The document presents three numerical methods - Trapezoidal, Simpson's 1/3, and Simpson's 3/8 - for approximating definite integrals. It analyzes the accuracy of the methods based on numerical examples evaluated in MATLAB. Simpson's 1/3 method was found to produce results closest to the exact values, with the minimum error. Graphs of the approximate solutions and errors visually demonstrate Simpson's 1/3 method's superior convergence compared to the other two methods.
A NEW STUDY OF TRAPEZOIDAL, SIMPSON’S1/3 AND SIMPSON’S 3/8 RULES OF NUMERICAL...mathsjournal
The main goal of this research is to give the complete conception about numerical integration including Newton-Cotes formulas and aimed at comparing the rate of performance or the rate of accuracy of Trapezoidal, Simpson’s 1/3, and Simpson’s 3/8. To verify the accuracy, we compare each rules demonstrating the smallest error values among them. The software package MATLAB R2013a is applied to determine the best method, as well as the results, are compared. It includes graphical comparisons mentioning these methods graphically. After all, it is then emphasized that the among methods considered, Simpson’s 1/3 is more effective and accurate when the condition of the subdivision is only even for solving a definite integral.
Applied Mathematics and Sciences: An International Journal (MathSJ)mathsjournal
The main goal of this research is to give the complete conception about numerical integration including
Newton-Cotes formulas and aimed at comparing the rate of performance or the rate of accuracy of
Trapezoidal, Simpson’s 1/3, and Simpson’s 3/8. To verify the accuracy, we compare each rules
demonstrating the smallest error values among them. The software package MATLAB R2013a is applied to
determine the best method, as well as the results, are compared. It includes graphical comparisons
mentioning these methods graphically. After all, it is then emphasized that the among methods considered,
Simpson’s 1/3 is more effective and accurate when the condition of the subdivision is only even for solving
a definite integral.
A NEW METHOD OF CENTRAL DIFFERENCE INTERPOLATIONmathsjournal
The document presents a new method of central difference interpolation that is derived from Gauss's third formula and another existing formula. It compares the new method to other common interpolation formulas through two example problems. For both examples, the new method is shown to provide results that are very close to the exact values and have lower error than the other existing interpolation formulas.
A NEW METHOD OF CENTRAL DIFFERENCE INTERPOLATIONmathsjournal
In Numerical analysis, interpolation is a manner of calculating the unknown values of a function for any conferred value of argument within the limit of the arguments. It provides basically a concept of estimating unknown data with the aid of relating acquainted data. The main goal of this research is to constitute a central difference interpolation method which is derived from the combination of Gauss’s third formula, Gauss’s Backward formula and Gauss’s forward formula. We have also demonstrated the graphical presentations as well as comparison through all the existing interpolation formulas with our propound method of central difference interpolation. By the comparison and graphical presentation, the new method gives the best result with the lowest error from another existing interpolationformula.
A NEW METHOD OF CENTRAL DIFFERENCE INTERPOLATIONmathsjournal
In Numerical analysis, interpolation is a manner of calculating the unknown values of a function for any
conferred value of argument within the limit of the arguments. It provides basically a concept of
estimating unknown data with the aid of relating acquainted data. The main goal of this research is to
constitute a central difference interpolation method which is derived from the combination of Gauss’s
third formula, Gauss’s Backward formula and Gauss’s forward formula. We have also demonstrated the
graphical presentations as well as comparison through all the existing interpolation formulas with our
propound method of central difference interpolation. By the comparison and graphical presentation, the
new method gives the best result with the lowest error from another existing interpolationformula.
A NEW METHOD OF CENTRAL DIFFERENCE INTERPOLATIONmathsjournal
The document presents a new method of central difference interpolation that is derived from Gauss's third formula and another existing formula. It compares the new method to other common interpolation formulas through two example problems. For both examples, the new method is shown to provide results that are very close to the exact values and have lower error than the other existing interpolation formulas.
A NEW STUDY TO FIND OUT THE BEST COMPUTATIONAL METHOD FOR SOLVING THE NONLINE...mathsjournal
The main purpose of this research is to find out the best method through iterative methods for solving the nonlinear equation. In this study, the four iterative methods are examined and emphasized to solve the nonlinear equations. From this method explained, the rate of convergence is demonstrated among the 1st degree based iterative methods. After that, the graphical development is established here with the help of the four iterative methods and these results are tested with various functions. An example of the algebraic equation is taken to exhibit the comparison of the approximate error among the methods. Moreover, two examples of the algebraic and transcendental equation are applied to verify the best method, as well as the level of errors, are shown graphically.
A NEW STUDY TO FIND OUT THE BEST COMPUTATIONAL METHOD FOR SOLVING THE NONLINE...mathsjournal
The main purpose of this research is to find out the best method through iterative methods for solving the nonlinear equation. In this study, the four iterative methods are examined and emphasized to solve the nonlinear equations. From this method explained, the rate of convergence is demonstrated among the 1st degree based iterative methods. After that, the graphical development is established here with the help of the four iterative methods and these results are tested with various functions. An example of the algebraic equation is taken to exhibit the comparison of the approximate error among the methods. Moreover, two examples of the algebraic and transcendental equation are applied to verify the best method, as well as the level of errors, are shown graphically.
