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.
Study of Correlation Theory with Different Views and Methodsamong Variables i...inventionjournals
Correlation among two numbers is an important concept and this relationship among two variables may be direct or indirect/inverse. Generally, correlation of two numbers is studyin statistics .The different types of correlation among numbers may be positively correlated, negatively correlated and perfectly correlated in statistics. Generally theserelationship is direct or indirect/inverse.So,in this paper it is tried to explore correlation among numbers/variables inunitary methods, ratio and proportion, variation methods.
The document contains 50 multiple choice questions related to regression analysis. Some key topics covered include: defining the slope and intercept of a regression line; interpreting the correlation coefficient and coefficient of determination; calculating regression equations and predicted values; and understanding how changing variables impacts regression coefficients.
Nirmal Singh presented a seminar on the Chi-Square test. The Chi-Square test is a non-parametric statistical test used to compare observed data with data we expect to obtain according to a specific hypothesis. It can be used with different types of measurements and does not require assumptions about the distribution of the attributes. The value of Chi-Square is calculated by summing the squares of the differences between observed and expected frequencies, divided by the expected frequencies. The degree of freedom, which is needed to compare the calculated Chi-Square value to table values, is determined by the number of rows and columns in the data less one. The Chi-Square test has become widely used for analyzing non-parametric data in statistical
The document discusses testing for independence between two variables using a contingency table and chi-square test. It explains how to set up a contingency table with observed and expected frequencies, and how to calculate the chi-square test statistic to determine if the variables are independent or dependent. An example is provided that tests if blood pressure is independent of jogging status using a contingency table and chi-square test.
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.
Study of Correlation Theory with Different Views and Methodsamong Variables i...inventionjournals
Correlation among two numbers is an important concept and this relationship among two variables may be direct or indirect/inverse. Generally, correlation of two numbers is studyin statistics .The different types of correlation among numbers may be positively correlated, negatively correlated and perfectly correlated in statistics. Generally theserelationship is direct or indirect/inverse.So,in this paper it is tried to explore correlation among numbers/variables inunitary methods, ratio and proportion, variation methods.
The document contains 50 multiple choice questions related to regression analysis. Some key topics covered include: defining the slope and intercept of a regression line; interpreting the correlation coefficient and coefficient of determination; calculating regression equations and predicted values; and understanding how changing variables impacts regression coefficients.
Nirmal Singh presented a seminar on the Chi-Square test. The Chi-Square test is a non-parametric statistical test used to compare observed data with data we expect to obtain according to a specific hypothesis. It can be used with different types of measurements and does not require assumptions about the distribution of the attributes. The value of Chi-Square is calculated by summing the squares of the differences between observed and expected frequencies, divided by the expected frequencies. The degree of freedom, which is needed to compare the calculated Chi-Square value to table values, is determined by the number of rows and columns in the data less one. The Chi-Square test has become widely used for analyzing non-parametric data in statistical
The document discusses testing for independence between two variables using a contingency table and chi-square test. It explains how to set up a contingency table with observed and expected frequencies, and how to calculate the chi-square test statistic to determine if the variables are independent or dependent. An example is provided that tests if blood pressure is independent of jogging status using a contingency table and chi-square test.
Siegel-Tukey test named after Sidney Siegel and John Tukey, is a non-parametric test which may be applied to the data measured at least on an ordinal scale. It tests for the differences in scale between two groups.
The test is used to determine if one of two groups of data tends to have more widely dispersed values than the other.
The test was published in 1980 by Sidney Siegel and John Wilder Tukey in the journal of the American Statistical Association in the article “A Non-parametric Sum Of Ranks Procedure For Relative Spread in Unpaired Samples “.
The chi-square test is used to determine if an observed frequency distribution differs from an expected theoretical distribution. It can test goodness of fit, independence of attributes, and homogeneity. The test involves calculating chi-square by taking the sum of the squares of the differences between observed and expected frequencies divided by expected frequencies. For the test to be valid, certain conditions must be met regarding sample size, expected frequencies, independence, and randomness. The test has some limitations such as not measuring strength of association and being unreliable with small expected frequencies.
This document provides an overview of line of best fit and linear regression. It defines key concepts like linear regression, residuals, and scatter plots. It explains how to determine the line of best fit for a data set by finding the line that minimizes the sum of the squared residuals. An example is worked through showing how to calculate the line of best fit and make a prediction based on that line. The document emphasizes that linear regression is useful for understanding relationships between variables in fields like business and medical research.
This document discusses linear regression analysis. It defines simple and multiple linear regression, and explains that regression examines the relationship between independent and dependent variables. The document provides the equations for linear regression analysis, and discusses calculating the slope, intercept, standard error of the estimate, and coefficient of determination. It explains that regression analysis is widely used for prediction and forecasting in areas like advertising and product sales.
This presentation describes the application of regression analysis in research, testing assumptions involved in it and understanding the outputs generated in the analysis.
The document discusses four nonparametric statistical tests:
1. The Wilcoxon Rank Sum Test (also called the Mann-Whitney U Test) compares the medians of two independent samples and is an alternative to the independent t-test.
2. The Wilcoxon Signed Rank Test compares the medians of two dependent/paired samples and is an alternative to the paired t-test. It calculates the differences between pairs, ranks their absolute values, and sums the ranks of positive differences.
3. The Kruskal-Wallis Test compares more than two independent samples.
4. The Runs Test examines the randomness of a single sample by counting runs, or streaks, of
This document provides information on chi-square tests and other statistical tests for qualitative data analysis. It discusses the chi-square test for goodness of fit and independence. It also covers Fisher's exact test and McNemar's test. Examples are provided to illustrate chi-square calculations and how to determine statistical significance based on degrees of freedom and critical values. Assumptions and criteria for applying different tests are outlined.
1. The document provides an overview of the key concepts in the Theory of Equations unit, including types of equations, methods for solving different types of equations, and properties of roots.
2. It discusses linear equations, simultaneous equations, quadratic equations and their solving methods like elimination, substitution, and factorization.
3. Examples of equation problems from commercial applications are also presented, involving linear, simultaneous and quadratic equations. Worked examples and practice problems are provided for each topic.
Dimensional analysis is a technique for simplifying physical problems by reducing the number of variables. It is useful for presenting experimental data, solving problems without exact theories, checking equations, and establishing relative importance of phenomena. Dimensional analysis refers to the physical nature and units of quantities. It can be used to develop equations for fluid phenomena, convert between unit systems, and reduce variables in experiments. Two common methods are Rayleigh's method, which determines expressions involving a maximum of four variables, and Buckingham's π-method, which arranges variables into dimensionless groups. Dimensional analysis provides a partial solution and requires understanding the relevant phenomena.
