This document proposes a modified Newton's method for solving nonlinear equations that uses harmonic mean. It begins by reviewing Newton's method and some existing variants that use arithmetic mean or other integration rules to modify Newton's method and achieve cubic convergence without using second derivatives. It then presents the new Harmonic-Simpson-Newton method, which replaces the arithmetic mean in an existing Simpson Newton's method with harmonic mean. The method is proven to have cubic convergence. Numerical examples are provided to compare the efficiency of the new method to other cubic convergent methods.
New Two-Step Method with Fifth-Order Convergence for Solving Nonlinear Equationsinventionjournals
Β
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 Engineering Research and Applications (IJERA) aims to cover the latest outstanding developments in the field of all Engineering Technologies & science.
International Journal of Engineering Research and Applications (IJERA) is a team of researchers not publication services or private publications running the journals for monetary benefits, we are association of scientists and academia who focus only on supporting authors who want to publish their work. The articles published in our journal can be accessed online, all the articles will be archived for real time access.
Our journal system primarily aims to bring out the research talent and the works done by sciaentists, academia, engineers, practitioners, scholars, post graduate students of engineering and science. This journal aims to cover the scientific research in a broader sense and not publishing a niche area of research facilitating researchers from various verticals to publish their papers. It is also aimed to provide a platform for the researchers to publish in a shorter of time, enabling them to continue further All articles published are freely available to scientific researchers in the Government agencies,educators and the general public. We are taking serious efforts to promote our journal across the globe in various ways, we are sure that our journal will act as a scientific platform for all researchers to publish their works online.
New Two-Step Method with Fifth-Order Convergence for Solving Nonlinear Equationsinventionjournals
Β
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 Engineering Research and Applications (IJERA) aims to cover the latest outstanding developments in the field of all Engineering Technologies & science.
International Journal of Engineering Research and Applications (IJERA) is a team of researchers not publication services or private publications running the journals for monetary benefits, we are association of scientists and academia who focus only on supporting authors who want to publish their work. The articles published in our journal can be accessed online, all the articles will be archived for real time access.
Our journal system primarily aims to bring out the research talent and the works done by sciaentists, academia, engineers, practitioners, scholars, post graduate students of engineering and science. This journal aims to cover the scientific research in a broader sense and not publishing a niche area of research facilitating researchers from various verticals to publish their papers. It is also aimed to provide a platform for the researchers to publish in a shorter of time, enabling them to continue further All articles published are freely available to scientific researchers in the Government agencies,educators and the general public. We are taking serious efforts to promote our journal across the globe in various ways, we are sure that our journal will act as a scientific platform for all researchers to publish their works online.
We disclose a simple and straightforward method of solving ordinary or linear partial differential equations of any order and apply it to solve the generalized Euler-Tricomi equation. The method is easier than classical methods and also didactic.
Date: Jan, 10, 202
Stereographic Circular Normal Moment Distributionmathsjournal
Β
Minh et al (2003) and Toshihiro Abe et al (2010) proposed a new method to derive circular distributions from the existing linear models by applying Inverse stereographic projection or equivalently bilinear transformation. In this paper, a new circular model, we call it as stereographic circular normal moment distribution, is derived by inducing modified inverse stereographic projection on normal moment distribution (Akin Olosunde et al (2008)) on real line. This distribution generalizes stereographic circular normal distribution (Toshihiro Abe et al (2010)), the density and distribution functions of proposed model admit closed form. We provide explicit expressions for trigonometric moments.
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
This Chapter is part of previous published ch.1 and ch.3 and its use for undergraduate students in physics department. also, you can use it for mathematical and Statistical courses and for those experimental courses of data fitting.
Elzaki transform homotopy perturbation method for solving porous medium equat...eSAT Journals
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Abstract In this paper, the ELzaki transform homotopy perturbation method (ETHPM) has been successfully applied to obtain the approximate analytical solution of the nonlinear homogeneous and non-homogeneous gas dynamics equations. The proposed method is an elegant combination of the new integral transform βELzaki Transformβ and the homotopy perturbation method. The method is really capable of reducing the size of the computational work besides being effective and convenient for solving nonlinear equations. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. A clear advantage of this technique over the decomposition method is that no calculation of Adomianβs polynomials is needed. Keywords: ELzaki transform, Homotopy perturbation method, non linear partial differential equation, and nonlinear gas dynamics equation
Adomian decomposition Method and Differential Transform Method to solve the H...IJERA Editor
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In this paper, we consider the AdomianDecomposition Method (ADM) and the Differential Transform Method (DTM) for finding approximate and exact solution of the heat equation with a power nonlinearity. Moreover, the reliability and performance of ADM and DTM. Numerical results show that these methods are powerful tools for solving heat equation with a power nonlinearity.
Fortran is a general-purpose programming language, mainly intended for mathematical computations in
science applications
this chapter is the third chapter
A Fifth-Order Iterative Method for Solving Nonlinear Equationsinventionjournals
Β
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
The influence of Rolling process on the porosity and wear behavior of Spray-f...IOSR Journals
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Spray forming, also called spray casting or spray deposition, is the inert gas atomization of a liquid metal stream into variously sized droplets. These droplets are interrupted by a substrate which collects and solidifies the droplets into coherent, near fully dense perform. The present work concerns with this techniques. This technology was applied to produce Al-6%Si-20%Pb alloys. The spray droplets were deposited over a copper substrate to achieve a disc shape perform. After spray deposition samples were rolled at room temperature on two high rolling mills and total porosity and wear characteristic of spray deposits were studies. The total porosity decreases with the increase in the percentage of reduction in thickness of the samples. Thickness of the samples in both middle and peripheral region. Wear testing of spray formed and rolled Al-Si-Pb alloy were investigated on a pin on disc type wear machine. Wear rate behavior with applied load and increase in percentage of reduction in thickness of samples was observed. Wear rate increased with increasing the load and decreased with increase in percentage of reduction in thickness of the sample.
Effect of Antioxidant status on liver following Atrazine exposure and its pro...IOSR Journals
Β
The efficacy of Andrographis paniculata (AP) extract was studied on atrazine induced hepatic damage in rats. Ethanolic extract of AP (150mg/kg body weight) was found to protect the male wistar rats from hepato toxic action of atrazine as evidence by significant reduction in the level of lipid peroxidation and increased the antioxidant defense system activity in the atrazine intoxicated rats. However, AP treatment ameliorated the effects of atrazine suggesting it as potential antioxidant against atrazine induced oxidative stress.
