1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 9 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. Newton Raphson X g(x) Initial Guess New Guess New Guess g’(x i )

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

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

9. Secant Method X g’(x) xixi x i-1 g(x i ) g(x i-1 )

10. Secant Method Newton Raphson Backward Divided Difference Secant

11. Secant Method X g(x) New Guess New Guess ~g’(x i ) Initial Guesses

12. Secant It converges very fast!! (when it does) Slower than Newton Raphson Two Initial Guesses required May not bracket the root

13. Modified Secant Fractional perturbation  x i

14. Modified Secant Method Newton Raphson Fractional Perturbation Modified Secant