The document provides objectives and examples for solving various types of linear equations and application problems involving linear equations. The key points are:
1. Linear equations are solved by simplifying expressions, gathering like terms, and isolating the variable. Rational equations are solved by determining excluded values, multiplying by the LCD, solving the resulting linear equation, and eliminating extraneous solutions.
2. Application problems are solved using a step-by-step process: analyzing the problem, making a diagram, determining unknowns, setting up an equation, solving the equation, and checking the solution.
3. Example application problems covered include number problems, geometry problems, digit problems, and money/coin problems. Real-life scenarios
The chi-square test is used to determine if there is a significant relationship between two nominal variables by comparing observed and expected frequencies. It can be used to examine relationships between categorical variables like education and income level or brand preference and gender. The null hypothesis states there is no relationship, while the alternative or research hypothesis states there is a relationship. Expected frequencies are calculated and chi-square is obtained using a formula. This value is then compared to a critical value based on the level of significance and degrees of freedom to determine whether to reject or accept the null hypothesis.
The Chi Square Test is a widely used non-parametric test that does not rely on assumptions about population parameters. It compares observed frequencies to expected frequencies specified by the null hypothesis. The Chi Square value is calculated by summing the squared differences between observed and expected values divided by the expected values. The Chi Square value is then compared to a critical value based on the degrees of freedom. Common applications include tests of goodness of fit, independence of variables, and homogeneity of proportions.
The chi-square test is used to determine if experimental data fits the results expected from genetic theory. It involves calculating an expected and observed count for each phenotype, then using a chi-square formula to determine how well the observed fits the expected. The result is compared to critical chi-square values from a table based on degrees of freedom and probability to determine if the null hypothesis should be rejected or failed to be rejected.
Hypothesis testing chi square goodness of fit testNadeem Uddin
This document provides examples of using the chi-square goodness of fit test to analyze categorical data. It first explains the procedure for conducting the test, which involves defining hypotheses, determining the level of significance, calculating the test statistic, identifying the critical region, and making a conclusion. Then it provides three examples that demonstrate applying the test to analyze the distribution of grades, whether grades differ from a historical pattern, and whether the probability of boy and girl births is equal.
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.
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.
The document discusses methods for finding the roots of polynomial equations, including Muller's method and Bairstow's method. Muller's method uses three points to derive the coefficients of a parabola and find an approximated root. Bairstow's method involves synthetically dividing a polynomial by a quadratic factor to find values of r and s that make the coefficients b1 and b0 equal to zero, through an iterative process. It provides an example of applying Bairstow's method to find the roots of a 5th order polynomial.
Numerical Study of Some Iterative Methods for Solving Nonlinear Equationsinventionjournals
In this paper we introduce, numerical study of some iterative methods for solving non linear equations. Many iterative methods for solving algebraic and transcendental equations is presented by the different formulae. Using bisection method , secant method and the Newton’s iterative method and their results are compared. The software, matlab 2009a was used to find the root of the function for the interval [0,1]. Numerical rate of convergence of root has been found in each calculation. It was observed that the Bisection method converges at the 47 iteration while Newton and Secant methods converge to the exact root of 0.36042170296032 with error level at the 4th and 5th iteration respectively. It was also observed that the Newton method required less number of iteration in comparison to that of secant method. However, when we compare performance, we must compare both cost and speed of convergence [6]. It was then concluded that of the three methods considered, Secant method is the most effective scheme. By the use of numerical experiments to show that secant method are more efficient than others.
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.
The document discusses various numerical methods for finding the roots or zeros of equations, including closed and open methods. Closed methods like bisection and false position trap the root within a closed interval by repeatedly dividing the interval in half. Open methods like Newton-Raphson and secant methods use information about the nonlinear function to iteratively refine the estimated root without being restricted to an interval. The document also covers methods for equations with multiple roots like Muller's method.
Siegel-Tukey test named after Sidney Siegel and John Tukey, is a non-parametric test which may be applied to the data measured at least on an ordinal scale. It tests for the differences in scale between two groups.
The test is used to determine if one of two groups of data tends to have more widely dispersed values than the other.
The test was published in 1980 by Sidney Siegel and John Wilder Tukey in the journal of the American Statistical Association in the article “A Non-parametric Sum Of Ranks Procedure For Relative Spread in Unpaired Samples “.
The chi-square test is used to determine if an observed frequency distribution differs from an expected theoretical distribution. It can test goodness of fit, independence of attributes, and homogeneity. The test involves calculating chi-square by taking the sum of the squares of the differences between observed and expected frequencies divided by expected frequencies. For the test to be valid, certain conditions must be met regarding sample size, expected frequencies, independence, and randomness. The test has some limitations such as not measuring strength of association and being unreliable with small expected frequencies.
This document provides an overview of line of best fit and linear regression. It defines key concepts like linear regression, residuals, and scatter plots. It explains how to determine the line of best fit for a data set by finding the line that minimizes the sum of the squared residuals. An example is worked through showing how to calculate the line of best fit and make a prediction based on that line. The document emphasizes that linear regression is useful for understanding relationships between variables in fields like business and medical research.
This document discusses linear regression analysis. It defines simple and multiple linear regression, and explains that regression examines the relationship between independent and dependent variables. The document provides the equations for linear regression analysis, and discusses calculating the slope, intercept, standard error of the estimate, and coefficient of determination. It explains that regression analysis is widely used for prediction and forecasting in areas like advertising and product sales.
This presentation describes the application of regression analysis in research, testing assumptions involved in it and understanding the outputs generated in the analysis.
The document discusses four nonparametric statistical tests:
1. The Wilcoxon Rank Sum Test (also called the Mann-Whitney U Test) compares the medians of two independent samples and is an alternative to the independent t-test.
2. The Wilcoxon Signed Rank Test compares the medians of two dependent/paired samples and is an alternative to the paired t-test. It calculates the differences between pairs, ranks their absolute values, and sums the ranks of positive differences.
3. The Kruskal-Wallis Test compares more than two independent samples.
4. The Runs Test examines the randomness of a single sample by counting runs, or streaks, of
This document provides information on chi-square tests and other statistical tests for qualitative data analysis. It discusses the chi-square test for goodness of fit and independence. It also covers Fisher's exact test and McNemar's test. Examples are provided to illustrate chi-square calculations and how to determine statistical significance based on degrees of freedom and critical values. Assumptions and criteria for applying different tests are outlined.
1. The document provides an overview of the key concepts in the Theory of Equations unit, including types of equations, methods for solving different types of equations, and properties of roots.
2. It discusses linear equations, simultaneous equations, quadratic equations and their solving methods like elimination, substitution, and factorization.