A NEW STUDY OF TRAPEZOIDAL, SIMPSON’S1/3 AND SIMPSON’S 3/8 RULES OF NUMERICAL...mathsjournal
The main goal of this research is to give the complete conception about numerical integration including Newton-Cotes formulas and aimed at comparing the rate of performance or the rate of accuracy of Trapezoidal, Simpson’s 1/3, and Simpson’s 3/8. To verify the accuracy, we compare each rules demonstrating the smallest error values among them. The software package MATLAB R2013a is applied to determine the best method, as well as the results, are compared. It includes graphical comparisons mentioning these methods graphically. After all, it is then emphasized that the among methods considered, Simpson’s 1/3 is more effective and accurate when the condition of the subdivision is only even for solving a definite integral.
Applied Mathematics and Sciences: An International Journal (MathSJ)mathsjournal
The main goal of this research is to give the complete conception about numerical integration including
Newton-Cotes formulas and aimed at comparing the rate of performance or the rate of accuracy of
Trapezoidal, Simpson’s 1/3, and Simpson’s 3/8. To verify the accuracy, we compare each rules
demonstrating the smallest error values among them. The software package MATLAB R2013a is applied to
determine the best method, as well as the results, are compared. It includes graphical comparisons
mentioning these methods graphically. After all, it is then emphasized that the among methods considered,
Simpson’s 1/3 is more effective and accurate when the condition of the subdivision is only even for solving
a definite integral.
A NEW METHOD OF CENTRAL DIFFERENCE INTERPOLATIONmathsjournal
The document presents a new method of central difference interpolation that is derived from Gauss's third formula and another existing formula. It compares the new method to other common interpolation formulas through two example problems. For both examples, the new method is shown to provide results that are very close to the exact values and have lower error than the other existing interpolation formulas.
A NEW METHOD OF CENTRAL DIFFERENCE INTERPOLATIONmathsjournal
In Numerical analysis, interpolation is a manner of calculating the unknown values of a function for any conferred value of argument within the limit of the arguments. It provides basically a concept of estimating unknown data with the aid of relating acquainted data. The main goal of this research is to constitute a central difference interpolation method which is derived from the combination of Gauss’s third formula, Gauss’s Backward formula and Gauss’s forward formula. We have also demonstrated the graphical presentations as well as comparison through all the existing interpolation formulas with our propound method of central difference interpolation. By the comparison and graphical presentation, the new method gives the best result with the lowest error from another existing interpolationformula.
A NEW METHOD OF CENTRAL DIFFERENCE INTERPOLATIONmathsjournal
In Numerical analysis, interpolation is a manner of calculating the unknown values of a function for any
conferred value of argument within the limit of the arguments. It provides basically a concept of
estimating unknown data with the aid of relating acquainted data. The main goal of this research is to
constitute a central difference interpolation method which is derived from the combination of Gauss’s
third formula, Gauss’s Backward formula and Gauss’s forward formula. We have also demonstrated the
graphical presentations as well as comparison through all the existing interpolation formulas with our
propound method of central difference interpolation. By the comparison and graphical presentation, the
new method gives the best result with the lowest error from another existing interpolationformula.
A NEW METHOD OF CENTRAL DIFFERENCE INTERPOLATIONmathsjournal
The document presents a new method of central difference interpolation that is derived from Gauss's third formula and another existing formula. It compares the new method to other common interpolation formulas through two example problems. For both examples, the new method is shown to provide results that are very close to the exact values and have lower error than the other existing interpolation formulas.
A NEW STUDY TO FIND OUT THE BEST COMPUTATIONAL METHOD FOR SOLVING THE NONLINE...mathsjournal
The main purpose of this research is to find out the best method through iterative methods for solving the nonlinear equation. In this study, the four iterative methods are examined and emphasized to solve the nonlinear equations. From this method explained, the rate of convergence is demonstrated among the 1st degree based iterative methods. After that, the graphical development is established here with the help of the four iterative methods and these results are tested with various functions. An example of the algebraic equation is taken to exhibit the comparison of the approximate error among the methods. Moreover, two examples of the algebraic and transcendental equation are applied to verify the best method, as well as the level of errors, are shown graphically.
A NEW STUDY TO FIND OUT THE BEST COMPUTATIONAL METHOD FOR SOLVING THE NONLINE...mathsjournal
The main purpose of this research is to find out the best method through iterative methods for solving the nonlinear equation. In this study, the four iterative methods are examined and emphasized to solve the nonlinear equations. From this method explained, the rate of convergence is demonstrated among the 1st degree based iterative methods. After that, the graphical development is established here with the help of the four iterative methods and these results are tested with various functions. An example of the algebraic equation is taken to exhibit the comparison of the approximate error among the methods. Moreover, two examples of the algebraic and transcendental equation are applied to verify the best method, as well as the level of errors, are shown graphically.
A NEW STUDY TO FIND OUT THE BEST COMPUTATIONAL METHOD FOR SOLVING THE NONLINE...mathsjournal
The main purpose of this research is to find out the best method through iterative methods for solving the
nonlinear equation. In this study, the four iterative methods are examined and emphasized to solve the
nonlinear equations. From this method explained, the rate of convergence is demonstrated among the 1st
degree based iterative methods. After that, the graphical development is established here with the help of
the four iterative methods and these results are tested with various functions. An example of the algebraic
equation is taken to exhibit the comparison of the approximate error among the methods. Moreover, two
examples of the algebraic and transcendental equation are applied to verify the best method, as well as the
level of errors, are shown graphically.
