Numerical Analysis lec-2 The Bisection Method

Hi, Today we'll learn about the most important topic of numerical analysis the bisection method.

It is very lengthy and slow process but it is the only process which we can surely find an approximate result and that's why it is the most important topic of numerical analysis so let's begin from the definition of bisection method.

*DEFINITION:
the bisection method are called the binary-search method is probably the most primitive for finding a real root and is describe as follows.

- it requires two starting values X0 and X1 for the solution such that:           F( X0 ) . F ( X1 ) < 0
then the equation :          F( X ) = 0
has at least one real root in the interval [ X0 , X1 ] we shall illustrate bisection method of graphically.

- in which X2 , X3 ......... denote successive mid points.










*Procedure of the Bisection method:
1- find X2, the new iterate using X2 = X0 + X1 / 2 and evaluate F ( X2 )
2- if F ( X2 ) = 0 then X2 is a root of F ( X )
3- if F ( X2 ) not equal to 0 then two things are possible 

(a) - if F ( X0 ) < 0 then we  compute the new iterates X3 = X0 + X2 / 2 and evaluate F ( X3 )

(b) - if F ( X1 ) * F ( X2 ) < 0 then we compute the new iterate X3 as X3 = X1 + X2 / 2

(solved Examples)

Note: if the interval is given then we directly apply into the function and solve the equation till the result becomes 0, but if interval is not given then first we find interval then solve the equation. don't forget to practice because without practice you'll learn nothing.