Preliminary Phytochemical Screening and Antibacterial Activity on Stem Bark E...IOSR Journals
Β
Ceiba Petandra is widely used in complementary medicine. Its stem bark extracts obtained via soxhlet extraction using n-hexane, acetone and ethanol gare alkaloids, Steroids, Saponins, Cardiac glycosides, triterpenes and tannins when screened phytochemically. These extracts were further investigated for their anti-bacterial activity against staphylococcus aureus klebsiella, pneumoniae and pseudomonas aeruginosa with ampicillina, as standard. Highest anti- bacterial activity with acetone extract at a concentration of 300mg/ml and least activity with ethanol extract at a concentration of 100mg/ml. n- Hexane extract showed no inhibition on the tested micro-organisms.
We disclose a simple and straightforward method of solving ordinary or linear partial differential equations of any order and apply it to solve the generalized Euler-Tricomi equation. The method is easier than classical methods and also didactic.
Date: Jan, 10, 202
Stereographic Circular Normal Moment Distributionmathsjournal
Β
Minh et al (2003) and Toshihiro Abe et al (2010) proposed a new method to derive circular distributions from the existing linear models by applying Inverse stereographic projection or equivalently bilinear transformation. In this paper, a new circular model, we call it as stereographic circular normal moment distribution, is derived by inducing modified inverse stereographic projection on normal moment distribution (Akin Olosunde et al (2008)) on real line. This distribution generalizes stereographic circular normal distribution (Toshihiro Abe et al (2010)), the density and distribution functions of proposed model admit closed form. We provide explicit expressions for trigonometric moments.
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
This Chapter is part of previous published ch.1 and ch.3 and its use for undergraduate students in physics department. also, you can use it for mathematical and Statistical courses and for those experimental courses of data fitting.
Elzaki transform homotopy perturbation method for solving porous medium equat...eSAT Journals
Β
Abstract In this paper, the ELzaki transform homotopy perturbation method (ETHPM) has been successfully applied to obtain the approximate analytical solution of the nonlinear homogeneous and non-homogeneous gas dynamics equations. The proposed method is an elegant combination of the new integral transform βELzaki Transformβ and the homotopy perturbation method. The method is really capable of reducing the size of the computational work besides being effective and convenient for solving nonlinear equations. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. A clear advantage of this technique over the decomposition method is that no calculation of Adomianβs polynomials is needed. Keywords: ELzaki transform, Homotopy perturbation method, non linear partial differential equation, and nonlinear gas dynamics equation
Adomian decomposition Method and Differential Transform Method to solve the H...IJERA Editor
Β
In this paper, we consider the AdomianDecomposition Method (ADM) and the Differential Transform Method (DTM) for finding approximate and exact solution of the heat equation with a power nonlinearity. Moreover, the reliability and performance of ADM and DTM. Numerical results show that these methods are powerful tools for solving heat equation with a power nonlinearity.
Fortran is a general-purpose programming language, mainly intended for mathematical computations in
science applications
this chapter is the third chapter
A Fifth-Order Iterative Method for Solving Nonlinear Equationsinventionjournals
Β
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
The influence of Rolling process on the porosity and wear behavior of Spray-f...IOSR Journals
Β
Spray forming, also called spray casting or spray deposition, is the inert gas atomization of a liquid metal stream into variously sized droplets. These droplets are interrupted by a substrate which collects and solidifies the droplets into coherent, near fully dense perform. The present work concerns with this techniques. This technology was applied to produce Al-6%Si-20%Pb alloys. The spray droplets were deposited over a copper substrate to achieve a disc shape perform. After spray deposition samples were rolled at room temperature on two high rolling mills and total porosity and wear characteristic of spray deposits were studies. The total porosity decreases with the increase in the percentage of reduction in thickness of the samples. Thickness of the samples in both middle and peripheral region. Wear testing of spray formed and rolled Al-Si-Pb alloy were investigated on a pin on disc type wear machine. Wear rate behavior with applied load and increase in percentage of reduction in thickness of samples was observed. Wear rate increased with increasing the load and decreased with increase in percentage of reduction in thickness of the sample.
Effect of Antioxidant status on liver following Atrazine exposure and its pro...IOSR Journals
Β
The efficacy of Andrographis paniculata (AP) extract was studied on atrazine induced hepatic damage in rats. Ethanolic extract of AP (150mg/kg body weight) was found to protect the male wistar rats from hepato toxic action of atrazine as evidence by significant reduction in the level of lipid peroxidation and increased the antioxidant defense system activity in the atrazine intoxicated rats. However, AP treatment ameliorated the effects of atrazine suggesting it as potential antioxidant against atrazine induced oxidative stress.
Preliminary Phytochemical Screening and Antibacterial Activity on Stem Bark E...IOSR Journals
Β
Ceiba Petandra is widely used in complementary medicine. Its stem bark extracts obtained via soxhlet extraction using n-hexane, acetone and ethanol gare alkaloids, Steroids, Saponins, Cardiac glycosides, triterpenes and tannins when screened phytochemically. These extracts were further investigated for their anti-bacterial activity against staphylococcus aureus klebsiella, pneumoniae and pseudomonas aeruginosa with ampicillina, as standard. Highest anti- bacterial activity with acetone extract at a concentration of 300mg/ml and least activity with ethanol extract at a concentration of 100mg/ml. n- Hexane extract showed no inhibition on the tested micro-organisms.
An Experimental Study on Strength Properties of Concrete When Cement Is Parti...IOSR Journals
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Ordinary Portland cement is recognized as major construction material throughout the world.
Cement is responsible for about 5% - 8% of global co2 emissions. Therefore the utilization of industrial waste
has been increased which is environment friendly. In this paper an attempt has been made to utilize sugarcane
bagasse ash (S.C.B.A) a residue from sugar industry as a supplementary cement replacement material.
