# Secant Bisection And Newton Raphson Methods Pdf

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- Comparative Study of Bisection, Newton-Raphson and Secant Methods of Root- Finding Problems
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Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. DOI: The study is aimed at comparing the rate of performance, viz-aviz, the rate of convergence of Bisection method, Newton-Raphson method and the Secant method of root-finding. The software, mathematica 9.

## Secant Method Matlab Code Pdf

To browse Academia. Skip to main content. By using our site, you agree to our collection of information through the use of cookies. To learn more, view our Privacy Policy. Log In Sign Up. Download Free PDF. Journal of Eng Download PDF. A short summary of this paper. The root finding problem is one of the most relevant computational problems.

It arises in a wide variety of practical applications in Physics, Chemistry, Biosciences, Engineering, etc. As a matter of fact, the determination of any unknown appearing implicitly in scientific or engineering formulas, gives rise to root finding problem [1]. Relevant situations in Physics where such problems are needed to be solved include finding the equilibrium position of an object, potential surface of a field and quantized energy level of confined structure [2]. The common root-finding methods include: Bisection, Newton-Raphson, False position, Secant methods etc.

Different methods converge to the root at different rates. That is, some methods are faster in converging to the root than others. The rate of convergence could be linear, quadratic or otherwise.

The higher the order, the faster the method converges [3]. The study is at comparing the rate of performance convergence of Bisection, Newton-Raphson and Secant as methods of root-finding. Obviously, Newton-Raphson method may converge faster than any other method but when we compare performance, it is needful to consider both cost and speed of convergence. An algorithm that converges quickly but takes a few seconds per iteration may take more time overall than an algorithm that converges more slowly, but takes only a few milliseconds per iteration [4].

Newton's method, on the other hand, requires one function and the derivative evaluation per iteration. It is often difficult to estimate the cost of evaluating the derivative in general if it is possible [1,[4][5]. It seem safe, to assume that in most cases, evaluating the derivative is at least as costly as evaluating the function [4]. Thus, we can estimate that the Newton iteration takes about two functions evaluation per iteration.

This disparity in cost means that we can run two iterations of the secant method in the same time it will take to run one iteration of Newton method. They concluded that Newton method is 7. The root always converges, though very slow in converging [5]. If so, stop.

The process is repeated until the root is found [5][6][7]. The method is probably the most popular technique for solving nonlinear equation because of its quadratic convergence rate. But it is sometimes damped if bad initial guesses are used [8][9]. It was suggested however, that Newton's method should sometimes be started with Picard iteration to improve the initial guess [9]. Newton Raphson method is much more efficient than the Bisection method. However, it requires the calculation of the derivative of a function as the reference point which is not always easy or either the derivative does not exist at all or it cannot be expressed in terms of elementary function [6, [7][8].

Furthermore, the tangent line often shoots wildly and might occasionally be trapped in a loop [6]. It was remarked in [1], that if none of the above criteria has been satisfied, within a predetermined, say, N, iteration, then the method has failed after the prescribed number of iteration.

In this case, one could try the method again with a different x o. Meanwhile, a judicious choice of x o can sometimes be obtained by drawing the graph of f x , if possible. However, there does not seems to exist a clear-cut guideline on how to choose a right starting point, x o that guarantees the convergence of the Newton-Raphson method to a desire root.

There are some functions that are either extremely difficult if not impossible or time consuming. The way out out of this, according to [1] is to approximate the derivative by knowing the values of the function at that and the previous approximation. Based on this definition, we show the rate of convergence of Newton and Secant methods of root-finding. Meanwhile, we may not border to show that of Bisection method, sequel to the fact that many literatures consulted are in agreement that Bisection method will always converge, and has the least convergence rate.

It was also maintained that it converges linearly [1][2][3][4][5][6][7]. Then, for every x in this interval, there exist a number, c between p and x such that! Now for the Newton method, we have seen that. By implication, the quadratic convergence we mean that the accuracy gets doubled at each iteration. In particular, the convergence is superlinear, but not quite quadratic.

This result only holds under some technical conditions, namely that f x be twice continuously differentiable and the root in question be simple. That is with multiplicity 1. If the initial values are not close enough to the root, then there is no guarantee that the Secant method converges [7]. The results are presented in Table 1 to 3. Table 1 shows the iteration data obtained for Bisection method with the aid of Mathematica 9.

This is in line with the findings of [4]. However, Newton's method requires the evaluation of both the function f x and its derivative at every iteration while Secant method only requires the evaluation of f x. Hence, Secant method may occasionally be faster in practice as in the case of our study. In Ref. So, on this premises also, we can claim that Secant method is faster than the Newton's method in terms of the rate of convergence.

This is sequel to the fact that it has a converging rate close to that of Newton-Raphson method, but requires only a single function evaluation per iteration.

We also concluded that though the convergence of Bisection is certain, its rate of convergence is too slow and as such it is quite difficult to extend to use for systems of equations.. Related Papers. By Trisna Darmawansyah. James F. By Jinhee Kwon. By Kar Heng Lee, Ph. Download pdf. Remember me on this computer.

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Skip to main content. Search form Search. Secant method code. Python Source Code: Secant Method. It is an iterative procedure involving linear interpolation to a root.

In numerical analysis , the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite-difference approximation of Newton's method. However, the secant method predates Newton's method by over years. The secant method is defined by the recurrence relation. As can be seen from the recurrence relation, the secant method requires two initial values, x 0 and x 1 , which should ideally be chosen to lie close to the root. Starting with initial values x 0 and x 1 , we construct a line through the points x 0 , f x 0 and x 1 , f x 1 , as shown in the picture above.

We look at three fundamental methods for finding roots of a function f: R → R. Method. Bisection. Newton. Secant. Input Req's • f ∈ C0([a, b]). • f ∈ C1(near.

## Comparative Study of Bisection, Newton-Raphson and Secant Methods of Root- Finding Problems

The convergce process in the bisection method is very slow. It depends only on the choice of end points of the interval [a,b]. The function f x does not have any role in finding the point c which is just the mid-point of a and b.

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### Secant method

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