Riccati matrix differential equation has long been known to be so difficult to solve analytically and/or numerically. In this connection, most of the recent studies are concerned with the derivation of the necessary conditions that ensure the existence of the solution. Therefore, in this paper, He’s Variational iteration method is used to derive the general form of the iterative approximate sequence of solutions and then proved the convergence of the obtained sequence of approximate solutions to the exact solution. This proof is based on using the mathematical induction to derive a general formula for the upper bound proved to be converging to zero under certain conditions.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
In solving real life assignment problem, we often face the state of uncertainty as well as hesitation due to varies uncontrollable factors. To deal with uncertainty and hesitation many authors have suggested the intuitionistic fuzzy representation for the data. In this paper, computationally a simple method is proposed to find the optimal solution for an unbalanced assignment problem under intuitionistic fuzzy environment. In conventional assignment problem, cost is always certain. This paper develops an approach to solve the unbalanced assignment problem where the time/cost/profit is not in deterministic numbers but imprecise ones. In this assignment problem, the elements of the cost matrix are represented by the triangular intuitionistic fuzzy numbers. The existing Ranking procedure of Varghese and Kuriakose is used to transform the unbalanced intuitionistic fuzzy assignment problem into a crisp one so that the conventional method may be applied to solve the AP . Finally, the method is illustrated by a numerical example which is followed by graphical representation and discussion of the finding.
This document summarizes research on applying differential geometry and optimization techniques to computer vision problems. Specifically, it develops a novel parameterization-based framework that views manifolds as collections of local coordinate charts. It carries out optimization in parameter space and projects the optimal vector back to the manifold. Newton-type algorithms are devised based on this approach and their local quadratic convergence is mathematically proven. The document reviews literature on Riemannian and non-Riemannian approaches to geometric optimization on manifolds, which has applications in problems like pose estimation from images.
Selecting the best stochastic systems for large scale engineering problemsIJECEIAES
Selecting a subset of the best solutions among large-scale problems is an important area of research. When the alternative solutions are stochastic in nature, then it puts more burden on the problem. The objective of this paper is to select a set that is likely to contain the actual best solutions with high probability. If the selected set contains all the best solutions, then the selection is denoted as correct selection. We are interested in maximizing the probability of this selection; P(CS). In many cases, the available computation budget for simulating the solution set in order to maximize P(CS) is limited. Therefore, instead of distributing these computational efforts equally likely among the alternatives, the optimal computing budget allocation (OCBA) procedure came to put more effort on the solutions that have more impact on the selected set. In this paper, we derive formulas of how to distribute the available budget asymptotically to find the approximation of P(CS). We then present a procedure that uses OCBA with the ordinal optimization (OO) in order to select the set of best solutions. The properties and performance of the proposed procedure are illustrated through a numerical example. Overall results indicate that the procedure is able to select a subset of the best systems with high probability of correct selection using small number of simulation samples under different parameter settings.
Numerical Investigation of Higher Order Nonlinear Problem in the Calculus Of ...IOSR Journals
This document presents a numerical investigation of higher order nonlinear problems in calculus of variations using the Adomian decomposition method (ADM). It introduces ADM for solving such problems by reducing them to systems of nonlinear algebraic equations. An example problem is provided and solved numerically using both the single-term Haar wavelet series method (STHWS) and ADM. Error calculations and graphs demonstrate that ADM provides more accurate solutions than STHWS, with less complexity and error. The document concludes that ADM performs better for this type of problem in calculus of variations.
A New Method Based on MDA to Enhance the Face Recognition PerformanceCSCJournals
A novel tensor based method is prepared to solve the supervised dimensionality reduction problem. In this paper a multilinear principal component analysis(MPCA) is utilized to reduce the tensor object dimension then a multilinear discriminant analysis(MDA), is applied to find the best subspaces. Because the number of possible subspace dimensions for any kind of tensor objects is extremely high, so testing all of them for finding the best one is not feasible. So this paper also presented a method to solve that problem, The main criterion of algorithm is not similar to Sequential mode truncation(SMT) and full projection is used to initialize the iterative solution and find the best dimension for MDA. This paper is saving the extra times that we should spend to find the best dimension. So the execution time will be decreasing so much. It should be noted that both of the algorithms work with tensor objects with the same order so the structure of the objects has been never broken. Therefore the performance of this method is getting better. The advantage of these algorithms is avoiding the curse of dimensionality and having a better performance in the cases with small sample sizes. Finally, some experiments on ORL and CMPU-PIE databases is provided.
Grey Multi Criteria Decision Making MethodsIJSRP Journal
Multi-Criteria Decision Making is the most well-known branch of decision making. In some cases, determining precisely the exact value of attributes is difficult and their values can be considered as Uncertain data. This paper presents two different Multi Criteria Decision Making methods based on grey numbers. The two methods are used to obtain the final ranking of the alternatives and select the best one under grey numbers. Finally, an illustrative example is presented and the results are analyzed.