3. Examples of equation problems from commercial applications are also presented, involving linear, simultaneous and quadratic equations. Worked examples and practice problems are provided for each topic.
Dimensional analysis is a technique for simplifying physical problems by reducing the number of variables. It is useful for presenting experimental data, solving problems without exact theories, checking equations, and establishing relative importance of phenomena. Dimensional analysis refers to the physical nature and units of quantities. It can be used to develop equations for fluid phenomena, convert between unit systems, and reduce variables in experiments. Two common methods are Rayleigh's method, which determines expressions involving a maximum of four variables, and Buckingham's π-method, which arranges variables into dimensionless groups. Dimensional analysis provides a partial solution and requires understanding the relevant phenomena.
The document provides objectives and examples for solving various types of linear equations and application problems involving linear equations. The key points are:
1. Linear equations are solved by simplifying expressions, gathering like terms, and isolating the variable. Rational equations are solved by determining excluded values, multiplying by the LCD, solving the resulting linear equation, and eliminating extraneous solutions.
2. Application problems are solved using a step-by-step process: analyzing the problem, making a diagram, determining unknowns, setting up an equation, solving the equation, and checking the solution.
3. Example application problems covered include number problems, geometry problems, digit problems, and money/coin problems. Real-life scenarios
The chi-square test is used to determine if there is a significant relationship between two nominal variables by comparing observed and expected frequencies. It can be used to examine relationships between categorical variables like education and income level or brand preference and gender. The null hypothesis states there is no relationship, while the alternative or research hypothesis states there is a relationship. Expected frequencies are calculated and chi-square is obtained using a formula. This value is then compared to a critical value based on the level of significance and degrees of freedom to determine whether to reject or accept the null hypothesis.
The Chi Square Test is a widely used non-parametric test that does not rely on assumptions about population parameters. It compares observed frequencies to expected frequencies specified by the null hypothesis. The Chi Square value is calculated by summing the squared differences between observed and expected values divided by the expected values. The Chi Square value is then compared to a critical value based on the degrees of freedom. Common applications include tests of goodness of fit, independence of variables, and homogeneity of proportions.
The chi-square test is used to determine if experimental data fits the results expected from genetic theory. It involves calculating an expected and observed count for each phenotype, then using a chi-square formula to determine how well the observed fits the expected. The result is compared to critical chi-square values from a table based on degrees of freedom and probability to determine if the null hypothesis should be rejected or failed to be rejected.
Hypothesis testing chi square goodness of fit testNadeem Uddin
This document provides examples of using the chi-square goodness of fit test to analyze categorical data. It first explains the procedure for conducting the test, which involves defining hypotheses, determining the level of significance, calculating the test statistic, identifying the critical region, and making a conclusion. Then it provides three examples that demonstrate applying the test to analyze the distribution of grades, whether grades differ from a historical pattern, and whether the probability of boy and girl births is equal.
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.
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.
The document discusses methods for finding the roots of polynomial equations, including Muller's method and Bairstow's method. Muller's method uses three points to derive the coefficients of a parabola and find an approximated root. Bairstow's method involves synthetically dividing a polynomial by a quadratic factor to find values of r and s that make the coefficients b1 and b0 equal to zero, through an iterative process. It provides an example of applying Bairstow's method to find the roots of a 5th order polynomial.
Numerical Study of Some Iterative Methods for Solving Nonlinear Equationsinventionjournals
In this paper we introduce, numerical study of some iterative methods for solving non linear equations. Many iterative methods for solving algebraic and transcendental equations is presented by the different formulae. Using bisection method , secant method and the Newton’s iterative method and their results are compared. The software, matlab 2009a was used to find the root of the function for the interval [0,1]. Numerical rate of convergence of root has been found in each calculation. It was observed that the Bisection method converges at the 47 iteration while Newton and Secant methods converge to the exact root of 0.36042170296032 with error level at the 4th and 5th iteration respectively. It was also observed that the Newton method required less number of iteration in comparison to that of secant method. However, when we compare performance, we must compare both cost and speed of convergence [6]. It was then concluded that of the three methods considered, Secant method is the most effective scheme. By the use of numerical experiments to show that secant method are more efficient than others.
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.
The document discusses various numerical methods for finding the roots or zeros of equations, including closed and open methods. Closed methods like bisection and false position trap the root within a closed interval by repeatedly dividing the interval in half. Open methods like Newton-Raphson and secant methods use information about the nonlinear function to iteratively refine the estimated root without being restricted to an interval. The document also covers methods for equations with multiple roots like Muller's method.
A Derivative Free High Ordered Hybrid Equation Solver Zac Darcy
Generally a range of equation solvers for estimating the solution of an equation contain the derivative of
first or higher order. Such solvers are difficult to apply in the instances of complicated functional
relationship. The equation solver proposed in this paper meant to solve many of the involved complicated
problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
A Novel Approach for the Solution of a Love’s Integral Equations Using Bernst...IOSRJM
In this paper a novel technique implementing Bernstein polynomials is introduced for the numerical solution of a Love’s integral equations. The Love’s integral equation is a class of second kind Fredholm integral equations, and it can be used to describe the capacitance of the parallel plate capacitor (PPC) in the electrostatic field.This numerical technique developed by Huabsomboon et al. bases on using Taylor-series expansion [1], [2], [3] and [4]. We compare the numerical solution using Bernstein technique with the numerical solution obtained by using Chebyshev expansion. It is shown that the numerical results are excellent.
This document discusses stability estimates for identifying point sources using measurements from the Helmholtz equation. It begins by introducing the inverse problem of determining source terms from boundary measurements of the electromagnetic field. It then proves identifiability, showing that a single boundary measurement can uniquely determine the sources. Next, it establishes a local Lipschitz stability result, demonstrating continuous dependence of the sources on the measurements. Finally, it proposes a numerical inversion algorithm using a reciprocity gap concept to quickly and stably identify multiple sources.
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
This document summarizes a study on evaluating the rate of convergence of the Newton-Raphson method. A computer program was coded in Java to calculate cube roots from 1 to 25 using Newton-Raphson. The lowest rate of convergence was for the cube root of 16, and the highest was for 3. The average rate of convergence was found to be 0.217920. Formulas for estimating the rate of convergence from successive approximations are also presented.
The document summarizes research on nonlinear correlation coefficients on manifolds. It defines a new nonlinear correlation coefficient called SEVP, proves some of its basic properties including that it ranges from 0 to 1. It discusses how to measure nonlinear correlation between variables on a manifold and reviews common dimensionality reduction methods for manifolds. The goal is to preserve nonlinear structure as much as possible by projecting onto the orthogonal complement of tangent spaces. An optimization problem is formulated to find the linear space with the largest angle to all tangent spaces, transforming it into an eigenvalue problem to solve.
