-E is outlined in Algorithm 2[13].
Algorithm 2(SGMRES-E(m, k)).
Input:A : the coefficient matrix; b : the right-hand side; x0 : an initial approximation; m : the
maximal dimension of the Krylov subspace; k : the number of harmonic Ritz vectors; ε : a
tolerance.
Output:An approximate solution x.
6. Advanced Computational Intelligence: An International Journal (ACII), Vol.3, No.2, April 2016
6
1. Apply one cycle of SGMRES(m) to generate Zm, Vm, Rm, xm, and rm.
2. Compute the eigenvalues and eigenvectors of the generalized eigenvalue problem (6) by
the QZ algorithm [11]. Let y1, y2, . . .,yk be k eigenvectors corresponding to k smallest
eigenvalues of (6).
3. Form k harmonic Ritz vectors xො୨ = Z୫y୨ , j = 1,2, … , k
4. Set x0 = xm, and compute r = r୫ , zଵ = r ‖r‖⁄ , zොଵ = Azଵ , rଵଵ = ‖zොଵ‖, vଵ = zොଵ/rଵଵ
5. Compute αଵ = vଵ
r , rଵ = r − αଵvଵ
6. For j = 2,3, … , m
6.1. If j ≤ m − kthen z୨ = v୨ିଵelse z୨ = xො୨ି୫ା୩
6.2. zො୨ = Az୨
6.3. For i = 1,2, … , j − 1
r୧୨ = v୧
zො୨zො୨ = zො୨ − r୧୨v୧
End For
6.4. r୨୨ = ฮzො୨ฮ , v୨ = zො୨ r୨୨⁄ , α୨ = v୨
r୨ିଵ
6.5.r୨ = r୨ିଵ − α୨v୨
6.6. If ฮr୨ฮ < ߝ then solve the linear triangular system
R୨t = α
with α = ൣ0, … ,0, α୩ାଵ, … , α୨൧
، and form the approximate solution x = x + Z୨t, and
stop.
7. Solve the linear triangular systemRmt=α with α = ൣαଵ, αଶ, … , α୨൧
and form the
approximate solution x୫ = x + Z୫t. Go to Step 2.
SupposeY୩ = ሾyଵ, yଶ, … , y୩ሿand let P୩L୩ = Y୩be the QR decomposition of Yk. Multiplying (4) by
Pkfrom the right yields
AZ୫P୩ = V୫R୫P୩
Let Q୩R୩
୬ୣ୵
= R୫P୩be the QR decomposition of RmPk. From the above equality, we obtain
AZ୫P୩ = V୫Q୩R୩
୬ୣ୵
Define
Z୩
୬ୣ୵
= Z୫P୩ , V୩
୬ୣ୵
= V୫Q୩
So, we have
AZ୩
୬ୣ୵
= V୩
୬ୣ୵
R୩
୬ୣ୵
where Z୩
୬ୣ୵
, V୩
୬ୣ୵
∈ R୬×୩
, and V୩
୬ୣ୵
has orthonormal columns, ܴ
௪
∈ ܴ×
is an upper
triangular matrix. This is aQR decomposition with k new vectors.We now restart the GMRES
method. Note that
x
୬ୣ୵
= x୫, r
୬ୣ୵
= r୫ = r − V୫V୫
r
Therefore, the kth
residual vector at the new cycle is
r୩
୬ୣ୵
= r
୬ୣ୵
− V୩
୬ୣ୵
൫(V୩
୬ୣ୵
)
r
୬ୣ୵
൯
Since
(V୩
୬ୣ୵
)
r
୬ୣ୵
= Q୩
V୫
r୫ = Q୩
V୫
൫r − V୫V୫
r൯ = 0
7. Advanced Computational Intelligence: An International Journal (ACII), Vol.3, No.2, April 2016
7
we obtain
r୩
୬ୣ୵
= r
୬ୣ୵
to extend the new QR factorization of order k (7) to obtain a QR factorization of order m, i.e., a
QR factorization with m vectors.