A Convex Hull Formulation for the Design of Optimal Mixtures.pdfChristine Maffla
This document summarizes a study on using convex hull formulations to design optimal mixtures. It presents a generalized modeling framework that integrates Generalized Disjunctive Programming (GDP) into Computer-Aided Mixture/blend Design via Hull Reformulation (HR). The methodology is applied to find an optimal solvent mixture that maximizes the solubility of ibuprofen. Two approaches for converting the GDP formulation into a mixed-integer nonlinear program are compared: the big-M (BM) approach and the hull reformulation (HR) approach. The results show that the HR approach has better computational efficiency and avoids numerical difficulties compared to the BM approach for solving mixture design problems.
EDGE DETECTION IN SEGMENTED IMAGES THROUGH MEAN SHIFT ITERATIVE GRADIENT USIN...ijscmcj
In this paper, we propose a new method for edge detection in obtained images from the Mean Shift iterative algorithm. The comparable, proportional and symmetrical images are de?ned and the importance of Ring Theory is explained. A relation of equivalence among proportional images are de?ned for image groups in equivalent classes. The length of the mean shift vector is used in order to quantify the homogeneity of the neighborhoods. This gives a measure of how much uniform are the regions that compose the image. Edge detection is carried out by using the mean shift gradient based on symmetrical images. The difference among the values of gray levels are accentuated or these are decreased to enhance the interest region contours. The chosen images for the experiments were standard images and real images (cerebral hemorrhage images). The obtained results were compared with the Canny detector, and our results showed a good performance as for the edge continuity.
Efficient approximate analytical methods for nonlinear fuzzy boundary value ...IJECEIAES
This paper aims to solve the nonlinear two-point fuzzy boundary value problem (TPFBVP) using approximate analytical methods. Most fuzzy boundary value problems cannot be solved exactly or analytically. Even if the analytical solutions exist, they may be challenging to evaluate. Therefore, approximate analytical methods may be necessary to consider the solution. Hence, there is a need to formulate new, efficient, more accurate techniques. This is the focus of this study: two approximate analytical methods-homotopy perturbation method (HPM) and the variational iteration method (VIM) is proposed. Fuzzy set theory properties are presented to formulate these methods from crisp domain to fuzzy domain to find approximate solutions of nonlinear TPFBVP. The presented algorithms can express the solution as a convergent series form. A numerical comparison of the mean errors is made between the HPM and VIM. The results show that these methods are reliable and robust. However, the comparison reveals that VIM convergence is quicker and offers a swifter approach over HPM. Hence, VIM is considered a more efficient approach for nonlinear TPFBVPs.
Analytical Hierarchical Process has been used as a useful methodology for multi-criteria decision making environments with substantial applications in recent years. But the weakness of the traditional AHP method lies in the use of subjective judgement based assessment and standardized scale for pairwise comparison matrix creation. The paper proposes a Condorcet Voting Theory based AHP method to solve multi criteria decision making problems where Analytical Hierarchy Process (AHP) is combined with Condorcet theory based preferential voting technique followed by a quantitative ratio method for framing the comparison matrix instead of the standard importance scale in traditional AHP approach. The consistency ratio (CR) is calculated for both the approaches to determine and compare the consistency of both the methods. The results reveal Condorcet- AHP method to be superior generating lower consistency ratio and more accurate ranking of the criterion for solving MCDM problems.
A comparative study of three validities computation methods for multimodel ap...IJECEIAES
The multimodel approach offers a very satisfactory results in modelling, diagnose and control of complex systems. In the modelling case, this approach passes by three steps: the determination of the model’s library, the validities computation and the establishment of the final model. In this context, this paper focuses on the elaboration of a comparative study between three recent methods of validities computation. Thus, it highlight the method that offers the best performances in term of precision. To achieve this goal, we apply, these three methods on two simulation examples in order to compare their performances.
A NEW STUDY TO FIND OUT THE BEST COMPUTATIONAL METHOD FOR SOLVING THE NONLINE...mathsjournal
The main purpose of this research is to find out the best method through iterative methods for solving the
nonlinear equation. In this study, the four iterative methods are examined and emphasized to solve the
nonlinear equations. From this method explained, the rate of convergence is demonstrated among the 1st
degree based iterative methods. After that, the graphical development is established here with the help of
the four iterative methods and these results are tested with various functions. An example of the algebraic
equation is taken to exhibit the comparison of the approximate error among the methods. Moreover, two
examples of the algebraic and transcendental equation are applied to verify the best method, as well as the
level of errors, are shown graphically.
This document presents a case study on a linear programming problem with fuzzy parameters for a bakery industry. It introduces the Neelachal Bakery Private Limited, which started in 1989 in Bhubaneswar, India as one of the first bakeries to use polythene packaging. It has grown to serve over 45 people daily from one bakery location with assets of Rs. 3 crores. The case study aims to formulate a fuzzy linear programming model to maximize the profit for Neelachal Bakery considering uncertain parameters like costs, supply, and demand. It will apply Robust's ranking technique to transform the fuzzy linear problem into a crisp one that can be solved using the simplex method.
This document summarizes a research paper that investigates techniques for solving linear programming problems when the decision parameters are fuzzy numbers. It begins with an abstract that overviews using Robust's ranking technique to transform fuzzy linear programming problems into crisp problems that can be solved using traditional methods. It then provides background on fuzzy linear programming research and defines key concepts like fuzzy numbers and arithmetic. The document outlines the proposed approach of using Robust's ranking to defuzzify parameters and allow for sensitivity analysis in the constraints. Finally, it presents a case study to illustrate applying the model.