A DERIVATIVE FREE HIGH ORDERED HYBRID EQUATION SOLVERZac Darcy
Generally a range of equation solvers for estimating the solution of an equation contain the derivative of
first or higher order. Such solvers are difficult to apply in the instances of complicated functional
relationship. The equation solver proposed in this paper meant to solve many of the involved complicated
problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
A Comparison Of Iterative Methods For The Solution Of Non-Linear Systems Of E...Stephen Faucher
This document compares iterative methods for solving nonlinear systems of equations, including Gauss-Seidel method. It describes Bisection method, Newton-Raphson method, Secant method, and False Position method. Bisection method converges linearly but is simple. Newton-Raphson converges faster if the initial guess is close to the root but may diverge otherwise. Secant method does not require derivatives but can fail if the function is flat. False Position combines features of Bisection and Secant methods. The document analyzes the algorithms and convergence properties of each method.
The following document presents some novel numerical methods valid for one and several variables, which using the fractional derivative, allow us to find solutions for some nonlinear systems in the complex space using real initial conditions. The origin of these methods is the fractional Newton-Raphson method, but unlike the latter, the orders proposed here for the fractional derivatives are functions. In the first method, a function is used to guarantee an order of convergence (at least) quadratic, and in the other, a function is used to avoid the discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible that the method has at most an order of convergence (at least) linear. Keywords: Iteration Function, Order of Convergence, Fractional Derivative.
This document discusses numerical methods for finding the roots of nonlinear equations. It covers the bisection method, Newton-Raphson method, and their applications. The bisection method uses binary search to bracket the root within intervals that are repeatedly bisected until a solution is found. The Newton-Raphson method approximates the function as a linear equation to rapidly converge on roots. Examples and real-world applications are provided for both methods.
A derivative free high ordered hybrid equation solverZac Darcy
Generally a range of equation solvers for estimating the solution of an equation contain the derivative of
first or higher order. Such solvers are difficult to apply in the instances of complicated functional
relationship. The equation solver proposed in this paper meant to solve many of the involved complicated
problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
This document provides an introduction to different types of equations including linear equations, simultaneous equations, and quadratic equations. It defines an equation as a statement of equality between two quantities. Linear equations are those where the highest power of the unknown is one. Simultaneous equations contain two or more unknowns and can be solved using substitution or row operations. Quadratic equations contain terms with powers of the unknown up to two and can be solved using factorization, completing the square, or the quadratic formula. Examples are provided for solving each type of equation. The objectives and end questions review solving these different equation types.
This document discusses stochastic partial differential equations (SPDEs). It outlines several approaches that have been used to solve SPDEs, including methods based on diffusion processes, stochastic characteristic systems, direct methods from mathematical physics, and substitution of integral equations. It also discusses using backward stochastic differential equations to study SPDEs and introduces notation for the analysis of an Ito SDE with inverse time. The document is technical in nature and outlines the mathematical frameworks and equations involved in solving SPDEs through various probabilistic methods.
Similar to A NEW STUDY TO FIND OUT THE BEST COMPUTATIONAL METHOD FOR SOLVING THE NONLINEAR EQUATION (20)
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints.
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,
a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We
will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal
argument in its original form to establish the cardinality of K between that of (N,R) respectively
A Positive Integer 𝑵 Such That 𝒑𝒏 + 𝒑𝒏+𝟑 ~ 𝒑𝒏+𝟏 + 𝒑𝒏+𝟐 For All 𝒏 ≥ 𝑵mathsjournal
According to Bertrand's postulate, we have 𝑝𝑛 + 𝑝𝑛 ≥ 𝑝𝑛+1. Is it true that for all 𝑛 > 1 then 𝑝𝑛−1 + 𝑝𝑛 ≥𝑝𝑛+1? Then 𝑝𝑛 + 𝑝𝑛+3 > 𝑝𝑛+1 + 𝑝𝑛+2where 𝑛 ≥ 𝑁, 𝑁 is a large enough value?
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,
a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We
will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal
argument in its original form to establish the cardinality of K between that of (N,R) respectively.
Moving Target Detection Using CA, SO and GO-CFAR detectors in Nonhomogeneous ...mathsjournal
systems in complex situations. A fundamental problem in radar systems is to automatically detect targets while maintaining a
desired constant false alarm probability. This work studies two detection approaches, the first with a fixed threshold and the
other with an adaptive one. In the latter, we have learned the three types of detectors CA, SO, and GO-CFAR. This research
aims to apply intelligent techniques to improve detection performance in a nonhomogeneous environment using standard
CFAR detectors. The objective is to maintain the false alarm probability and enhance target detection by combining
intelligent techniques. With these objectives in mind, implementing standard CFAR detectors is applied to nonhomogeneous
environment data. The primary focus is understanding the reason for the false detection when applying standard CFAR
detectors in a nonhomogeneous environment and how to avoid it using intelligent approaches.
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints.
The Impact of Allee Effect on a Predator-Prey Model with Holling Type II Func...mathsjournal
There is currently much interest in predator–prey models across a variety of bioscientific disciplines. The focus is on quantifying predator–prey interactions, and this quantification is being formulated especially as regards climate change. In this article, a stability analysis is used to analyse the behaviour of a general two-species model with respect to the Allee effect (on the growth rate and nutrient limitation level of the prey population). We present a description of the local and non-local interaction stability of the model and detail the types of bifurcation which arise, proving that there is a Hopf bifurcation in the Allee effect module. A stable periodic oscillation was encountered which was due to the Allee effect on the
prey species. As a result of this, the positive equilibrium of the model could change from stable to unstable and then back to stable, as the strength of the Allee effect (or the ‘handling’ time taken by predators when predating) increased continuously from zero. Hopf bifurcation has arose yield some complex patterns that have not been observed previously in predator-prey models, and these, at the same time, reflect long term behaviours. These findings have significant implications for ecological studies, not least with respect to examining the mobility of the two species involved in the non-local domain using Turing instability. A spiral generated by local interaction (reflecting the instability that forms even when an infinitely large
carrying capacity is assumed) is used in the model.
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal argument in its original form to establish the cardinality of K between that of (N,R) respectively.