In SGMRES-DR, we have
z୨ାଵ
୬ୣ୵
. j = k, k + 1, … , m − 1asz୨ାଵ
୬ୣ୵
= v୨
୬ୣ୵
Then, the expansion of (7) to a QR factorization of order m can be done as follows: the vector
Az୩ାଵ
୬ୣ୵
= Av୩
୬ୣ୵
is orthogonalized against V୩
୬ୣ୵
and normalized to give v୩ାଵ
୬ୣ୵
, after which R୩ାଵ
୬ୣ୵
is
formed from R୩
୬ୣ୵
. With z୨ାଵ
୬ୣ୵
= v୨
୬ୣ୵
, j = k, k + 1, … , m − 1 , the process is iterated until a QR
factorization of order m is obtained. The implementation of SGMRES-DR is outlined in
Algorithm 3[13].
Algorithm 3 (SGMRES-DR(m, k)).
Input: A : the coefficient matrix; b : the right-hand side; x0 : an initial approximation; m : the
maximal dimension of the Krylov subspace; k : the number of harmonic Ritz vectors; ε : a
tolerance.
Output: An approximate solution x.
1. Apply one cycle of SGMRES(m) to generate Zm, Vm, Rm, xm, and rm.
2. Compute the eigenvalues and eigenvectors of the generalized eigenvalue problem (6) by
the QZ algorithm [11]. Set Y୩ = ሾyଵ, yଶ, … , y୩ሿ, where yଵ, yଶ, … , y୩are k eigenvectors
corresponding to k smallest eigenvalues of (6).
3. Compute the QR decomposition of Y୩:P୩L୩ = Y୩and the QR decomposition of
R୫P୩: Q୩R୩
୬ୣ୵
= R୫P୩.
4. Set Z୩
୬ୣ୵
= Z୫P୩and V୩
୬ୣ୵
= V୫Q୩.
5. Let Z୩ = Z୩
୬ୣ୵
, V୩ = V୩
୬ୣ୵
, R୩ = R୩
୬ୣ୵
, x = x୫, r = r୫, r୩ = r
6. For j = k + 1, k + 2, … , m
6.1. z୨ = v୨ିଵ
6.2.zො୨ = Az୨
6.3. For i = 1,2, … , j − 1
r୧୨ = v୧
zො୨zො୨ = zො୨ − r୧୨v୧
End For
6.4.r୨୨ = ฮzො୨ฮ , v୨ = zො୨ r୨୨⁄ , α୨ = v୨
r୨ିଵ
6.5. r୨ = r୨ିଵ − α୨v୨
6.6. If ฮr୨ฮ < ߝ, then solve the linear triangular system
R୨t = α
With α = ሾ0, … ,0, α୩ାଵ, … , α୫ሿ
, and form the approximate solution x୫ = x + Z୫t.
Go to Step 2.
8. Advanced Computational Intelligence: An International Journal (ACII), Vol.3, No.2, April 2016
8
If the eigenvector yiat Step 2 in Algorithm 3 is complex, the matrix Ykshould include both the real
and imaginary parts of yi. In this situation, we may have to tune k eigenvectors at a restart.
After running one cycle of SGMRES(m), we reach toZ୫ = ሾr ‖r‖⁄ , V୫ିଵሿ. Based on [12], the
matrix V୫
Z୫in the generalized eigenvalue problem (5) can be formulated as
V୫
Z୫ =
ۏ
ێ
ێ
ۍ
vଵ
vଶ
⋮
v୫
ے
ۑ
ۑ
ې
ሾr ‖r‖⁄ V୫ିଵሿ =
ۏ
ێ
ێ
ێ
ێ
ێ
ێ
ێ
ۍ
vଵ
r
‖r‖
1
vଶ
r
‖r‖
0 1
⋮
⋮
v୫
r
‖r‖
0 ⋱
⋱ 1
0
ے
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ې
=
ۏ
ێ
ێ
ێ
ێ
ێ
ێ
ێ
ۍ
vଵ
r
‖r‖
1
vଶ
rଵ
‖r‖
0 1
⋮
⋮
v୫
r୫ିଵ
‖r‖
0 ⋱
⋱ 1
0
ے
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ې
Here, the next identity: for all j > i, v୨
r୧ = v୨
r୨ିଵ, has been used. Becausev୨
r୨ିଵ, j = 1,2, … m ,
has been generated in applying one cycle of SGMRES (m), the matrix V୫
Z୫without additional
inner products of vectors can be formed with order n.