The present study investigates with the strength and durability properties of the concrete when the
cement is replaced with S.C.B.A in different proportions. In the first stage the S.C.B.A (B.A.1) is partially
replaced in the percentages of 0 to 40% in increasing steps of 5%. Further in the second stage the S.C.B.A
(B.A.2) is heated up to 8500
c for about 8 hours in a muffle furnace and is replaced in the percentages of 0 to
40% in increasing steps of 5%. Cubes, Cylinders and Beams are casted and tests are conducted and
compressive strength, split tensile strength and flexural strength obtained for both the conditions.
Improving Sales in SME Using Internet MarketingIOSR Journals
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Abstract : In Indonesia, SMEs are the backbone of the Indonesian economy. Number of SMEs until 2011 to
reach around 52 million. SMEs in Indonesia is very important for the economy because it accounts for 60% of
GDP and 97% of the workforce holds. But access is limited to financial institutions only 25% or 13 million
SMEs who have access to financial institutions. Indonesian government, SMEs, through the Department of
Cooperatives and SMEs, in each province or regency / city.
Although Small and Medium Enterprises (SMEs) is driving the nation's economy, but in reality many of
the problems SMEs are still entangled. The main thing to note is the ability of SMEs to access a wider market.
Because of the ability to change and adapt to a changing environment will determine the existence of small
businesses in the nation's economy. In the end, the existence of small businesses that have high competitiveness
will strengthen the nation's economy as a whole. Thus, in this study will use an appropriate technology tools
that can provide assistance in introducing products through internet and increase sales in each SME
This study uses a sample of students at the State University of Malang that can make a significant
contribution in the small and medium businesses that are being initiated by students.
Keywords: Small Medium Enterprise, Internet Marketing, Sales Improvement
Vibration Analysis of Cracked Rotor Using Numerical ApproachIOSR Journals
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In general rotating machines have wide applications in systems, plants, vehicles, and industries. Every rotating machine uses shaft as power transforming unit. It is very dangerous to operate the machine with the presence of crack in the shaft. The growth of the crack is dangerous to operate and may lead to catastrophic failure. It is to be detected at earlier stages. In this paper relation between vibration amplitude and on the crack depth was developed, this helps in determine the depth of the crack by measuring the vibration amplitudes. To develop the relation equation strain energy density function was used. By observing the generated curves amplitude of vibration increases with respect to the depth of the crack due to reduction in stiffness of the shaft.
An Efficient Approach for Requirement Traceability Integrated With Software R...IOSR Journals
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Abstract: Traceability links between requirements of a system and its source code are helpful in reducing
system conception effort. During software updates and maintenance, the traceability links become invalid since
the developers may modify or remove some features of the source code. Hence, to acquire trustable links from a
system source code, a supervised link tracing approach is proposed here. In proposed approach, IR techniques
are applied on source code and requirements document to generate baseline traceability links. Concurrently,
software repositories are also mined to generate validating traceability links i.e. Histrace links which are then
called as experts. Now a trust model named as DynWing is used to rank the different types of experts. DynWing
dynamically assigns weights to different types of experts in ranking process. The top ranked experts are then fed
to the trust model named as Trumo. Trumo validates the baseline links with top ranked experts and finds the
trustable links from baseline links set. While validating the links, Trumo is capable of discarding or re-ranking
the experts and finds most traceable links. The proposed approach is able to improve the precision and recall
values of the traceability links.
Index Terms: Traceability, requirements, features, source code, repositories, experts.
Asifβs Equation of Charge Variation and Special RelativityIOSR Journals
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The theory of special relativity plays an important role in the modern theory of classical electromagnetism. Considering deeply the effect of Special relativity in Electromagnetism, when a charge particle moves with high speed as comparable to the speed of light in vacuum tube or in space under influence of electromagnetic field, its mass varies under Lorentz transformation [1].The question arises that does its charge vary under Lorentz transformation? In this paper, Asif's equation of charge variation demonstrates the variation of electric charge under Lorentz transformation. The more sophisticated view of electromagnetism expressed by electromagnetic fields in moving inertial frame can be achieved by considering some relativistic effect including charge as well. One can easily achieve the mass-energy relation from Asifβs equation of charge variation as proved in this paper.
Cataloging Of Sessions in Genuine Traffic by Packet Size Distribution and Ses...IOSR Journals
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Abstract: Cataloging traffic keen on precise network applications is vital for application-aware network
organization and it turn into more taxing because modern applications incomprehensible their network
behaviors. Whereas port number-based classifiers work merely for a little renowned application and signaturebased
classifiers are not significant to encrypted packet payloads, researchers are inclined to classify network
traffic rooted in behaviors scrutinized in network applications. In this document, a session level Flood
Cataloging (SLFC) approach is proposed to organize network Floods as a session, which encompasses of
Floods in the equal discussion. SLFC initially classifies flood into the analogous applications by packet size
distribution (PSD) and subsequently faction Floods as sessions by port locality. With PSD, each Flood is
distorted into a set of points in a two-Dimension space and the remoteness among all Flood and the
representatives of preselected applications are calculated. The Flood is predicted as the application having a
least distance. Meanwhile, port locality is accustomed to cluster Floods as sessions since an application often
uses successive port statistics surrounded by a session. If flood of a session are categorized into diverse
applications, an arbitration algorithm is invoked to make the improvement.
Keywords: Flood Cataloging; session grouping; session Cataloging; packet size distribution
Analytical Determination of Antihestamine drugs Pure and its pharmaceutical P...IOSR Journals
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A quick and convenient method has been developed for the micro estimation of antihistamine drugs. 1-5 mg of sample is allowed to react with 10 ml of 0.02 M, N-bromosaccharin solution. Unconsumed reagent can be accurately titrated with 5 ml of 15% KI and 0.02N sodium thiosulphate solution using starch as indica-tor. SD and CV was calculated for reproducible and accurate result. The accuracy of the method is within + 1 % and possible course of reaction was suggested on the basis of stoichiometry as well as find reaction product.