Moving Target Detection Using CA, SO and GO-CFAR detectors in Nonhomogeneous ...mathsjournal
Modernization of radar technology and improved signal processing techniques are necessary to improve detection systems in complex situations. A fundamental problem in radar systems is to automatically detect targets while maintaining a
desired constant false alarm probability. This work studies two detection approaches, the first with a fixed threshold and the
other with an adaptive one. In the latter, we have learned the three types of detectors CA, SO, and GO-CFAR. This research
aims to apply intelligent techniques to improve detection performance in a nonhomogeneous environment using standard
CFAR detectors. The objective is to maintain the false alarm probability and enhance target detection by combining
intelligent techniques. With these objectives in mind, implementing standard CFAR detectors is applied to nonhomogeneous
environment data. The primary focus is understanding the reason for the false detection when applying standard CFAR
detectors in a nonhomogeneous environment and how to avoid it using intelligent approaches
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints
Modified Alpha-Rooting Color Image Enhancement Method on the Two Side 2-D Qua...mathsjournal
Color in an image is resolved to 3 or 4 color components and 2-Dimages of these components are stored in separate channels. Most of the color image enhancement algorithms are applied channel-by-channel on each image. But such a system of color image processing is not processing the original color. When a color image is represented as a quaternion image, processing is done in original colors. This paper proposes an implementation of the quaternion approach of enhancement algorithm for enhancing color images and is referred as the modified alpha-rooting by the two-dimensional quaternion discrete Fourier transform (2-D QDFT). Enhancement results of this proposed method are compared with the channel-by-channel image enhancement by the 2-D DFT. Enhancements in color images are quantitatively measured by the color enhancement measure estimation (CEME), which allows for selecting optimum parameters for processing by thegenetic algorithm. Enhancement of color images by the quaternion based method allows for obtaining images which are closer to the genuine representation of the real original color.
An Application of Assignment Problem in Laptop Selection Problem Using MATLABmathsjournal
The assignment – selection problem used to find one-to- one match of given “Users” to “Laptops”, the main objective is to minimize the cost as per user requirement. This paper presents satisfactory solution for real assignment – Laptop selection problem using MATLAB coding.
The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, pnormal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed
functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed
functions and preservation theorems.
Cubic Response Surface Designs Using Bibd in Four Dimensionsmathsjournal
Response Surface Methodology (RSM) has applications in Chemical, Physical, Meteorological, Industrial and Biological fields. The estimation of slope response surface occurs frequently in practical situations for the experimenter. The rates of change of the response surface, like rates of change in the yield of crop to various fertilizers, to estimate the rates of change in chemical experiments etc. are of
interest. If the fit of second order response is inadequate for the design points, we continue the
experiment so as to fit a third order response surface. Higher order response surface designs are sometimes needed in Industrial and Meteorological applications. Gardiner et al (1959) introduced third order rotatable designs for exploring response surface. Anjaneyulu et al (1994-1995) constructed third order slope rotatable designs using doubly balanced incomplete block designs. Anjaneyulu et al (2001)
introduced third order slope rotatable designs using central composite type design points. Seshu babu et al (2011) studied modified construction of third order slope rotatable designs using central composite
designs. Seshu babu et al (2014) constructed TOSRD using BIBD. In view of wide applicability of third
order models in RSM and importance of slope rotatability, we introduce A Cubic Slope Rotatable Designs Using BIBD in four dimensions.
The caustic that occur in geodesics in space-times which are solutions to the gravitational field equations with the energy-momentum tensor satisfying the dominant energy condition can be circumvented if quantum variations are allowed. An action is developed such that the variation yields the field equations
and the geodesic condition, and its quantization provides a method for determining the extent of the wave packet around the classical path.
Approximate Analytical Solution of Non-Linear Boussinesq Equation for the Uns...mathsjournal
For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy perturbation transform method(HPTM). The solution is compared with the exact solution. The comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer
parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height of water table. The results resemble well with the physical phenomena.
Common Fixed Point Theorems in Compatible Mappings of Type (P*) of Generalize...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings
under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy
metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
Mathematics subject classifications: 45H10, 54H25
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
- The document presents a probabilistic algorithm for computing the polynomial greatest common divisor (PGCD) with smaller factors.
- It summarizes previous work on the subresultant algorithm for computing PGCD and discusses its limitations, such as not always correctly determining the variant τ.
- The new algorithm aims to determine τ correctly in most cases when given two polynomials f(x) and g(x). It does so by adding a few steps instead of directly computing the polynomial t(x) in the relation s(x)f(x) + t(x)g(x) = r(x).
Table of Contents - September 2022, Volume 9, Number 2/3mathsjournal
Applied Mathematics and Sciences: An International Journal (MathSJ ) aims to publish original research papers and survey articles on all areas of pure mathematics, theoretical applied mathematics, mathematical physics, theoretical mechanics, probability and mathematical statistics, and theoretical biology. All articles are fully refereed and are judged by their contribution to advancing the state of the science of mathematics.
Brand Guideline of Bashundhara A4 Paper - 2024khabri85
It outlines the basic identity elements such as symbol, logotype, colors, and typefaces. It provides examples of applying the identity to materials like letterhead, business cards, reports, folders, and websites.
Creative Restart 2024: Mike Martin - Finding a way around “no”Taste
Ideas that are good for business and good for the world that we live in, are what I’m passionate about.
Some ideas take a year to make, some take 8 years. I want to share two projects that best illustrate this and why it is never good to stop at “no”.
A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.pptHenry Hollis
The History of NZ 1870-1900.
Making of a Nation.
From the NZ Wars to Liberals,
Richard Seddon, George Grey,
Social Laboratory, New Zealand,
Confiscations, Kotahitanga, Kingitanga, Parliament, Suffrage, Repudiation, Economic Change, Agriculture, Gold Mining, Timber, Flax, Sheep, Dairying,
Elevate Your Nonprofit's Online Presence_ A Guide to Effective SEO Strategies...TechSoup
Whether you're new to SEO or looking to refine your existing strategies, this webinar will provide you with actionable insights and practical tips to elevate your nonprofit's online presence.
How to Manage Reception Report in Odoo 17Celine George
A business may deal with both sales and purchases occasionally. They buy things from vendors and then sell them to their customers. Such dealings can be confusing at times. Because multiple clients may inquire about the same product at the same time, after purchasing those products, customers must be assigned to them. Odoo has a tool called Reception Report that can be used to complete this assignment. By enabling this, a reception report comes automatically after confirming a receipt, from which we can assign products to orders.
🔥🔥🔥🔥🔥🔥🔥🔥🔥
إضغ بين إيديكم من أقوى الملازم التي صممتها
ملزمة تشريح الجهاز الهيكلي (نظري 3)
💀💀💀💀💀💀💀💀💀💀
تتميز هذهِ الملزمة بعِدة مُميزات :
1- مُترجمة ترجمة تُناسب جميع المستويات
2- تحتوي على 78 رسم توضيحي لكل كلمة موجودة بالملزمة (لكل كلمة !!!!)
