2. Application of Bisection Method Presenting: c

3. Presenters : Faisal Zubi

4. Oldest known approximation Problem Numerical Computation; Root - finding Problem: Eq. :- Root Approximate 1700 B.C. traced back Intermediate Value Theorem----1 st technique BISECTION Method z Find_ root ‘x’ ? Eq.—form : f(x) = 0 given ‘f’ Number ‘x’ -> zero of ‘f’ Kind of Binary Search Efficient than stepwise search

5. m. Function----continuous. F(x) +/- [ _,_ ] F(x) = 0. Put ‘x’ and When root lies :-

6. Bisection Method : ‘ f ’ __ continuous function l x3  f(x3) = 0 f(x 1 ) & f(x 2 ) ________ +/- Procedure __ work only when Remember [x 1,x 2 ] f(x1) & f(x2) opposite sign

7. x 3  midpoint [x 1,x 2 ] = 1 / 2 (x1 + x2) f (x3) = 0 ___ x = x3  ROOT if else f(x3) ------ + /+, - / - f(x1) or f(x2) if f(x3) & f(x1) ___ same sign x  (x3,x2) set x1 = x3 if f(x3) & f(x1) ___ opposite sign x  (x1,x3) set x2 = x3 z

8. Drawback : Slow in converging i.e. # of iterations ___ quite large Property : Always converges  solution z

9. Demonstration g x1 f(x1) x2 f(x2) _ +

10. k x3 f(x3) _ x1 f(x1) x2 f(x2) +

11. g x4 f(x4) x3 f(x3) _ x1 f(x1) x2 f(x2) +

12. m x5 f(x5) x4 f(x4) x3 f(x3) _ x1 f(x1) x2 f(x2) +

13. Application t Zero degree Celsius With time, temp. Temp. 0 C Machine Shut down _ operation time 9 hrs. _ Machine switch off _ Temp. 0 C ‘machine’ if

14. r x (time,hr) 0 3 5 7 9 11 13 15 f(x) (temp. Celsius) - 30 - 25 - 21.5 - 16 - 12.5 - 07 - 3.5 + 0 2 Table of the machine:

15. s From divided difference method Get the function of the plant P7(x) = -30 + (x-0)(1.66) + (x-0)(x-3)(0.068) + (x-0)(x-3)(x-5)(0.0170)+ (x-0)(x-3)(x-5)(x-7)(-0.0098) + (x-0)(x-3)( x-5)(x-7)(x-9)(0.0025) + (x-0)(x-3)(x-5)(x-7)(x-9)(x-11)(-0.0002) + (x-0)(x-3)(x-5)(x-7)(x-9)(x-11)(x-13)(-0.0000066). K Where ‘K’……..any constant, representing the environmental temp.

16. -3.5 -07 -12.5 -16 -21.5 -25 -30 +2 3 57911 1315 l hrs. temp. [ 0,15 ]

17. w Atif Abbas