Effective Leadership-Employee Retention-Work Life Balance: A Cyclical ContinuumIOSR Journals
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It is an interactive paper which has been made to explain the association between leadership and its impact on employeeβs work-life and employee retention. Both the expressions have gained a lot of magnitude in these dayβs Organizations. Leadership has been taken up as a foundation in the study as it builds up the essence of our Organizations; Work-life balance has also been addressed as a basic issue for the employees as well the employers. The study undertaken tries to explain in a theoretical and a very interactive way how effective leadership can help balancing the work-life of employees and consecutively leads to help the Organization identify its quality employees and tap their efficiencies for a long-term through employee retention
Abstract: The increasing amount of applications using digital multimedia technologies has accentuated the need to provide copyright protection to multimedia data. This paper reviews one of the data hiding techniques - digital image watermarking. Through this paper we will explore some basic concepts of digital image watermarking techniques.Two different methods of digital image watermarking namely spatial domain watermarking and transform domain watermarking are briefly discussed in this paper. Furthermore, two different algorithms for a digital image watermarking have also been discussed. Also the comparision between the different algorithms,tests performed for the robustness and the applications of the digital image watermarking have also been discussed.
Analysis Of Lpg Cylinder Using Composite MaterialsIOSR Journals
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This paper aims is innovation of alternative materials of Liquid petroleum gas (LPG). So, the finite
element analysis of Liquefied Petroleum Gas (LPG) cylinders made of Steel and Fiber Reinforced Plastic (FRP)
composites has been carried out. Finite element analysis of composite cylinder subjected to internal pressure is
performed. Layered shell element of a versatile FE analysis package ANSYS (version 11.0) has been used to
model the shell with FRP composites.
A number of cases are considered to study the stresses and deformations due to pressure loading inside the
cylinder. First, the results of stresses and deformation for steel cylinders are compared with the analytical
solution available in literature in order to validate the model and the software. The weight savings are also
presented for steel, Glass Fiber Reinforced Plastic (GFRP) composites LPG cylinders. Variations of stresses
and deformations throughout the cylinder made of steel and GFRP are studied.
Low Power FPGA Based Elliptical Curve CryptographyIOSR Journals
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Abstract: Cryptography is the study of techniques for ensuring the secrecy and authentication of the information. The development of public-key cryptography is the greatest and perhaps the only true revolution in the entire history of cryptography. Elliptic Curve Cryptography is one of the public-key cryptosystem showing up in standardization efforts, including the IEEE P1363 Standard. The principal attraction of elliptic curve cryptography compared to RSA is that it offers equal security for a smaller key-size, thereby reducing the processing overhead. As a Public-Key Cryptosystem, ECC has many advantages such as fast speed, high security and short key. It is suitable for the hardware of implementation, so ECC has been more and more focused in recent years. The hardware implementation of ECC on FPGA uses the arithmetic unit that has small area, small storage unit and fast speed, and it is an extremely suitable system which has limited computation ability and storage space.[1][2] The modular arithmetic division operations are carried out using conditional successive subtractions, thereby reducing the area. The system is implemented on Vertex-Pro XCV1000 FPGA. Index Terms β VHDL, FSM, FPGA, Elliptic Curve Cryptography.
The Comparison of theMaterials in Styles of Iranian Architecture and its Effe...IOSR Journals
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During the history,different elements have been determined as influential factors affecting the
architecture in different areas. The historical events and political alterations as well as religious and
economical changes can directly lead to the architecture style. One of the historical countries with rich
architectural history that can be increasingly exposed to more alterations is Iran. In general the architecture
styles in Iran can be categorised in six groups. These groups can be divided in two periods before and after
Islamβs emergence in Iran. The βParsiβ and βPartiβ architecture styles belong to the former period and
βRaziβ, βKhorasaniβ, βIsfahaniβ and βAzariβ were common in the latter period, after Islam. Such alterations
brought in a variety of architecture styles, in this country, due to theoretical alterations. Furthermore, some
novel architectural styles were resulted from a number of physical conditions which had also effects on the
theoretical architecture. The current research intends to put an emphasize on the alterations in materials used
in two historical periods of Iran, βAchaemenidβ Empire (550β330 BCE) and The βSassanidβ Empire (224 CE
to 651 CE) resulting in the changes in Iranian architecture. It also aims to explore the differences and the
reasons of changes in the materials used for constructions and the influence that these changes had on the
architectural style in the above mentioned periods
Uranium Concentrations measurement for Ground Water and Soil Samples in Al-Na...IOSR Journals
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In this work, Uranium concentrations in groundwater and soil samples from some areas of Al-Najaf city, Iraq were measured solid state nuclear track detector (CR-39) detector technique. The uranium concentration in groundwater samples was found to vary from (1.617782869Β±0.04342) ΞΌglβ1 to (5.079235843Β±0.137827) ΞΌglβ1, while in soil samples vary from (0.093558Β±0.000938) ppm to (0.184325Β±0.017511) ppm.
This study showed that levels of uranium concentration in groundwater and soil of study area of were compared with safe limit values recommended by WHO and UNSCEAR below the safe limit
bT-Locally Closed Sets and bT-Locally Continuous Functions In Supra Topologic...IOSR Journals
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The aim of this paper is to introduce a decompositions namely supra bT- locally closed sets and define supra bT-locally continuous functions. This paper also discussed some of their properties.
Enhancing Security of Multimodal Biometric Authentication System by Implement...IOSR Journals
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Abstract : Conventional personal identification techniques for instance passwords, tokens, ID card and PIN
codes are prone to theft or forgery and thus biometrics isa solution thereto. Biometrics is the way of recognizing
and scrutinizing the physical traits of a person. Automated biometrics verification caters as a conducive and
legitimate method, but there must be an assurance to its cogency. Furthermore, in most of the cases unimodal
biometric recognition is not able to meet the performance requirements of the applications. According to recent
trends, recognition based on multimodal biometrics is emerging at a greater pace. Multimodal biometrics
unifies two or more biometric traits and thus the issues that emerge in unimodal recognition can be mitigated in
multimodal biometric systems. But with the rapid ontogenesis of information technology, even the biometric
data is not secure. Digital watermarking is one such technique that is implemented to secure the biometric data
from inadvertent or premeditated attacks.This paper propounds an approach that is projected in both the
directions of improving the performance of biometric identification system by going multimodal and, increasing
the security through watermarking. The biometric traits are initially transformed using Discrete Wavelet and
Discrete Cosine Transformation and then watermarked using Singular Value Decomposition. Scheme depiction
and presented outcomes justifies the effectiveness of the scheme.
