1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 6 Roots of Equations Bracketing Methods

2. Last Time - Accuracy and Precision Accuracy Precision

3. Last Time - Truncation Errors vivi titi t i+1 v i+1 True Slope Approximate Slope Truncation errors due to using approximation in place of exact solution

4. Last Time - Roundoff Errors A=  d 2

5. Last Time - Error Definition E t =true value - approximation True Error  t = (E t /True Value)100% Relative True Error

6. Last Time - Error Definition Approximate Relative Error Iteration Relative Error

7. Last Time - The Taylor Series Predict value of a function at one point in terms of the function value and its derivatives at another point

8. Last Time - Taylor’s Theorem Error of Order (x i+1 – x i ) n+1

9. Last Time - Numerical Differentiation Forward DifferenceBackward DifferenceCentral Difference

10. The Problem Analytic Solution

11. The Problem To design the parachute: v=10 m/s t=3 sec m=64 kg g=9.81 c=? CANNOT rearrange to solve for c

12. The Problem Define Function c must satisfy c is the ROOT of the equation

13. Objectives Master methods to compute roots of equations Assess reliability of each method Choose best method for a specific problem

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

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

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

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

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

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

20. Bisection Method Check Convergence Root = If Error

21. 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

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

23. Bisection - Flowchart Loop x old =x x=(x l +x u )/2 Error=100*abs(x-x old )/x Sign=f(x l )*f(x r ) Sign x u =xx l =xError=0 Error<E all ROOT=x FALSE <0>0

24. Pseudo Code