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.
A Study on Differential Equations on the Sphere Using Leapfrog Methodiosrjce
In this article, a new method of analysis of the differential equations on the sphere using Leapfrog
method is presented. To illustrate the effectiveness of the Leapfrog method, an example of the differential
equations on the sphere has been considered and the solutions were obtained using methods taken taken from
the literature [17] and Leapfrog method. The obtained discrete solutions are compared with the exact solutions
of the differential equations on the sphere. Solution graphs for the differential equations on the sphere have
been presented in the graphical form to show the efficiency of this Leapfrog method. This Leapfrog method can
be easily implemented in a digital computer and the solution can be obtained for any length of time
An improvised similarity measure for generalized fuzzy numbersjournalBEEI
Similarity measure between two fuzzy sets is an important tool for comparing various characteristics of the fuzzy sets. It is a preferred approach as compared to distance methods as the defuzzification process in obtaining the distance between fuzzy sets will incur loss of information. Many similarity measures have been introduced but most of them are not capable to discriminate certain type of fuzzy numbers. In this paper, an improvised similarity measure for generalized fuzzy numbers that incorporate several essential features is proposed. The features under consideration are geometric mean averaging, Hausdorff distance, distance between elements, distance between center of gravity and the Jaccard index. The new similarity measure is validated using some benchmark sample sets. The proposed similarity measure is found to be consistent with other existing methods with an advantage of able to solve some discriminant problems that other methods cannot. Analysis of the advantages of the improvised similarity measure is presented and discussed. The proposed similarity measure can be incorporated in decision making procedure with fuzzy environment for ranking purposes.
A View From The Bridge Essay ConclusionFelicia Clark
The document discusses the steps involved in requesting writing assistance from HelpWriting.net. It outlines 5 steps: 1) Create an account with a password and email. 2) Complete a 10-minute order form providing instructions, sources, and deadline. 3) Review bids from writers and choose one. 4) Review the completed paper and authorize payment. 5) Request revisions to ensure satisfaction, with a refund offered for plagiarized work.
The document outlines a 5-step process for requesting and receiving writing assistance from the HelpWriting.net website. Students must first create an account, then submit a request form providing instructions and deadlines. Writers will bid on the request and students can choose a writer, make a deposit, and receive a completed paper to review and pay for upon approval.
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IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
In solving real life assignment problem, we often face the state of uncertainty as well as hesitation due to varies uncontrollable factors. To deal with uncertainty and hesitation many authors have suggested the intuitionistic fuzzy representation for the data. In this paper, computationally a simple method is proposed to find the optimal solution for an unbalanced assignment problem under intuitionistic fuzzy environment. In conventional assignment problem, cost is always certain. This paper develops an approach to solve the unbalanced assignment problem where the time/cost/profit is not in deterministic numbers but imprecise ones. In this assignment problem, the elements of the cost matrix are represented by the triangular intuitionistic fuzzy numbers. The existing Ranking procedure of Varghese and Kuriakose is used to transform the unbalanced intuitionistic fuzzy assignment problem into a crisp one so that the conventional method may be applied to solve the AP . Finally, the method is illustrated by a numerical example which is followed by graphical representation and discussion of the finding.
This document summarizes research on applying differential geometry and optimization techniques to computer vision problems. Specifically, it develops a novel parameterization-based framework that views manifolds as collections of local coordinate charts. It carries out optimization in parameter space and projects the optimal vector back to the manifold. Newton-type algorithms are devised based on this approach and their local quadratic convergence is mathematically proven. The document reviews literature on Riemannian and non-Riemannian approaches to geometric optimization on manifolds, which has applications in problems like pose estimation from images.
Selecting the best stochastic systems for large scale engineering problemsIJECEIAES
Selecting a subset of the best solutions among large-scale problems is an important area of research. When the alternative solutions are stochastic in nature, then it puts more burden on the problem. The objective of this paper is to select a set that is likely to contain the actual best solutions with high probability. If the selected set contains all the best solutions, then the selection is denoted as correct selection. We are interested in maximizing the probability of this selection; P(CS). In many cases, the available computation budget for simulating the solution set in order to maximize P(CS) is limited. Therefore, instead of distributing these computational efforts equally likely among the alternatives, the optimal computing budget allocation (OCBA) procedure came to put more effort on the solutions that have more impact on the selected set. In this paper, we derive formulas of how to distribute the available budget asymptotically to find the approximation of P(CS). We then present a procedure that uses OCBA with the ordinal optimization (OO) in order to select the set of best solutions. The properties and performance of the proposed procedure are illustrated through a numerical example. Overall results indicate that the procedure is able to select a subset of the best systems with high probability of correct selection using small number of simulation samples under different parameter settings.
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Multi-Criteria Decision Making is the most well-known branch of decision making. In some cases, determining precisely the exact value of attributes is difficult and their values can be considered as Uncertain data. This paper presents two different Multi Criteria Decision Making methods based on grey numbers. The two methods are used to obtain the final ranking of the alternatives and select the best one under grey numbers. Finally, an illustrative example is presented and the results are analyzed.