For other cycles, we have Z୫ = ሾzଵ, … , z୩, v୩, … , v୫ିଵሿ
So, V୫
Z୫ =
ۏ
ێ
ێ
ێ
ێ
ێ
ۍ
vଵ
⋮
v୩
v୩ାଵ
⋮
v୫
ے
ۑ
ۑ
ۑ
ۑ
ۑ
ې
ሾzଵ, … , z୩, v୩, … , v୫ିଵሿ =
ۏ
ێ
ێ
ێ
ێ
ێ
ێ
ۍ
vଵ
zଵ
⋮
⋯
⋮
vଵ
z୩
⋮
v୩
zଵ
v୩ାଵ
zଵ
⋮
⋮
v୩
z୩
v୩ାଵ
z୩
1
0 1
v୩ାଶ
zଵ
⋮
v୫
zଵ
⋮
⋮
⋯
v୩ାଶ
z୩
⋮
v୫
z୩
0 ⋱
⋱ 1
0 ے
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ې
Thus, km additional inner products of vectors with order n to form V୫
Z୫ is required.
3. THE PROPOSED METHOD
As mentioned in the previous section, methods for solving linear system Ax = b start with an
initial vector x, and iteratively try to change the values of these vectors in order to reduce the
estimation error. In this methods, unfortunately, the process of calculating the amount of change
of the vector x is a time consuming process. Therefore, in this paper, we will use an intelligent
heuristic method to quickly predict the amount of change of the vector x in each iteration. The
proposed heuristic is as follows: “The value of each dimension d of vector x which in c previous
iterations of the algorithm steadily decreases (increases) is likely to decrease (increase) further in
the next iteration.” So, without much calculation, we can perform intelligent changes in the
direction of each dimension of vector x. By this assumption, in practice dramatic convergence
will be reached.
By changing the step 6 of algorithm 1, the Improved SGMRES(m) (ISGMRES(m)) is obtained.
The implementation of ISGMRES(m) is outlined in Algorithm 4.
9. Advanced Computational Intelligence: An International Journal (ACII), Vol.3, No.2, April 2016
9
Algorithm 4:The template of ISGMRES(m)
1. Step 1 to step 5 of algorithm 1.
2. Decrease (increase) the value of each dimension d of vector x which in c previous
iterations of the algorithm steadily decrease (increase), by β ∗ xᇱ
(d), where βis a real
number that is called the update rate and xᇱ
(d)is the amount of change in the value of
vector xin dimension d in the before iteration.
3. Goto step 1.
By changing the step 8 of algorithm 2, the Improved SGMRES-E(m,k) (ISGMRES-E(m,k)) can
be obtained. The implementation of ISGMRES-E(m,k) is outlined in Algorithm 5.
Algorithm 5: The template of ISGMRES-E(m,k)
1. Step 1 to step 7 of algorithm 2.
2. Decrease (increase) the value of each dimension d of vector x which in c previous
iterations of the algorithm steadily decrease (increase), by β ∗ xᇱ
(d), where βis a real
number that is called the update rate and xᇱ
(d)is the amount of change in the value of
vector xin dimension d in the before iteration.
3. Goto step 1.
By changing the step 8 of algorithm 3, the Improved SGMRES-DR(m,k) (ISGMRES-DR(m,k)) is
obtained. The implementation of ISGMRES-DR(m,k) is outlined in Algorithm 6.