#فهم_ماكو_درخ
3- دقة الكتابة والصور عالية جداً جداً جداً
4- هُنالك بعض المعلومات تم توضيحها بشكل تفصيلي جداً (تُعتبر لدى الطالب أو الطالبة بإنها معلومات مُبهمة ومع ذلك تم توضيح هذهِ المعلومات المُبهمة بشكل تفصيلي جداً
5- الملزمة تشرح نفسها ب نفسها بس تكلك تعال اقراني
6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
واخيراً هذهِ الملزمة حلالٌ عليكم وإتمنى منكم إن تدعولي بالخير والصحة والعافية فقط
كل التوفيق زملائي وزميلاتي ، زميلكم محمد الذهبي 💊💊
🔥🔥🔥🔥🔥🔥🔥🔥🔥
A NEW STUDY TO FIND OUT THE BEST COMPUTATIONAL METHOD FOR SOLVING THE NONLINEAR EQUATION
1. Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 2/3, September 2019
DOI : 10.5121/mathsj.2019.6302
15
A NEW STUDY TO FIND OUT THE BEST
COMPUTATIONAL METHOD FOR SOLVING THE
NONLINEAR EQUATION
Mir Md. Moheuddin1
, Md. Jashim Uddin2
and Md. Kowsher3*
1
Dept. of CSE, Atish Dipankar University of Science and Technology, Dhaka-1230,
Bangladesh.
2,3
Dept. of Applied Mathematics, Noakhali Science and Technology University,
Noakhali-3814, Bangladesh.
ABSTRACT
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.
KEYWORDS
Newton-Raphson method, False Position method, Secant method, Bisection method, Non-
linear equations and Rate of convergence.
1. INTRODUCTION
In numerical analysis, the most appearing problem is to determine the equation’s root in the form
of f(x) = 0 rapidly where the equations of the types of f(x) = 0 is known as Algebraic or
Transcendental according to as
f(x) = a0xn+a1xn−1+. .... +an−1x +an,(a≠0),
where the expression of f(x) being an algebraic or transcendental or a combination of both.If f(x)
= 0
at the point x= α, then α is acquainted as the root of the given equation if and only if f(α)=0. The
equations of f(x) = 0 is an algebraic equation of degree k such as x3-3x2+12=0, 2x3-3x-6=0 are
some kinds of algebraic equations of two and three degree whereas if f(x) takes on small number
of function including trigonometric, logarithmic and exponential etc then f(x) = 0 is a
Transcendental equations, for instance, x sin x + cos x = 0, log x – x+3=0 etc are the
transcendental equations. In both cases, the coefficients are known as numerical equations if it is
pure numbers. Besides, several degrees of algebraic functions are solved by well-known methods
that are common to solve it. but problems are when we are trying to solve the higher degree or
transcendental functions for which there are no direct methods of how to get the best solution
easily. Using approximate methods are a facile way for these sorts of equations. Numerical
2. Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 2/3, September 2019
16
methods are applied to impart formative solutions of the problems including nonlinear equations.
Various methods for discovering the nonlinear equation’s root exists. For example, the Graphical
method, Bisection method, Regula-falsi method, Secant method, Iteration method, Newton-
Raphson method, Bairstow’s method and Graeffe’s root squaring method and Newton’s-iterative
method, etc. Several formulae of the numerical method is described in the books of S.S. Sastry
[3], R.L. Burden [1], A.R. Vasishtha [2] as well as there are lots of paper involving nonlinear
Algebraic, Transcendental equations like improvement in Newton-Rapson method for nonlinear
equations applying modified Adomian Decomposition method [4], on modified Newton method
for solving a nonlinear algebraic equations by mid-point [10], New three-steps iterative method
for solving nonlinear equations [7], A New two-step method for solving nonlinear equations [9],
as well as a new method for solving nonlinear equations by Taylor expansion [8]. Besides, some
investigations [5, 6, 16] have shown a better way to interpret the algebraic and transcendental
equation’s root whereas Mohammad Hani Almomani et al. [11] presents a method for selecting
the best performance systems. What is more, in [12], Zolt ´an Kov´acs et al. analyzed in
understanding the convergence and stability of the Newton-Raphson method. In [13], Fernando
Brambila Paz et al. calculated and demonstrated the fractional Newton-Raphson method
accelerated with Aitken’s method. Robin Kumar et al. [14] examined five numerical methods for
their convergence. In [15] Nancy Velasco et al. implemented a Graphical User Interface in Matlab
to find the roots of simple equations and more complex expressions. However, Kamoh Nathaniel
Mahwash et al. [16] represents the two basic methods of approximating the solutions of nonlinear
systems of algebraic equations. In [17], Farooq Ahmed Shah et al. suggest and analyze a new
recurrence relation which creates the iterative methods of higher order for solving nonlinear
equation f (x) = 0. Also, M. A. Hafiz et al. [18] propound and analyse a new predictor-corrector
method for solving a nonlinear system of equations using the weight combination of mid-point,
Trapezoidal and quadrature formulas.
In our working procedure, we have discussed and also compared with the existing some iterative
methods, for example, Newton’s method, False Position method, Secant method as well as
Bisection method to achieve the best method among them. This research paper based on analyzing
the best method to impart the best concept for solving the nonlinear equations. With the help of
this paper, we can easily perceive how to determine the approximate roots of non-linear equations
speedily. Moreover, we demonstrated some function to obtain an effective method which gives a
smaller level of iteration among the mentioned methods.
SCOPES AND OBJECTIVES OF THE STUDY
The main scopes of the study are to evaluate the best method for solving nonlinear equations
applying numerical methods. The objectives are given below;
Finding the rate of convergence, proper solution, as well as the level of errors of the
methods.
Comparing the existed methods to achieve the best method for solving nonlinear
equations.
2. MATERIALS AND METHODS
Due to these works, the following methods is discussed such as Newton-Raphson method, False
Position method, Secant method and Bisection method.
2.1. Newton-Raphson Method
Newton-Raphson method is the famous numerical method that is almost applied to solve the
3. Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 2/3, September 2019
17
equation’s root and it helps to evaluate the roots of an equation f(x) = 0 where f(x) is considered
to have a derivative of f'(x). The basic idea of Newton-Raphson
Figure 1. Newton-Raphson method
Also, Newton method can be achieved from the Taylor’s series expansion of the function f(x) at
the point 𝑥0 as follows:
Since (𝑥−𝑥0) is small, we can neglect second, third and higher degree terms in (𝑥−𝑥0) and thus
we obtain
(𝑥)=𝑓(𝑥0)+(𝑥−𝑥0)𝑓′(𝑥0)……..(1)
Putting f(x) = 0 in (1) we get,
Generalizing, we get the general formula
Which is called Newton-Raphson formula.
Example: Solve the equation f(x) = 𝑥3+2𝑥2+10𝑥−20=0 using Newton’s method.