A Convex Hull Formulation for the Design of Optimal Mixtures.pdfChristine Maffla
This document summarizes a study on using convex hull formulations to design optimal mixtures. It presents a generalized modeling framework that integrates Generalized Disjunctive Programming (GDP) into Computer-Aided Mixture/blend Design via Hull Reformulation (HR). The methodology is applied to find an optimal solvent mixture that maximizes the solubility of ibuprofen. Two approaches for converting the GDP formulation into a mixed-integer nonlinear program are compared: the big-M (BM) approach and the hull reformulation (HR) approach. The results show that the HR approach has better computational efficiency and avoids numerical difficulties compared to the BM approach for solving mixture design problems.
EDGE DETECTION IN SEGMENTED IMAGES THROUGH MEAN SHIFT ITERATIVE GRADIENT USIN...ijscmcj
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the four iterative methods and these results are tested with various functions. An example of the algebraic
equation is taken to exhibit the comparison of the approximate error among the methods. Moreover, two
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A New Analysis Of Approximate Solutions For Numerical Integration Problems With Quadrature-Based
1. Pure and Applied Mathematics Journal
2020; 9(3): 46-54
http://www.sciencepublishinggroup.com/j/pamj
doi: 10.11648/j.pamj.20200903.11
ISSN: 2326-9790 (Print); ISSN: 2326-9812 (Online)
A New Analysis of Approximate Solutions for Numerical
Integration Problems with Quadrature-based Methods
Mir Md. Moheuddin1, *
, Muhammad Abdus Sattar Titu2
, Saddam Hossain3
1
Department of CSE (Mathematics), Atish Dipankar University of Science and Technology (ADUST), Dhaka, Bangladesh
2
Department of Mathematics (General Science), Mymensingh Engineering College (MEC), Mymensingh, Bangladesh
3
Department of Basic Science (Mathematics), World University of Bangladesh (WUB), Dhaka, Bangladesh
Email address:
*
Corresponding author
To cite this article:
Mir Md. Moheuddin, Muhammad Abdus Sattar Titu, Saddam Hossain. A New Analysis of Approximate Solutions for Numerical Integration
Problems with Quadrature-based Methods. Pure and Applied Mathematics Journal. Vol. 9, No. 3, 2020, pp. 46-54.
doi: 10.11648/j.pamj.20200903.11
Received: May 4, 2020; Accepted: June 9, 2020; Published: June 20, 2020
Abstract: In this paper, we mainly propose the approximate solutions to solve the integration problems numerically using
the quadrature method including the Trapezoidal method, Simpson’s 1/3 method, and Simpson’s 3/8 method. The three
proposed methods are quite workable and practically well suitable for solving integration problems. Through the MATLAB
program, our numerical solutions are determined as well as compared with the exact values to verify the higher accuracy of the
proposed methods. Some numerical examples have been utilized to give the accuracy rate and simple implementation of our
methods. In this study, we have compared the performance of our solutions and the computational attempt of our proposed
methods. Moreover, we explore and calculate the errors of the three proposed methods for the sake of showing our
approximate solution’s superiority. Then, among these three methods, we analyzed the approximate errors to prove which
method shows more appropriate results. We also demonstrated the approximate results and observed errors to give clear idea
graphically. Therefore, from the analysis, we can point out that only the minimum error is in Simpson’s 1/3 method which will
beneficial for the readers to understand the effectiveness in solving the several numerical integration problems.
Keywords: Numerical Integration, Trapezoidal Method, Simpson’s One-Third Method, Simpson’s Three-eighth’s Method
1. Introduction
Numerical integration is the procedure of finding the
approximate value of a definite integral. It is the approximate
calculation of an integral with the help of numerical methods.
In numerical integration, quadrature is a word that means the
integration of a function that has a single variable and
cubature to mean the numerical computation of multiple
integrals. There are so many methods are available for
numerical integration. Newton-Cotes formulas which are also
known as quadrature formulas is a numerical integration
technique that approximates a function at a sequence of
regularly spaced intervals. Numerical methods are commonly
used for solving mathematical problems that are applied in
science and engineering where it is difficult or even
impossible to obtain exact solutions. It plays a vital role to
solve mathematical problems in the numerical integration
field. There are lots of numerical integral problems whose
solutions cannot be achieved in closed form by using well
known analytical methods, where we have to use the
numerical methods to get the approximate solution of the
definite integral problems under the prescribed conditions.
Besides, there are various types of practical numerical
methods for solving integral problems. Numerical techniques
are important to evaluate the solutions of common and
complicated problems easily. Nowadays the methods of
numerical integrations based on information theory. Which
developed to simulate information systems such as computer-
controlled systems, communication systems, etc.