Algorithm 6: The template of ISGMRES-E(m,k)
1. Step 1 to step 7 of algorithm 3.
2. Decrease (increase) the value of each dimension d of vector x which in c previous
iterations of the algorithm steadily decrease (increase), by β ∗ xᇱ
(d), where βis a real
number that is called the update rate and xᇱ
(d)is the amount of change in the value of
vector xin dimension d in the before iteration.
3. Goto step 1.
4. Experimental results
To investigate the performance of proposed methods, several experiments on a number of
standard matrices of the University of Florida has been carried out. In all these experiments, the
vector x0 is initialized by zero, and computational error ‖ܾ − ܣ ∗ ‖ݔ ‖ܾ‖⁄ intended. In Table 1,
SGMRES(m) and ISGMRES(m) are compared. According to Table 1, Figure1, and Figure 2, the
performance of ISGMRES(m) is better than SGMRES(m).
Table 1: Comparison of the results of SGMRES(m) and ISGMRES(m)
Problem m error SGMRES ISGMRES
iteration c ߚ iteration
Tols1090 20 0.2375 200 10 0.05 144
Tols1090 20 0.1847 500 10 0.007 458
Zenios 30 0.9585 200 5 0.00006 84
Zenios 30 0.9584 500 5 0.00007 252
Qh148 20 0.9898 200 5 0.005 200
Qh148 20 0.9898 500 5 0.9 257
10. Advanced Computational Intelligence: A
Figure 1: Comparison of convergence of SGMRES(m) and ISGMRES(m) on Zenios matrix with up to 200
Figure 2: Comparison of convergence of SGMRES(m) and ISGMRES(m) on Zenios matrix with up to 500
In Table 2, SGMRES-E(m,k) and ISGMRES
and Figure 4, the performance of ISGMRES
Table 2: Comparison of the results of SGMRES
Problem m k
Tols1090 20 5
Tols1090 20 5
Zenios 30 10
Zenios 30 10
Qh148 20 5
Qh148 20 5
Advanced Computational Intelligence: An International Journal (ACII), Vol.3, No.2, April
Figure 1: Comparison of convergence of SGMRES(m) and ISGMRES(m) on Zenios matrix with up to 200
iteration
f convergence of SGMRES(m) and ISGMRES(m) on Zenios matrix with up to 500
iteration
E(m,k) and ISGMRES-E(m,k) is compared. According to Table 2,Figure3,
and Figure 4, the performance of ISGMRES-E(m,k) is better than SGMRES-E(m,k).
e 2: Comparison of the results of SGMRES-E(m,k) and ISGMRES-E(m,k)
error SGMRES-
E
ISGMRES-
iteration c ߚ
0.2603 200 20 0.8
0.2063 500 20 0.4
0.9574 200 5 0.00005
0.9570 500 10 0.0005
0.9758 200 1 0.006
0.9750 500 15 0.1
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Figure 1: Comparison of convergence of SGMRES(m) and ISGMRES(m) on Zenios matrix with up to 200
f convergence of SGMRES(m) and ISGMRES(m) on Zenios matrix with up to 500
E(m,k) is compared. According to Table 2,Figure3,
E(m,k)
-E
iteration
128
408
182
500
190
500
11. Advanced Computational Intelligence: A
Figure 3: Comparison of convergence of SGMRES
Figure 4: Comparison of convergence
In Table 3, SGMRES-DR(m,k) and ISGMRES
3,Figure5, and Figure 6, the performance of ISGMRES
DR(m,k).
