numerical analysis. 3. solutions of equations in one variable
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Numerical Analysis
Dr Houssem Bouchekara
3. Solutions of Equations in One Variable
Bisection Method
Bisection Method
Bisection Method
Bisection Method
Bisection Method
Bisection Method
Example
Example
Example
Bisection Method
Bisection Method
Bisection Method
Homework 1
Exercise 1
Exercise 2
Fixed Point Iteration
Example
Fixed Point Iteration
Example
Show that the function g(x) bellow has a unique fixed point on the interval [-1,1]
Example
Example
Example
Fixed Point Iteration
Fixed Point Iteration
Example
Example
Example
Fixed Point Iteration
Fixed Point Iteration
Homework 2
Exercise 1
Newton's Method
Newton's (or the Newton-Raphson) method is one of the most powerful and well-known numerical methods for
solving a root-finding problem.
There are many ways of introducing Newton's method:
1. If we only want an algorithm, we can consider the
technique graphically, as is often done in calculus.
2. Another possibility is to derive Newton's method as a
technique to obtain faster convergence than offered by
other types of functional iteration.
3. A third means of introducing Newton's method, is based
on Taylor polynomials.
Newton's Method
Newton's Method
Newton's Method
Newton's Method
Newton's Method
Newton's Method
In fact, this is the functional iteration technique that was used to give the rapid
convergence we saw in part (e) of the fixed point iteration Example ..
Example
Consider the function f(x)=cos(x) –x. Approximate a root of f using:
(a) A fixed point method
(b) Newton’s method
Example
A solution of this root finding problem is also a solution to the fixed point
problem g(x) = cos(x).
The graph in Figure 8 implies that a single fixed-point p lies in [0, /2].
Example
Newton's Method
Newton's Method
The Secant Method
The Secant Method
The Secant Method
Example
Example
Example
Modified Newton's method
Modified Newton's method
Newton's method
Example
Find the root of the following function
(a)Using Newton's method.
(b)Using the modified Newton's method.
Example
Example
Find the root of the following function
(a)Using Newton's method.
(b)Using the modified Newton's method.
Example
Homework 3
Exercise 1
Exercise 2
Use Newton's method and the modified Newton-Raphson method
to find a solution accurate to within 10-5 to the problem