From the literature view, we can realize that many
researchers have tried to achieve good results and accuracy
quickly by using numerical methods. Mazzia et al. described
and compared some numerical quadrature rules with the aim
of preserving the MLPG solution accuracy and at the same
2. 47 Mir Md. Moheuddin et al.: A New Analysis of Approximate Solutions for Numerical Integration
Problems with Quadrature-based Methods
time reducing its computational cost [1]. Different numerical
rules are interpreted to count an approximate value of a
definite integral to a given accuracy in the book of Davis et
al., J. Douglas Fairs et al., S. S. Sastry, and L. Ridgway Scott
[2, 3, 4, 5]. Joulaian, M. et al. proposed a numerical approach
suitable for integrating broken elements with a low number
of integration points [6]. Also, Md. Jashim Uddin et al.
presented the complete conception about numerical
integration including Newton-Cotes formulas and compared
the rate of performance or the rate of accuracy of
Trapezoidal, Simpson’s 1/3, and Simpson’s 3/8 [7]. Jüttler, B.
et al. presented a detailed case study of different quadrature
schemes for isogeometric discretizations of partial
differential equations on closed surfaces with Loop’s
subdivision scheme [8]. ur Rehman, M. et al. developed a
numerical method to obtain approximate solutions for a
certain class of fractional differential equations [9]. However,
Thiagarajan, V. et al. demonstrated the application of AW
integration scheme in the context of the Finite Cell Method
which must perform numerical integration over arbitrary
domains without meshing [10]. Gerry Sozio discussed a
detailed summary of different techniques of numerical
integration [11]. Tornabene, F. et al. have investigated and
compared the accuracy and convergence behavior of two
different numerical approaches based on Differential
Quadrature (DQ) and Integral Quadrature (IQ) methods,
respectively, when applied to the free vibration analysis of
laminated plates and shells [12]. Rajesh Kumar Sinha et al.
have worked to evaluate an integral polynomial discarding
Taylor series [13]. Mahboubi, A. et al. presented an efficient
method for automatically computing and proving bounds on
some definite integrals inside the Coq formal system [14].
Yang WY et al. have explained into detail the Romberg rule,
Richardson’s extrapolation, Gauss quadrature, Euler method,
and so forth [15]. Concepcion Ausin, M. compared various
numerical integration producers and examined about more
advanced numerical integration procedures [16].
What’s more, Chakraborty, S. presented optimal numerical
integration schemes for a family of Polygonal Finite
Elements with Schwarz–Christoffel (SC) conformal mapping
with improved convergence properties [17]. Chapra SC
showed in Applied Numerical Methods with MATLAB to
learn and apply numerical methods to solve problems in
engineering and science [18]. Mettle, F. O. propounded a
numerical integration method that provides improved
estimates as compared to the Newton-Cotes methods of
integration [19]. Smyth, G. K. focused on the process of
approximating a definite integral from values of the integrand
when exact mathematical integration is not available [20].
Furthermore, Marinov, T. et al. used the three basic methods
which are the Midpoint, the Trapezoidal, and Simpson’s rules
[21]. Kwasi A. et al. proposed a numerical integration
method using polynomial interpolation that provides
improved estimates as compared to the Newton-Cotes
methods of integration [22]. Also, Mir Md. Moheuddin et al.
discussed to find out the best method through iterative
methods for solving the nonlinear equation [23]. Md. Jashim
Uddin et al. presented a central difference interpolation
method which is derived from the combination of Gauss’s
third formula, Gauss’s Backward formula and Gauss’s
forward formula [24]. Zheng, C. et al. considered the
Numerical computation of a nonlocal diffusion equation on
the real axis in their paper [25].
In the present work, we discussed the numerical
approximate solutions based on Newton-cotes methods for
solving the definite integral problems. We have analyzed our
numerical results obtained by the three proposed methods
which are compared with the exact values. We have also
graphically represented our discussion which obtained by our
proposed methods and errors as well. Here we find a more
accurate result comparing with exact results. Finally, we
make our decision that Simpson’s 1/3 rule is very close to
analytical solutions.
The paper is organized as follows: in section 2, we
introduce an overview of the numerical evaluation
procedures that are utilized in the next section. Section 3
shows some numerical examples to get accuracy. In section 4
we highlight the discussion of our results. Finally, section 5
includes a conclusion and recommendations of our tasks.
2. Numerical Evaluation Procedures
In this section we utilize three different procedures to solve
numerical integration problems which are shown below:
2.1. Procedure -1
In numerical analysis, the trapezoidal rule or method is an
idea for approximating the definite integral, the average of
the left and right sums as well as usually imparts a better
approximation than either does individually.
=
Also
,
we
know
from
the
well
-
known
Newton
-
Cotes
general
quadrature
formula
that
I = ℎ + ∆ + −
∆
!
+ − +
∆
!
+ − +
!!
− 3
∆
!
+ ⋯ $ (1)
Now, putting n = 1 in the above formula (1) and neglecting
the second and higher difference we get,
%
'(
= h [ +
!
∆ ]
= h [ +
!
! − )]
=
!
ℎ [ + ! ]
Similarly, % y dx
' (
./
=
!
ℎ ! + )
⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
3. Pure and Applied Mathematics Journal 2020; 9(3): 46-54 48
%
' (
' 0! (
=
!
ℎ 0! + )
Adding these all integrals, we get,
%
' (
=
(
[ 2 !++ + ........+ 0!) + ]
This rule is acquainted as the trapezoidal rule.
2.2. Procedure -2
In numerical integration, the Simpson’s 1/3 rule is a
numerical scheme for discovering the integral %
1
2
within
some finite limits a and b. Simpson’s 1/3 rule approximates f
(x) with a polynomial of degree two p (x), i.e a parabola
between the two limits a and b, and then searches the integral
of that bounded parabola which is applied to exhibit the
approximate integral %
1
2
. Besides, Simpson’s one-third
rule is a tract of trapezoidal rule therein the integrand is
approximated through a second-order polynomial.
Also, we
know
from
the
well
-
known
Newton
-
Cotes
general
quadrature
formula
that
I = ℎ + ∆ + −
∆
!