Table 3: Comparison of the results of SGMRES
Problem m k
Tols1090 20 5
Tols1090 20 5
Zenios 30 10
Zenios 30 10
Qh148 20 5
Qh148 20 5
Advanced Computational Intelligence: An International Journal (ACII), Vol.3, No.2, April
Figure 3: Comparison of convergence of SGMRES-E(m,k) and ISGMRES-E(m,k) on Tols1090 matrix
with up to 200 iteration
Figure 4: Comparison of convergence of SGMRES-E(m,k) and ISGMRES-E(m,k) on Tols1090 matrix
with up to 500 iteration
DR(m,k) and ISGMRES-DR(m,k) is compared. According to Table
3,Figure5, and Figure 6, the performance of ISGMRES-DR(m,k) is better than SGMRES
le 3: Comparison of the results of SGMRES-DR(m,k) and ISGMRES-DR(m,k)
error SGMRES-
DR
ISGMRES-DR
iteration c ߚ
0.3292 200 10 0.3
0.0327 500 5 0.005
0.9576 200 10 0.00005
0.9575 500 10 0.0005
0.9758 200 5 0.04
0.9757 500 5 0.02
onal Journal (ACII), Vol.3, No.2, April 2016
11
E(m,k) on Tols1090 matrix
E(m,k) on Tols1090 matrix
DR(m,k) is compared. According to Table
DR(m,k) is better than SGMRES-
DR(m,k)
DR
iteration
119
355
182
500
164
453
12. Advanced Computational Intelligence: A
Figure 5: Comparison of convergence of SGMRES
Figure 6: Comparison of convergence of SGMRES
It should be noted that the computational complexity of each iteration of ISGMRES(m),
ISGMRES-E(m), and ISGMRES
SGMRES(m), ISGMRES-E(m), and ISGMRES
As it can be observed from the above tables and figures, the proposed methods compared with
corresponding standard methods for finding solutions with the same error require a lesser number
of iterations.
5. CONCLUSIONS
Based on the mathematical techniques, to improve the performance of SGMRES(m), SGMRES
E(m, k) and SGMRES-DR(m, k) methods, [5]
computational complexity and sometimes execution time and n
vector multiplication greatly increase, a modified technique is proposed. Although, improving the
Advanced Computational Intelligence: An International Journal (ACII), Vol.3, No.2, April
Figure 5: Comparison of convergence of SGMRES-DR(m,k) and ISGMRES-DR(m,k) on Qh148 matrix
with up to 200 iteration
gence of SGMRES-DR(m,k) and ISGMRES-DR(m,k) on Qh148 matrix
with up to 500 iteration
It should be noted that the computational complexity of each iteration of ISGMRES(m),
E(m), and ISGMRES-DR(m) is approximately equal to each iteration of
E(m), and ISGMRES-DR(m), respectively.
As it can be observed from the above tables and figures, the proposed methods compared with
corresponding standard methods for finding solutions with the same error require a lesser number
Based on the mathematical techniques, to improve the performance of SGMRES(m), SGMRES
DR(m, k) methods, [5] - [8] - [7] - [13], that each of which have their own
computational complexity and sometimes execution time and number of operations in the matrix
vector multiplication greatly increase, a modified technique is proposed. Although, improving the
onal Journal (ACII), Vol.3, No.2, April 2016
12
DR(m,k) on Qh148 matrix
DR(m,k) on Qh148 matrix
It should be noted that the computational complexity of each iteration of ISGMRES(m),
DR(m) is approximately equal to each iteration of
As it can be observed from the above tables and figures, the proposed methods compared with
corresponding standard methods for finding solutions with the same error require a lesser number
Based on the mathematical techniques, to improve the performance of SGMRES(m), SGMRES-
[13], that each of which have their own
umber of operations in the matrix
vector multiplication greatly increase, a modified technique is proposed. Although, improving the
13. Advanced Computational Intelligence: An International Journal (ACII), Vol.3, No.2, April 2016
13
efficiency of these algorithms by using the concepts and techniques of artificial intelligence has
received little attention, in this paper, an intelligent heuristic method that has very little
computational complexity is used to improve the performance of procedures SGMRES(m),
SGMRES-E(m, k) and SGMRES-DR(m, k).The numerical results obtained from implementation
of the proposed method on several University of Florida standard matrixes confirm the efficiency
of the proposed method.
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