Keywords: Discrete Cosine Transform (DCT), Discrete Wavelet Transform (DWT), Multimodal biometrics,
Singular Value Decomposition, Watermarking
βComparitive Study of Prevalence of Hyperlactatemia in HIV / AIDS Patients re...IOSR Journals
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Hyperlactatemia is one of the important metabolic abnormalities in HIV infected patients. The
prevalence of hyperlactatemia in natural course of HIV disease is approximately about 2%. Aim of this study is
to estimate the prevalence of hyperlactatemia in HIV patients receiving two antiretroviral regimens, advocated
by NACO by monitoring the plasma lactate levels. This study was taken up with 200 patients to compare the
prevalence of hyperlactatemia of two commonly used NACO regimens (zidovudine+ lamivudine+ nevirapine)
Vs (stavudine+ lamivudine+ nevirapine). The plasma lactate levels were estimated between 9th to 18thmonth
after initiation of antiretroviral therapy. The comparision and correlation between plasma lactate levels, CD4
counts and haemoglobin percentage in both regimens was done. There was statistically significant rise in the
plasma lactate levels (p<0.05) in both regimens. The increase in plasma lactate levels is more in stavudine
group compared to zidovudine group. There was low degree of positive correlation between plasma lactate and
haemoglobin in Stavudine group but negative correlation between Plasma lactate and CD4 counts in both
groups. More focus is needed on Pharmacovigilance of NRTIs induced hyperlactatemia especially Stavudine.
Dust Interception Capacity And Alteration Of Various Biometric And Biochemica...IOSR Journals
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The dust accumulation capacity of Ficus carica L. was evaluated from eight different sites in and around Multan. The impact of dust accumulation was observed via various biometric attributes (leaf area, leaf fresh and dry weights) and biochemical attributes (chlorophyll contents, carotenoids & ascorbic acid) from leaves of F. carica. The maximum dust accumulation was occurred in the plants growing at Road sides while, the minimum dust was found on plants growing at Bahauddin Zakariya University. Dust accumulation has caused a significant effect on almost all foliage and biochemical attributes of F. carica. A positive correlation was found between dust accumulation and biometric attributes in F. carica. Biochemical responses had shown an inconsistency as chlorophylls (a, b & total), carotenoids decreased and ascorbic acid contents increased with an increase in dust accumulation. A negative correlation was found between dust deposition and chlorophyll contents. Whereas, accumulation of ascorbic acid was associated with a decline in pigment contents
International Journal of Computational Engineering Research(IJCER)ijceronline
Β
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.
ABSTRACT: In this paper, we construct new classes of derivative-free of tenth-order iterative methods for solving nonlinear equations. The new methods of tenth-order convergence derived by combining of theSteffensen's method, the Kung and Traubβs of optimal fourth-order and the Al-Subaihi's method. Several examples to compare of other existing methods and the results of new iterative methods are given the encouraging results and have definite practical utility.
Overviewing the techniques of Numerical Integration.pdfArijitDhali
Β
In this presentation we discuss the ways of integrating a function using trapezoidal, simpsons 1/3,3/8 and boole's method. We also discussed its error and analysised other ways to solve it than mention.
Derivation and Application of Six-Point Linear Multistep Numerical Method for...IOSR Journals
Β
A six-step Continuous Block method of order (5, 5, 5, 5, 5, 5) T is proposed for direct solution of the second (2nd) order initial value problems. The main method and additional ones are obtained from the same continuous interpolant derived through interpolation and collocation procedures. The methods are derived by interpolating the continuous interpolant at π₯ = π₯π+π , π = 6 and collocating the first and second derivative of the
continuous interpolant at π₯π+π , π = 0 and π = 2, 3, β¦ 5 respectively. The stability properties of the methods are discussed and the stability region shown. The methods are then applied in block form as simultaneous numerical integrators. Two numerical experiments are given to illustrate the efficiency of the new methods.
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A NEW STUDY TO FIND OUT THE BEST COMPUTATIONAL METHOD FOR SOLVING THE NONLINE...mathsjournal
Β
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
Β
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
Β
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.
Elzaki transform homotopy perturbation method for solving porous medium equat...eSAT Journals
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Abstract In this paper, we apply a new method called ELzaki transform homotopy perturbation method (ETHPM) to solve porous medium equation. This method is a combination of the new integral transform βELzaki transformβ and the homotopy perturbation method. The nonlinear term can be easily handled by homotopy perturbation method. The porous medium equations have importance in engineering and sciences and constitute a good model for many systems in various fields. Some cases of the porous medium equation are solved as examples to illustrate ability and reliability of mixture of ELzaki transform and homotopy perturbation method. The results reveal that the combination of ELzaki transform and homotopy perturbation method is quite capable, practically well appropriate for use in such problems and can be applied to other nonlinear problems. This method is seen as a better alternative method to some existing techniques for such realistic problems. Key words: ELzaki transform, Homotopy perturbation method, non linear partial differential equation, and porous medium equation
We give an elementary exposition of a method to obtain the infinitesimal point symmetries of Lagrangians.Besides, we exhibit the Lanczos approach to Noetherβs theorem to construct the first integral associated with each symmetry.
MSC 2010:49S05, 58E30, 70H25, 70H33
Numerical solution of fuzzy differential equations by Milneβs predictor-corre...mathsjournal
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The study of this paper suggests on dependency problem in fuzzy computational method by using the numerical solution of Fuzzy differential equations(FDEs) in Milneβs predictor-corrector method. This method is adopted to solve the dependency problem in fuzzy computation. We solve some fuzzy initial value problems to illustrate the theory.
