1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 7 Roots of Equations Bracketing Methods

2. Last Time The Problem Define Function c must satisfy c is the ROOT of the equation

3. Last Time Classification Methods BracketingOpen Graphical Bisection Method False Position Fixed Point Iteration Newton-Raphson Secand

4. Last Time Graphical Methods c f(c) v=10 m/s t=3 sec m=65 kg g=9.81

5. Last Time Graphical Methods No Roots Even Number of Roots Lower and Upper Bounds of interval yield values of same sign

6. Last Time Graphical Methods Lower and Upper Bounds of interval yield values of opposite sign Odd number of Roots

7. Last Time Bisection Method Choose Lower, x l and Upper x u guesses that bracket the root xlxl xuxu

8. Last Time Bisection Method Calculate New Estimate x r and f(x r ) xlxl xuxu x r =0.5(x l +x u )

9. Last Time Bisection Method Define New Interval that Brackets the Root Check sign of f(x l )*f(x r ) and f(x u )*f(x r ) xlxl xuxu Previous Guess xuxu

10. Last Time Bisection Method Repeat until convergence xlxl xuxu Previous Guess x r =0.5(x l +x u )

11. Last Time Bisection Method Check Convergence Root = If Error

12. Objectives Master methods to compute roots of equations Assess reliability of each method Choose best method for a specific problem REGULA FALSI Method (False Position)

13. False Position Method xlxl xuxu x r =0.5(x l +x u ) Recall Bisection Method No consideration on function values

14. False Position Method f(x l ) f(x u ) xlxl xuxu xrxr NEW ESTIMATE

15. False Position Method f(x l ) f(x u ) xlxl xuxu xrxr

16. False Position Method f(x l ) f(x u ) xlxl xuxu xrxr From Similar Triangles

17. False Position Method

19. Add and subtract New Estimate

20. Loop x old =x r Error=100*abs(x-x old )/x r Sign=f(x l )*f(x r ) Sign x u =x r f u =f(x u ) x l =x r f l =f(x l ) Error=0 Error<E all ROOT=x r FALSE <0>0 f u =f(x u ), f l =f(x l )

21. False Position Typically Faster Convergence than Bisection

22. False Position Not Efficient in this Case