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
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Bisection Method
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Bisection Method
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Bisection Method
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Bisection Method
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Bisection Method
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Bisection Method
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Example
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Example
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Example
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Bisection Method
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Bisection Method
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Bisection Method
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Homework 1
Exercise 1
Exercise 2
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Fixed Point Iteration
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Example
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Fixed Point Iteration
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Example
Show that the function g(x) bellow has a unique fixed point on the interval [-1,1]
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Example
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Example
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Example
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Fixed Point Iteration
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Fixed Point Iteration
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Example
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Example
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Example
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Fixed Point Iteration
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Fixed Point Iteration
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Homework 2
Exercise 1
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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.
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Newton's Method
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Newton's Method
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Newton's Method
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Newton's Method
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Newton's Method
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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 ..
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Example
Consider the function f(x)=cos(x) –x. Approximate a root of f using:
(a) A fixed point method
(b) Newton’s method
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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].
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Example
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Newton's Method
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Newton's Method
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The Secant Method
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The Secant Method
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The Secant Method
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Example
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Example
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Example
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Modified Newton's method
Modified Newton's method
Newton's method
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Example
Find the root of the following function
(a)Using Newton's method.
(b)Using the modified Newton's method.
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Example
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Example
Find the root of the following function
(a)Using Newton's method.
(b)Using the modified Newton's method.
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Example
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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