Comparative analysis of x^3+y^3=z^3 and x^2+y^2=z^2 in the Interconnected Sets Vladimir Godovalov
Β
This paper introduces an innovative technique of study z^3-x^3=y^3 on the subject of its insolvability in integers. Technique starts from building the interconnected, third degree sets: A3={a_nβa_n=n^3,nβN}, B3={b_nβb_n=a_(n+1)-a_n }, C3={c_nβc_n=b_(n+1)-b_n } and P3={6} wherefrom we get a_n and b_n expressed as figurate polynomials of third degree, a new finding in mathematics. This approach and the results allow us to investigate equation z^3-x^3=y in these interconnected sets A3 and B3, where z^3β§x^3βA3, yβB3. Further, in conjunction with the new Method of Ratio Comparison of Summands and Pascalβs rule, we finally prove inability of y=y^3. After we test the technique, applying the same approach to z^2-x^2=y where we get family of primitive z^2-x^2=y^2 as well as introduce conception of the basic primitiveness of z^'2-x^'2=y^2 for z^'-x^'=1 and the dependant primitiveness of z^'2-x^'2=y^2 for co-prime x,y,z and z^'-x^'>1.
The finite difference method can be considered as a direct discretization of differential equations but in finite element methods, we generate difference equations by using approximate methods with piecewise polynomial solution. In this paper, we use the Galerkin method to obtain the approximate solution of a boundary value problem. The convergence analysis of these solution are also considered.
1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 7, Issue 4 (Jul. - Aug. 2013), PP 93-97
www.iosrjournals.org
www.iosrjournals.org 93 | Page
Modified Newton's method using harmonic mean for solving
nonlinear equations
J. Jayakumarand Kalyanasundaram M.
Department of Mathematics, Pondicherry Engineering College, Pondicherry 605 014, India.
Abstract: Different modifications in the Newtonβs method with cubic convergence have become popular
iterative methods to find the roots of non-linear equations. In this paper, a modified Newtonβs method for
solving a single nonlinear equation is proposed. This method uses harmonic mean while using Simpsonβs
integration rule, thus replacing π β²
π₯ in the classical Newtonβs method. The convergence of the proposed
method is found to be order three. Numerical examples are provided to compare e the efficiency of the method
with few other cubic convergent methods.
Keywords: Non-linear equation, Iterative Methods, Newtonβs Method, Third order convergence.
I. Introduction
A frequently occurringand most important problem of applied mathematicsis to find a root of the
equation
π π₯ = 0 (1)
wheref βΆ I β R β R for an open interval I is a scalar function.
Newtonβs method is one of the famous iterative methods to solve equation (1). It is well known that it has
quadratic convergence. The iterative formula of Newtonβs method is given by
π₯ π+1
β
= π₯ π β
π π₯ π
πβ² π₯ π
, π = 0, 1 ,2 β¦ (2)
By improving Newtonβs method, Chebyshev [1] derived a cubic convergent iterative method which requires
computing of second derivative of the function (1). A prolific advancement in this area has happened in the last
decade where lots of modifications in Newtonβs method are proposed without using second derivative. First
among this is arithmetic mean Newtonβs method [2] derived using the trapezoid rule. Hasanov et al [3] modified
Newtonβs method with a third order convergent method by using Simpsonβs rule. Nedzhibov [4] gave several
classes of iterative methods using different quadrature rules. Frontini et al [5] extended the results of [2] and got
methods of order three independent of the integration formula used. Ozban [6] derived some new variants of
Newtonβs method by using Harmonic mean instead of arithmetic mean and the midpoint integration instead of
the trapezoidal formula. Using Newtonβs theorem for the inverse function, Homeier [7] derived a new class of
Newton-type methods. Babajee et al [8] listed eight different variants of Newtonβs method and the relationship
between them. They analyzed the properties of arithmetic mean Newtonβs method and explain the third order
accuracy via the Tailorβs expansion. Kou Jisheng [9] gave a new variant of Newtonβs method from the
Harmonic mean and midpoint Newtonβs method [6]. Tibor et al [10] suggested geometric mean Newtonβs
method for simple and multiple roots and showed this method is of order three for simple roots and order one
for multiple roots. Ababneh [11] proposed a modified Newtonβs method based on contra harmonic mean. The
advantage of all the above methods is that they use only the first derivative of the function and has cubic
convergence.
In this paper,we propose amodification in the method of[3] including Harmonic mean instead of
Arithmetic meanwhile using the Simpsonβs 1/3 rule of integration for the equation (3). The proposed new
method has the advantage of evaluating only the firstderivativeand less number of iterations to achieve third
order accuracy.In Section II, we present some definitions related to our study. In Section III, some known
variants of Newtonβs method are discussed. Section IV presents the new method and its analysis of
convergence. Finally, Section V gives numerical results and a discussion is carried out in section VI.
II. Definitions
Definition 2.1 [2]: Let πΌ β π ,π₯ π β π , n = 0, 1, 2, οΌ. Then the sequence {π₯ π }is said to converge to πΌ
if
πππ
πββ
π₯ π β πΌ = 0
If, in addition, there exist a constantπ β₯ 0, an integer π0 β₯ 0 and p β₯ 0 such that for all π > 0,
π₯ π+1 β πΌ β€ π π₯ π β πΌ P
2. Modified Newton's method using harmonic mean for solving nonlinear equations
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then {π₯ π } is said to converge to πΌwith order at least p. If p = 2 or 3, the convergence is said to be quadratic or
cubic respectively.
Definition 2.2 [2]: Let πΌ be a root of the function (1)and suppose that π₯ πβ1, π₯ π and π₯ π+1 are three successive
iterations closer to the root πΌ. Then, the computational order of convergence (COC) πcan be approximated using
the formulaο² β
ln (π₯ π+1βπΌ)/(π₯ πβπΌ)
ln (π₯ πβπΌ)/(π₯ πβ1βπΌ)
.
III. Known Variants of Newtonβs Method
Let πΌ be a simple root of a sufficiently differentiable function π(π₯). Consider the numerical solution of
the function π π₯ = 0. From the Newtonβs theorem, we have
π π₯ = π π₯ π + π β²
π‘ ππ‘
π₯
π₯ π
. (3)
Arithmetic Mean Newtonβs method (AN): By using the trapezoidal rule in (3), we obtain
π β² π‘ ππ‘
π₯
π₯ π
β
(π₯βπ₯ π )
2π
πβ² π₯ π + 2 π β² π₯ π β
ππ π₯ π
π πβ² π₯ π
πβ1
π=1 + πβ²
π₯ . (4)
From equations (3) and (4), a new approximation π₯n+1 for π₯is obtained and for m = 1, we get
π₯ π+1 = π₯ π β
2π π₯ π
πβ² π₯ π +πβ² π₯ π+1
β , π = 0, 1, 2 β¦ (5)
Equation (5) is the arithmetic mean Newtonβs method [2]. Since equation (5) is implicit, we can overcome this
implicit nature by the use of Newtonβs iterative step (2).
