WHICH OF THESE i.e., NEWTON RAPHSON METHOD, MULLER METHOD AND CHEBYSHEV METHOD GIVES THE MOST EFFICIENT ROOTS FOR A GIVEN EQUATION BY ITERATION? NEWTON RAPHSON METHOD, MULLER METHOD AND CHEBYSHEV METHOD (Math Problem Sample)

Mathematics Research PaperWHICH OF THESE i.e., NEWTON RAPHSON METHOD, MULLER METHOD AND CHEBYSHEV METHOD GIVES THE MOST EFFICIENT ROOTS FOR A GIVEN EQUATION BY ITERATION?NEWTON RAPHSON METHOD, MULLER METHOD AND CHEBYSHEV METHODNAME1/12/2019
INTRODUCTION
In everyday happenings, calculations are becoming essential. Making proper predictions helps in obtaining the correct solution and conclusion. The variations in the results obtained in calculations dependents on the approximations errors from the right answer. Therefore, there is a relationship between the method of analysis used and the accuracy of the solution or the approximation errors of the solutions obtained. As such, it is important to take keen consideration in selecting the appropriate method of analysis or running your program to get an accurate solution and thus making the right conclusion. In this research paper, I endeavor to look at three methods used in a mathematical approximation of equations roots. They include the Newton Raphson method, Regula Falisis method, and the Chebyshev method. This research will involve performing the three methods using the same equation and assessing the variations of the results with the accurate result from P-inspire calculator. Basing on the results and the approximation errors, I will be able to conclude which of the three methods is most efficient in finding the root of equations. My research paper will be divided into the background section, methodology section, and analysis and conclusion sections.
BACKGROUND INFORMATION
THE NEWTON-RAPHSON METHOD
The Newton-Raphson Method is a powerful method of solving complicated algebraic functions numerically. In other words, it is a technique used in finding successively better approximations to the roots or zeroes of real-valued function (Shodor, 2009). This technique uses the knowledge of linear approximation, therefore, expending a combination of instructions to approximate unit root basing on the function, the functions derivative and the original value of x. Newton-Raphson Method employs the use of iterative methods to reach a unit root function. The initially chosen value of x determines the specific root that the iteration method traces. The function is given as:
Whereby the currently known x-value is given byxn, the function value at xn given byf(xn ), the denominator f(xn ), as the derivative atxn. While the value is located in the equation is given by xn+1. Basically, the derivative value of the function f(xn ) is represented as f(x)/dx as such implying that the function f(xn)f(xn) is equivalent to dx. Therefore;
dx Will be closer to zero with the increase in some iteration that is run (Shodor, 2009).
A historical perspective of the Newton-Raphson method reveals that the technique was first developed by Newton in 1669 even though published later. The work was at the moment referred to as the Newton Method. Twenty years after the Newton's Method, Raphson Joseph made a development related to Newton's work, specifically for polynomials of degree 3, 4, 5 up to 10.Raphson computed an enhanced estimate g+x from an estimate g for x root. His approach was by setting pg+x=0, expanding, doing away with xk with K ≥ 2 and lastly solving for x. Like Newton, Raphson was not aware of the relationship between the derivative and his approach. Simpson and Euler identified the relationship fifty years later as such extending Newton's method beyond polynomial equations (Ypma, 1995).
CHEBYSHEV METHOD
The Chebyshev Method is an approximation method for functions mainly used by physicists and engineers. The method was originally founded by P.I Chebyshev in an attempt to solve a mechanical linkage problem.
Chebyshev Approximation