EEE DEPARTMENT

Course Code: EEE 203

Course Title: Numerical Analysis for Electrical Engineering

ASSIGNMENT :03

Name of the assignment: Comparative analysis among the root finding methods.

Submitted To: TAHSEEN KAMAL

Prepared by: SANIUL ISLAM (ID: 2006-3-80-004)

MEHRIN HOSSAIN (ID: 2006-3-80-015) MD. TOUSIF TANIM (ID: 2006-3-80-010)

Date of Submission: 26-06-2008

PREFACE

The aim of this assignment is to develop a concept of similarities and differences between some root finding methods. In this assignment, we have discussed about the Bisection method, False position, Newton-Raphson method and the Secant method. Here we have demonstrated the comparison between several root finding methods and have also showed their graphical representations.We have also discovered the easiest and the most difficult root finding methods- the one through which we get closer to the answer fastest and the one which includes complex calculations and lengthy procedure.

Bisection Method: If we have given interval (XL, XU), from the interval we can find f (XL) & f (XU). The value of one root should be positive and the other one should be negative. We then calculate c = (XL+XU)/2 and hence we calculate f(c).Then we have used sign rule for finding new average value.

Bisection method has a lengthy procedure for finding root. This method converges at a fairly slow speed. We have to perform iteration many times in order to find the root. The mathematical calculation of this method is easier than False position or Regula Falsi method.

False position: The Regula Falsi method (also called the method of false position) is a slightly more sophisticated method for finding the zero of a function. The bisection method used the midpoint of the interval (XL, XU) as the next iterate. A better approximation is obtained if we find the point (c,0) by using False position or Regula Falsi method.

Here,

We can find root more quickly in Regula Falsi method then Bisection method. But we have to do more complex calculation for Regula Falsi method then Bisection method.

Newton-Raphson method : Newton-Raphson method takes less iteration and less time to find out the root than the Bisection and Regula Falsi methods. If f(x), f '(x), f '' (x) are continuous near a root p. It starts from a single point (a, f (a)) and makes a straight-line approximation to the function f which is tangent to f at the point (a, f (a)). In order to do this, the procedure needs to know the derivative of the function f. We can find the root by this formula:

In order to converge, there is a condition f(x)* f ''(x) < [f '(x)]2 which has to be satisfied. However, convergence is not guaranteed. If the derivative is very small or zero, then the method will not proceed further.

Here, we find the root from less iteration in Newton-Raphson method than the Secant method. But we have to do more complex calculation for the Newton-Raphson method than the Secant method. In Newton-Raphson method we have to calculate both f(x) & f' (x), hence this method consumes more time.

Secant method: It is a root finding algorithm which assumes a function to be approximately linear in the region of interest. Each improvement is taken as the point where the approximating line crosses the axis. The secant method retains only the most recent estimate, so the root does not necessarily remain bracketed. 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.We can find the root by this formula :

Similarities between the root finding methods:

All the methods were developed in order to find out the root of a function. All of these methods are used in numerical analysis. All the numerical root-finding methods use iteration, producing a sequence of numbers, converging towards a limit ( called the "fixed point") which is a root. The first values of this series are initial guesses. The method computes subsequent values based on the old ones and the function f.

a)By using all of these methods, we get the approximate result not the actual one. Thus, there are some absolute and relative errors in these methods.

b)In numerical analysis, the False position method or Regula Falsi method is a root finding algorithm that combines features from the Bisection method and the Secant method.

c)Like the Bisection method, the False Position method starts with two points a0 and b0 such that f (a0) and f (b0) are of opposite signs,

which implies by the intermediate value theorem that, the function f has a root in the interval [a0, b0].

d)The recurrence formula of the Secant method can be derived from the formula of the Newton's method.

e)The Secant method is just like the Newton’s method but, instead of following the tangent line to the x-axis we follow the secant line based on the previous 2 iterates.

Comparison & difference between the root finding methods:

a)We can find root more quickly in Regula Falsi method than the Bisection method. But we have to do more complex calculation for Regula Falsi method than the Bisection method.

b)We find the root from less iteration in Newton-Raphson method than the Secant method. But we have to do more complex calculation for Newton-Raphson method than the Secant method. In Newton-Raphson method we have to calculate both f(x) & f' (x), so this method takes more time.

c) If we compare Newton's method with the secant method, we see that Newton's method converges faster. However, Newton's method requires the evaluation of both f and its derivative at every step, while the secant method only requires the evaluation of f. Therefore, the Secant method may well be faster in practice.

d)Replacing the derivative in Newton's method with a finite difference, we get the Secant method. This method does not require the computation (nor the existence) of a derivative, but the price is slower convergence (the order is approximately 1.6).

e)The bisection method is less efficient than the Newton's method but it is much less likely to show odd behavior.

f) Newton's method assumes the function f to have a continuous derivative. Newton's method may not converge if we start too far away from a root. However, if it does converge, it is faster than the Bisection method.

g)The Secant method does not require the root to remain bracketed like the Bisection method does, and hence it does not always converge.

h)The False Position method uses the same formula as the Secant method. However, it does not apply the formula on xn−1 and xn, like the Secant method, but on xn and on the last iterate xk such that f (xk) and f (xn) have a different sign. This means that the False Position method always converges.

i) If we compare Newton's method with the Secant method, we find

that the Newton's method converges faster.

CONCLUSION:

In this assignment, we learnt various root finding methods which use different ways to find the iteration. Through this task we came across to know the best possible method among the mentioned ones which gives us the fastest result, i.e. closest to the root. The methods have been compared in various ways and the relative error in each method had been estimated. We came across the conclusion that the Newton Raphson method is the best root finding method among all the mentioned methods. We also came to know that some of the methods include complex calculations while others had simple but elaborate calculations. Some methods had complex calculations but requires less time in finding the roots where as others had easier calculations but consumed more time.

REFERANCE

1. math.fullerton.edu/mathews/n2003/FixedPointMod

2. en.wikipedia.org/wiki/ Bisection _ method

3. mathworld.wolfram.com/ Method of FalsePosition

4. mathworld.wolfram.com/ Newton s Method .html

5. http://en.wikipedia.org/wiki/Secant_method

6. Numerical methods for engineers (Steven C Chapra).