Harmonic Mean Newtonβs method (HN): In equation (5), if we use the harmonic mean instead of arithmetic
mean, we obtain harmonic mean Newtonβs method [6]
π₯ π+1 = π₯ π β
π π₯ π πβ² π₯ π +πβ² π₯ π+1
β
2πβ² π₯ π πβ² π₯ π+1
β , π = 0, 1, 2 β¦ (6)
whereπ₯ π+1
β
is calculated from (2).
Midpoint Newtonβs method (MN):If the integral in (3) is approximated using the midpoint integration rule
instead of trapezoidal rule, we get the midpoint Newtonβs method [6]
π₯ π+1 = π₯ π β
π π₯ π
πβ² π₯ π +π₯ π+1
β
2
, n = 0, 1, 2,β¦
wherexn+1
β
is calculated from (2).
Trapezoidal-Newtonβs Method (TN): Simplifying and computing π₯n+1from equations (3) and (4), one
obtains a new approximation for m = 2 as
π₯ π+1 = π₯ π β
4π π₯ π
π β² π₯ π +2π β² π₯ π +1
β +π₯ π
2
+π β² π₯ π+1
β
, π = 0, 1, 2 β¦ . (7)
where π₯ π+1
β
is calculated from (2). The method (7) is called Trapezoidal-Newtonβs method [4].
Simpson-Newtonβs Method (SN): The Simpsonβs
1
3
integration rule given by
π β² π‘ ππ‘
π₯
π₯ π
β
π₯βπ₯ π
6
[πβ²(π₯ π ) + 4π β²
π₯+π₯ π
2
+ πβ²
π₯ ] , (8)
is used to find the value of the integral in (3), then we get Simpson-Newtonβs method [3].Simplifying and
computing for π₯n+1from equation (8), one obtains a new approximation
π₯ π+1 = π₯ π β
6π π₯ π
π β² π₯ π +4π β² π₯ π +1+ π₯ π
2
+π β² π₯ π+1
, π = 0, 1, 2 β¦ . (9)
Since equation (9) is implicit, replacing xn+1 with xn+1
β
(calculated from (2)), we get
π₯ π+1 = π₯ π β
6π π₯ π
π β² π₯ π+1
β +4π β² π₯ π+1
β +π₯ π
2
+π β² π₯ π
, π = 0, 1, 2 β¦. (10)
IV. Modified Newtonβs Method and its analysis ofconvergence
We obtain the present method as follows: Rewrite equation (10) as
π₯ π+1 = π₯ π β
3π π₯ π
π β²(π₯ π )+π β² (π₯ π+1
β )
2
+ 2π β² π₯ π+1
β +π₯ π
2
3. Modified Newton's method using harmonic mean for solving nonlinear equations
www.iosrjournals.org 95 | Page
Wereplace harmonic mean [6] instead of arithmetic mean in the first part of the denominator. Then we obtain a
modified Newtonβs method calledHarmonic-Simpson-Newtonβs method (HSN)
π₯ π+1 = π₯ π β
3π π₯ π
2πβ² π₯ π πβ² π₯ π+1
β
πβ² π₯ π +πβ² π₯ π+1
β +2π β² π₯ π+1
β +π₯ π
2
(11)
whereπ₯ π+1
β
is calculated from (2).
In the following theorem, we prove the cubic convergence of the present method(11).
Theorem 4.1:Let πΌ β πΌ, be a simple zero of a sufficiently differentiablefunction π: πΌ β π β π for an open
intervalI. If π₯0 is sufficiently nearertoπΌ, then the method (11) has third order convergence.
Proof.Let πΌ be a simple zero of function π π₯ = 0. (That is, π πΌ = 0and πβ²
πΌ β 0).
By expanding π π₯ π and πβ² π₯ π by Taylor seriesaboutπΌ, we obtain
π π₯ π = πβ²
πΌ [π π + π2 π π
2
+ π3 π3
3
+ π π π
4
β¦ ] (12)
πβ²
π₯ π = πβ²
πΌ [1 + 2π2 π π + 3π3 π π
2
+ 4π4 π π
3
+ π π π
4
β¦ ] (13)
where π π =
π π πΌ
π! πβ² πΌ
, π = 2, 3, β¦and π π = π₯ π β πΌ .
Dividing equation (12) and (13), we get
π π₯ π
πβ² π₯ π
= π π β π2 π π
2
+ 2 π2
2
β π3 π π
3
+ π(π π
4
). Hencefrom (2), we get
π₯ π+1
β
= πΌ + π2 π π
2
+ 2 π3 β π2
2
π π
3
+ π(π π
4
) (14)
Again expandingπβ² π₯ π+1
β
about πΌand using (14),we get
πβ²
π₯ π+1
β
= πβ²
πΌ [1+2c2
2
en
2
+ O(en
3
)]. (15)
From (13) and (15), we have
πβ²
π₯ π + πβ²
π₯ π+1
β
= 2πβ²
πΌ [1+c2en + (c2
2
+
3
2
c3)en
2
+ O(en
3
)], (16)
2πβ²
π₯ π πβ²
π₯ π+1
β
= 2πβ²
Ξ± 2
[1+2c2en + (2c2
2
+ 3c3)en
2
+ O(en
3
)]. (17)
Dividing (17) by (16), we get
2πβ² π₯ π πβ²(π₯ π+1
β )
πβ² π₯ π +πβ²(π₯ π+1
β )
= πβ²
Ξ± [1+c2en +
3
2
c3en
2
+ O(en
3
)]. (18)
Further, we have
xn+1
β +xn
2
= Ξ± +
1
2
en +
1
2
c2en
2
+ c3 β c2
2
en
3
+ O en
4
. (19)
Again expanding πβ² π₯ π+1
β +π₯ π
2
by Tailorβs series about πΌ and using (19), we get
2πβ² xn+1
β +xn
2
= 2πβ²
Ξ± [1+c2en + c2
2
+
3
4
c3 en
2
+ O(en
3
)]. (20)
Adding (18) and (20), we get
2πβ² π₯ π πβ²(π₯ π+1
β )
πβ² π₯ π +πβ²(π₯ π+1
β )
+ 2πβ² π₯ π+1
β +π₯ π
2
= 3πβ²
Ξ± [1 + c2en +
2
3
c2
2
+ c3 en
2
+ O(en
3
)] . (21)
Then, from equations(12)and (21), we get
3π π₯ π
2πβ² π₯ π πβ² (π₯ π+1
β )
πβ² π₯ π +πβ² (π₯ π+1
β )
+2π β² π₯ π+1
β +π₯ π
2
= π π β
2
3
π2
2
π π
3
+ π π π
4
. (22)
The above equation (22) is substituted in (11)to get
π₯ π+1 = π₯ π β [π π β
2
3
π2
2
π π
3
+ π π π
4
].Sinceπ₯ π = π π + πΌ, we finally obtain that
π π+1 =
2
3
π2
2
π π
3
+ π π π
4
.This shows that the method (11) has third order convergence.
