Numerical Analysis Lecture 5 Chapter 2 Solution of

Numerical Analysis Lecture 5

Chapter 2 Solution of Non-Linear Equations

Introduction Bisection Method Regula-Falsi Method of iteration Newton - Raphson Method Muller’s Method Graeffe’s Root Squaring Method

Bisection Method (Bolzano)

Regula-Falsi Method of false position

Here, we choose two points xn and xn -1 such that f (xn) and f (xn-1) are of opposite signs. Intermediate value property suggests that the graph of y = f (x) crosses the x-axis between these two points and therefore, the root lies between these two points.

Example Using Regula-Falsi method, find the real root of the following equation correct, to three decimal places: x log 10 x =1. 2

Solution Let f (x) = x log 10 x – 1. 2 f (2) = – 0. 5979, f (3) = 0. 2314. Since f (2) and f (3) are of opposite signs, the real root lies between x 1 = 2, x 2 = 3.

The first approximation is obtained from

Since f (x 2) and f (x 3) are of opposite signs, the root of f (x) = 0 lies between x 2 and x 3. Now, the second approximation is given by

Answer ! Thus, the root of the given equation correct to three decimal places is 2. 740

Method of Iteration

METHOD OF ITERATION can be applied to find a real root of the equation f (x) = 0 by rewriting the same in the form, (2. 3) Example, f (x) = cos x – 2 x + 3 = 0. It can be rewritten as

Also, f ’(x) and f ’’(x) do not vanish in (0, 1) and f (x) and f ’’(x) will have the same sign at x = 1. Therefore, we take the first approximation x 0 = 1, and using N-R method, we get

The second approximation is The required root is 0. 853.

Example Find a real root of the 3 equation x – 1 = 0 using Newton - Raphson method, correct to four decimal places.

Numerical Analysis Lecture 5