1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 5 Approximations, Errors and The Taylor Series

2. Objectives Distinguish between algorithm errors and roundoff errors. Introduce Taylor Theorem Calculate the numerical error of a finite difference formula for derivatives

3. Significant Figures

4. Designate the reliability of a numerical value The significant digits of a number are those that can be used with confidence

5. Accuracy and Precision Accuracy Precision

6. 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

7. Roundoff Errors A=  d 2

8. Roundoff Errors A1=  d 1 2 A2=  d 2 2 d1=1.0d2=1.000001 A1=3.14A2=3.14000628

9. Error Definition True Value = Approximation + Error Numerical Error E t =true value - approximation Does not account for order of magnitude

10. Error Definition True Value = Approximation + Error Relative Error = True Error/True Value  t = (True Error/True Value)100%

11. Approximate Error Numerical Methods: the true value is not known apriori

12. Approximate Error Numerical Methods: approximate procedures Mclaurin Series Exact

13. Approximate Error

14. The Taylor Series Predict value of a function at one point in terms of the function value and its derivatives at another point

15. Taylor’s Theorem

16. Error of Order (x i+1 – x i ) n+1

17. Numerical Differentiation

18. First Divided Difference

19. Forward Difference vivi titi t i+1 v i+1 True Slope Approximate Slope

20. Backward Difference

21. Central Difference Forward Backward Central

22. Homework Problems 4.5 and 4.6 Due Wednesday Sept 10