Given that, f(x)= 𝑥3+2𝑥2+10𝑥−20=0
∴𝑓′(𝑥)=3𝑥2+4𝑥+10
Apply Newton’s method to the equation, correct to five decimal places.
Now we can see that f(1) = - 7 < 0 and f(2) = 16 > 0.
Therefore the root lies between 1 and 2.
We know that, Newton’s formula as follows,
4. Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 2/3, September 2019
18
Solving the right hand side yields
Convergence of Newton-Raphson method:
Newton’s method is given below,
This is really an iteration method where
2.2. False Position method
False position method, which is a very old algorithm for solving a root finding problem. It is
called trial and error technique that exerts test values for the sake of variable and after that
adjusting this test value in imitation of the outcome. This formula is also called as "guess and
check". The basic concept of this method is demonstrated in the following figure,
5. Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 2/3, September 2019
19
Figure 2. False Position method
Consider the equation f(x) = 0, where f(x) is continuous. In this method we choose two points a
and b such that f(a) and f(b) be of opposite signs (a < b). So the curve of y = f(x) will meet the x-
axis between A(a, f(a)) and B(b, f(b)). Hence a root lies between these two points. Now the
equation of the chord joining the points A(a, f(a)) and B(b, f(b)) is given below
The point of intersection of the chord equation for the x-axis imparts the first approximation 𝑥0
for the root of f(x) = 0, that is achieved by substituting y = 0 in (1) we get,
If f(a) and f(𝑥0) be of opposite signs, so its root is between a and 𝑥0. Then replacing b by 𝑥0 in
(2), we obtain the next approximation 𝑥1. But if f(a) and f(𝑥0) are same sign, so f(𝑥0) and f(b) be
of opposite signs, therefore its root is between 𝑥0 and b. Again, replacing a by 𝑥0 in (2) we find
𝑥1. This work is continued till the root is achieved its desired accuracy.The genearl formula is
Example: Solve the equation f(x) = 𝑥3−3𝑥−5=0 by the method of False Position method.
Solution:
Let f(x) = 𝑥3−3𝑥−5=0
Since f(2) = -3<0
F(3) = 13>0
The root lies between 2.1875 and 3 such that a = 2.1875 and b = 3 etc
.
Convergence of False Position method:
We know that,
6. Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 2/3, September 2019
20
In the equation f(x) = 0, if the function f(x) is convex in the interval (𝑥0,𝑥1) that takes a root then
one of the points 𝑥0, or 𝑥1 is always fixed and the other point varies with n. If the point 𝑥0 is fixed
, then the function f(x) is approximated by the straight line passing through the points (𝑥0,𝑓0) as
well as (𝑥 𝑛,𝑛), n = 1,2,……
Now, the error equation (2) from the convergence of secant method becomes
From equation (1), we get
Where C = C𝜀0 is the asymptotic error constant. Hence, the false Position method has Linear rate
of Convergence.
2.3. Secant Method
The secant method is just a variation from the Newton’s method. The basic idea is shown below,
Figure 3. Secant method
The Newton method of solving a nonlinear equation f(x) = 0 is given by the iterative formula.
One of the drawbacks of the Newton’s method is that one has to evaluation the derivative of the
function. To overcome these drawbacks, the derivative of the function f(x) is approximated as
Putting equation (2) into (1), we get,
7. Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 2/3, September 2019
21
The above equation is called the secant method. This is like Newton-Raphson method but
requires two initial guess.
Example: Solve the equation f(x) = 𝑥3−2𝑥−5=0 using secant method.
Let the two initial approximations be given by
𝑥−1=2 and 𝑥0= 3
The root lies between 2 and 3.
Convergence of Secant method:
We know that,
Where C = ½ 𝑓′′(𝜁)/𝑓′(𝜁) and the higher powers of 𝜀 𝑛 are neglected.
8. Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 2/3, September 2019
22
Neglecting the negative sign, we get the rate of convergence for the Secant method is p = 1.618.
2.4. Bisection Method
The bisection method is very facile and perhaps always faithful to trace a root in numerical
schemes for solving non-linear equations. It is frequently applied to acquire of searching a root
value of a function and based on the intermediate value theorem (IVT). This method always
succeeds due to it is an easier way to perceive. The basic concept of this method is given below;
Figure 4. Bisection method
This Bisection method states that if f(x) is continuous which is defined on the interval [a, b]
satisfying the relation f(a) f(b) < 0 with f(a) and (b) of opposite signs then by the intermediate
value theorem(IVT) ,it has at least a root or zero of f(x) = 0 within the interval [a, b]. It consists of
finding two real numbers a and b, having the interval [a, b], as well as the root, lies in a ≤ x ≤ b
whose length is, at each step, half of its original interval length. This procedure is repeated until
the interval imparts required accuracy by virtue of the requisite root is achieved. While it has
more than one root on the interval (a, b) ,as a result, the root must be unique. Moreover, the root
converges linearly and very slowly. But when f(𝑥0) is very much nearly zero or is very small
then we can stop the iteration. Therefore, the general formula is
9. Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 2/3, September 2019
23
So at least one root of the given equation lies between 1 and 2.
The root lies between 1 and 1.5 etc.
Convergence of Bisection method:
2.5. Comparison Of The Rate Of Convergence Among The Four Method
The rate of convergence is a very essential issue for the sake of solution of algebraic and
transcendental equations, due to the rate of convergence of any numerical method evaluates the
speed of the approximation to the solution of the problems.
Following table demonstrates the comparison of rate of convergence.
Table 1. Comparison of the rate of convergence
From this table, we see that Newton-Raphson has a better rate of convergence than other methods
i.e. 2.
10. Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 2/3, September 2019
24
3. ANALYSIS AND COMPARISON
3.1. Graphical Development Of The Approximation Root To Achieve The Best
Method
The graphical development of the four method is illustrated as follows,
Figure 5. Graphical development among the four methods. The function used is 𝑥2-30 with the interval [5
6] and the desired error of 0.0001.
From the above graph, for the Newton-Raphson method, in each iteration, the graph demonstrates
the approximate root with only 3 iterations as it converges more rapidly and accurately whereas
other method delays to give desire roots of the equation. In the Bisection method, it requires a
large number of iterations to meet the desired root as convergent slowly. For the Secant method,
it imparts a level of convergence but not as compared to the Newton method. The False Position
method converges more slowly due to it needs at least 10 iterations to compare to the Newton and
secant method respectively.
Both algorithms are tested with several functions taking as data for error 0.0001 and the desired
accuracy. The results gained for the approximate root, error, and iteration are summarized in table
2.6.1 for the Newton-Raphson method, in table 2.6.2 for False-Position method, in table 2.6.3 for
11. Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 2/3, September 2019
25
Secant method and in table 2.6.4 for Bisection method. The Newton-Raphson imparts the small
number of iteration correctly as long as the initial values enclose the desired root.