+ − +
∆
!
+ − +
!!
− 3
∆
!
+ ⋯ $ (2)
Now, putting n = 2 in the formula (2) and neglecting the
third and higher difference we get,
%
' (
= h [2 + 2∆ +
4
0
∆ ]
= h [2 + 2 ! − ) +
!
( − 2 !+ )]
=
!
h (y + 4y!+ y )
Similarly, %
' (
' (
=
!
ℎ( + 4 + )
⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
%
' (
' 0 }(
=
!
ℎ( 0 + 4 0!+ )
When n is even.
Adding these all integrals, we obtain,
%
' (
+%
' (
' (
+ ⋯ + % . /
' 0 }(
=
!
ℎ [( + )+ 4( ! + + ⋯+ 0! + 2( +
+ ⋯ + 0 )]
Or, %
' (
=
(
[(y0 + yn) + 4(y1 + y3 +..... + yn-1)+2(y2 +
y4 +... + yn-2)].
This formula is known as Simpson’s one-third rule. If the
number of sub-divisions of the interval is even then this
method is only applied.
2.3. Procedure -3
Simpson’s three-eight rule is a process for approximating a
definite integral by evaluating the integrand at finitely many
points and based upon a cubic interpolation rather than a
quadratic interpolation. The difference is Simpson’s 3/8
method applies a third-degree polynomial (cubic) to calculate
the curve.
Further, we
know
from
the
well
-
known
Newton
-
Cotes
general
quadrature
formula
that
I = ℎ + ∆ + −
∆
!
+ − +
∆
!
+ − +
!!
− 3
∆
!
+ ⋯ $ (3)
Putting n = 3 in the formula (3) and neglecting all
differences above the third, we get,
%
' (
= ℎ [3 +
8
∆ +(
9
−
8
)
∆
!
+(
:!
− 27 +
9
∆
!
]
= h [3 +
8
! − +
8
( − 2 !+ )+
:
( −
3 +3 !0 ]
%
' (
=
:
ℎ( + 3 !+3 + )
Similarly, %
'=(
' (
=
:
ℎ( + 3 +3 + =)
⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
%
' (
' 0 (
=
:
ℎ( 0 + 3 0 +3 0! + )
Adding these all integrals, we get,
%
' (
+%
'=(
' (
+ ⋯ + %
' (
' 0 (
=
:
h [(yo
+ yn) + 3(y1 + y2 + y4 + y5 +.... + yn-1) + 2(y3 + y6 +... + yn-3)]
This formula is known as Simpson’s three-eighths rule.
3. Numerical Examples
In this section, we take into account three numerical
examples in order to verify the accuracy of our propounded
methods. With the help of this method, we evaluate the
desired numerical approximations as well as errors to
demonstrate which numerical methods converge better to
exact value. we have given a more robust analysis of our
proposed methods as well. All of the calculations are
achieved by MATLAB R2013a software. Numerical
solutions and errors are calculated and the outcomes are
introduced graphically.
Example-1: we consider the numerical integration problem
is % sin − log + D
!.
.
. The numerical
approximations and errors using three proposed methods are
represented in tables 1-2 and the graphs of the
approximations solutions, as well as errors, are exhibited in
8. 53 Mir Md. Moheuddin et al.: A New Analysis of Approximate Solutions for Numerical Integration
Problems with Quadrature-based Methods
4. Discussion of Results
Tables 1, 3, and 5 present a comparison between
approximate results and exact solution with three proposed
methods and tables 2, 4, and 6 impart the numerical errors.
Graphically representations for each example are shown in
figures 1-2, figures 3-4, figures 5-6 where even subdivision
like 2, 4, 6, 8, 10 of Simpson’s 1/3 method display more
correct solution and fewer errors as well. From the above
tables and graphs, we can figure out that the numerical solution
converges at a quicker rate only for Simpson’s 1/3 method
when the condition of the subdivision is even. On the other
hand, approximate solution in case of other methods tends to
converge slowly. Among the three proposed rules, Simpson’s
1/3 rule delivers very good accuracy as well as less error for
each example when compared to the exact solution. Finally,
we observe that Simpson’s 1/3 method’s solution is more
accurate and appropriate than the other mentioned method’s
solution and Simpson’s 1/3 method is the most effective
method for solving definite integral problems.
5. Conclusion and Recommendatons
5.1. Conclusion
In this article, Trapezoidal, Simpson’s 1/3, Simpson’s 3/8
methods are used for solving integral problems numerically. To
gain the desired accuracy level of numerical solutions we have
taken some subdivisions sequentially where even subdivisions
for Simpson’s 1/3 method give good agreement in comparison
with the exact values. Comparing the approximate results and
errors of each problem, Trapezoidal, Simpson’s 3/8 methods
became less accurate because of their very slow convergence.
Besides, Simpson’s 1/3 methods solution is more convergent,
even it is faster when compared to the other proposed methods.
Lastly, it was found usually that Simpson’s 1/3 method is the
best useful and reliable rule in finding the approximate solutions
for various integral problems.
5.2. Recommendations
In future research, we have the plan to develop these
methods for analyzing and forecasting value among the
competitors in economics, business, network systems, etc.
Also, we plan to study more with numerical methods and
updating our solutions with lots of analyzing factor.
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