1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 8 Roots of Equations Open 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 Bisection Method Repeat until convergence xlxl xuxu x r =0.5(x l +x u )

5. Last Time False Position Method f(x l ) f(x u ) xlxl xuxu xrxr

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

7. Last Time Convergence

8. Objectives OPEN Methods –Fixed Point Iteration –Newton Raphson –Secant

9. Open Methods Bracketing Methods Two Initial Estimates Needed that bracket the root Always Converge Open Methods ONE Initial Estimate Needed Sometimes Diverge

10. Fixed Point Iteration X root x is a root if f(x) = 0

11. Fixed Point Iteration X X f 1 (X) f 2 (X) +x+x +x+x f 1 (X)f 2 (X) root

12. Fixed Point Iteration X f 1 (X) f 2 (X) f 1 (X)f 2 (X) root x is a root if f 1 (x) = f 2 (x)

13. Fixed Point Iteration X f 1 (X) f 2 (X) Initial Guess New Guess root New Guess

14. Fixed Point Iteration X f 1 (X) f 2 (X) Initial Guess New Guess root New Guess Method Diverges

15. Condition for Convergence X f 1 (X) f 2 (X) New Guess

16. Newton Raphson X g(x) Initial Guess New Guess New Guess g’(x i )

17. Newton Raphson

18. Inflection Point in Vicinity of Root

19. Newton Raphson Persistent Oscillations near local max or min

20. Newton Raphson Initial guess close to root jumps several roots away

21. Newton Raphson Zero Derivative

22. Newton Raphson No Convergence Criteria Depends on Nature of Function Depends on Initial Guess Use Initial Guess Sufficiently Close to Root It converges very fast!! (when it does)

23. Homework 5.7 5.19 6.4 6.8 (a),(b),(c) Due Date: September 22