Table 2. Data obtained with the Newton-Raphson Method
In the False position method, it imparts a lot of iteration number to meet the desired root.
Table 3. Data obtained with the False Position Method
We apply the secant method to obtain the result with a few numbers of iterations.
Table 4. Data obtained with the Secant Method
Similarly, the Bisection method demonstrates the roots slowly as long as the initial values enclose
the desired root as well as gives a large number of iteration.
12. Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 2/3, September 2019
26
Table 5. Data obtained with the Bisection Method
3.2. Comparison Of The Approximate Error
Figure 6. The approximate error is plotted against the number of iterations (1-20).
From the above graphical development and comparison, it is recommended that Newton-
Raphson’s method is the fastest in the rate of convergence compared with the other methods.
4. TEST AND RESULTS TO VERIFY THE BEST METHOD
Following examples interpret the result achieved by the four methods to solve nonlinear equations
such as algebraic and transcendental equations.
Problem 1. Consider the following equation [2 3],
13. Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 2/3, September 2019
27
Table 7. Number of iterations and the results obtained by the four method
Moreover, we can look at the following approximate error graph gained from the above four
methods which illustrate the change of errors from one method to another method.
Figure 7. The graph of NRM Figure 8. The graph of FPM
14. Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 2/3, September 2019
28
Figure 9. The graph of SM Figure 10. The graph of BM
Comparing the above graph, we see that Newton-Raphson Method is a better method than others.
Similarly,
Problem 2. Consider the following equation [-1 -2],
15. Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 2/3, September 2019
29
Table 8. Number of iterations and the results obtained by the four method
Moreover, we can also look at the following approximate error graph gained from the above four
methods which illustrate the change of errors from one method to another method.
Figure 11. The graph of NRM Figure 12. The graph of FPM
Figure 13. The graph of SM Figure 14. The graph of BM
Comparing the above graph, we see that Newton-Raphson Method is a better method than others.
5. CONCLUSION
In this research works, we have analyzed and compared the four iterative methods for solving
non-linear equations that assist to present a better performing method. The MATLAB results are
also delivered to check the appropriateness of the best method. With the help of the approximate
error graph, we can say that the Newton’s method is the most robust and capable method for the
sake of solving the nonlinear equation. Besides, Newton’s method gives a lesser number of
iterations concerning the other methods and it shows less processing time. Ultimately, it is
consulted steadily that Newton’s method is the most effective and accurate owing to solving the
nonlinear problems.
16. Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 2/3, September 2019
30
REFERENCES
[1] Richard L. Burden, J.douglas Faires, Annette M. burden, “Numerical Analysis”, Tenth Edition, ISBN:
978-1-305-25366-7.
[2] R. Vasishtha, Vipin Vasishtha, “Numerical Analysis”, Fourth Edition.
[3] S.S. Sastry, “Introduction Methods of Numerical Analysis”, Fourth Edition, ISBN: 81-203-2761-6.
[4] Shin Min Kang, Waqas Nazeer, “Improvements in Newton-Rapson Method for Nonlinear equations
using Modified Adomain Decomposition Method”, International Journal of Mathematical Analysis,
Vol. 9, 2015, no. 39, 1919.1928.
[5] Muhammad Aslam Noor, Khalida Inayat Noor, Eisa Al-Said and Muhammad Waseem, “Some New
Iterative methods for Nonlinear Equations, Hindawi Publishing Corporation.
[6] Okorie Charity Ebelechukwu, ben Obakpo Johnson, Ali Inalegwu Michael, akuji Terhemba fideli,
“Comparison of Some Iterative methods of Solving Nonlinear Equations”, International Journal of
Theroretical and Applied Mathematics.
[7] Ogbereyivwe Oghovese, Emunefe O. John, “New Three-Steps Iterative Method for Solving Nonlinear
Equations, IOSR Journal of Mathematics.
[8] Masoud Allame, Nafiseh azad, “A new method for solving nonlinear equations by Taylor expansion”,
Application of Mathematics and Computer engineering.
[9] Jishe Feng, “A New Two-step Method for solving Nonlinear equations”, International Journal of
Nonlinear Science”, Vol.8(2009) No.1pp.40-44.
[10] Masoud Allame and Nafiseh Azad, “On Modified Newton Method for solving a Nonlinear Algebraic
equation by Mid-Point”, World applied science Journal 17(12): 1546-1548, 2012.
[11] Mohammad Hani Almomani, Mahmoud Hasan Alrefaei, Shahd Al Mansour, “A method for selecting
the best performance systems”, International Journal of Pure and Applied Mathematics.
[12] Zolt ´an Kov´acs, “Understanding convergence and stability of the Newton-Raphson method”,
https://www.researchgate.net/publication/277475242.
[13] Fernando Brambila Paz, Anthony Torres Hernandez, Ursula Iturrarán-Viveros, Reyna Caballero Cruz,
“Fractional Newton-Raphson Method Accelerated with Aitken’s Method”.
[14] Robin Kumar and Vipan, “Comparative Analysis of Convergence of Various Numerical Methods”,
Journal of Computer and Mathematical Sciences, Vol.6(6),290-297, June 2015, ISSN 2319-8133
(Online).
[15] Nancy Velasco, Dario Mendoza, Vicente Hallo, Elizabeth Salazar-Jácome and Victor Chimarro,
“Graphical representation of the application of the bisection and secant methods for obtaining roots of
equations using Matlab”, IOP Conf. Series: Journal of Physics: Conf. Series 1053 (2018) 012026 doi
:10.1088/1742-6596/1053/1/012026.
[16] Kamoh Nathaniel Mahwash, Gyemang Dauda Gyang, “Numerical Solution of Nonlinear Systems of
Algebraic Equations”, International Journal of Data Science and Analysis2018; 4(1): 20-23
http://www.sciencepublishinggroup.com/j/ijdsa.
17. Applied Mathematics and Sciences: An International Journal (MathSJ), Vol. 6, No. 2/3, September 2019
31
[17] Farooq Ahmed Shah, and Muhammad Aslam Noor, “Variational Iteration Technique and Some
Methods for the Approximate Solution of Nonlinear Equations”, Applied Mathematics& Information
Sciences Letters An International Journal, http://dx.doi.org/10.12785/amisl/020303.
[18] M. A. Hafiz & Mohamed S. M. Bahgat, “An Efficient Two-step Iterative Method for Solving System
of Nonlinear Equations”, Journal of Mathematics Research; Vol. 4, No. 4; 2012 ISSN 1916-9795 E-
ISSN 1916-9809.