V. Numerical Examples
In this section, we present the results of numerical calculations to compare the efficiency of Harmonic-
Simpson-Newtonβs method (HSN). We compare HSN with Newtonβs method (NM) and other cubic convergent
4. Modified Newton's method using harmonic mean for solving nonlinear equations
www.iosrjournals.org 96 | Page
methods given in section III. We note thatfunctionπ5 π₯ is not convergent inAN method if x0=ο1 even when 100
iterations are used, but convergent in the present method.Stopping criterion for the iterations has been taken
as xn+1 β xn + f(xn+1) < Ξ΅,where π = 10β14
.
Different functions and number of iterations to find their roots are given in Table1. Computational order of
convergence (COC) and the number of function evaluation (NOFE) are given in Table 2. Numerical
computations are performed in the Matlab software with double precision accuracy and the results are presented.
Table 1
π π₯ X0
Number of iterations Root
NM AN MN TN SN HSN
π1 π₯ = x3
+ x2
β 2 1.5 6 5 4 5 5 4 πΌ =1
3 8 6 6 6 6 5
π2 π₯ = Sin(x) + x Cos(x) 1.3 6 5 5 5 5 5 πΌ = 2.028757838110434
3 6 5 4 4 4 4
π3 π₯ = x
2
β e
x
β 3x + 2 3 7 5 5 5 5 5 πΌ = 0.257530285439861
5 9 7 6 6 6 6
π4 π₯ = x e
x
β 1 3 9 7 6 7 6 6 πΌ = 0.567143290409784
4.5 11 8 8 8 8 7
π5 π₯ = x
3
β 10 3 6 5 5 5 5 4 πΌ = 2.154434690031884
ο1 7 NC 6 11 7 5
π6 π₯ = e
x
+ x β 20 10 13 9 9 9 9 8 πΌ = 2.842438953784447
5 8 6 4 6 6 5
Table 2
π π₯ X0
COC NOFE
NM AN MN TN SN HSN NM AN MN TN SN HSN
π1 π₯ 1.5 2.00 2.99 2.75 2.99 2.99 2.80 12 15 12 20 20 16
3 2.00 2.98 2.99 2.99 2.99 2.80 16 18 18 24 24 20
π2 π₯ 1.3 1.99 3.82 3.17 3.07 3.12 3.08 12 15 15 20 20 20
3 1.99 4.10 2.71 2.88 2.60 2.70 12 15 12 16 16 16
π3 π₯ 3 2.00 3.04 3.39 2.91 2.66 2.78 14 15 15 20 20 20
5 2.00 3.01 3.29 3.20 3.16 2.69 18 21 18 24 24 24
π4 π₯ 3 2.00 2.99 2.88 2.99 2.84 2.93 18 21 18 28 24 24
4.5 2.00 2.96 2.99 2.98 2.99 2.91 22 24 24 32 32 28
π5 π₯ 3 2.00 2.99 2.99 2.99 2.99 2.81 12 15 15 20 20 16
ο1 2.00 NC 2.91 3.01 3.02 2.95 14 NC 18 44 36 20
π6 π₯ 10 2.00 2.93 3.00 2.98 2.99 2.94 26 27 27 36 28 32
5 2.00 2.99 3.00 2.99 2.99 2.94 16 18 12 24 24 20
NC β Not Convergent, X0ο initial point
VI. Discussion
The basic advantage of the present method is that it is a variant of Newtonβs method based on
Newtonβs theorem. Comparing to other cubic convergent methods, our method converges faster and takes less
number of iterations. Further,this method does not require the computation of second or higher order derivatives
although it produces third order convergence. This method can easily be extended for equations having multiple
roots and for multi-variable case.
References
[1] M.K. Jain, S.R.K. Iyengar, R.K. Jain, Numerical Methods for Scientific and Engineering Computation, New Age International,6th
edition, 2012.
[2] S. Weerakoon. T.G.I. Fernando, A variant of Newtonβs method with accelerated third-order convergence, Applied Mathematics
Letters 13 (8)(2000) 87-93.
5. Modified Newton's method using harmonic mean for solving nonlinear equations
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[3] V. I. Hasanov, I. G. Ivanov, G. Nedjibov, A new modification of Newtonβs method, Applied Mathematics and Engineering 27
(2002) 278 -286.
[4] G. Nedzhibov, On a few iterative methods for solving nonlinear equations. Application of Mathematics in Engineering and
Economicsβ28, in: Proceeding of the XXVIII Summer school Sozopolβ 02, pp.1-8, Heron press, Sofia, 2002.
[5] M. Frontini, E. Sormoni, Some variants of Newtonβs method with third order convergence, Applied Mathematics and Computation
140 (2003) 419-426.
[6] A.Y. Ozban, Some new variants of Newtonβs method, Applied Mathematics Letters 17 (2004) 677-682.
[7] H.H.H. Homeier, On Newton type methods with cubic convergence, Journal of Computational and Applied Mathematics 176
(2005) 425-432.
[8] D.K.R. Babajee, M.Z. Dauhoo, An analysis of the properties of the variants of Newtonβs method with third order convergence,
Applied Mathematics and Computation 183 (2006) 659β684.
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Mathematics 205 (2007) 1 β 5.
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