1. NUMERICAL ERRORStudent Notes ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier

3. Objectives • To understand error terms • Become familiar with notation and techniques used in this course

4. Approximation and ErrorsSignificant Figures • 4 significant figures • 1.845 • 0.01845 • 0.0001845 • 43,500 ? confidence • 4.35 x 104 3 significant figures • 4.350 x 104 4 significant figures • 4.3500 x 104 5 significant figures

5. Accuracy and Precision • Accuracy - how closely a computed or measured value agrees with the true value • Precision - how closely individual computed or measured values agree with each other • number of significant figures • spread in repeated measurements or computations

6. Accuracy and Precision increasing accuracy increasing precision

7. Error Definitions • Numerical error - use of approximations to represent exact mathematical operations and quantities • true value = approximation + error • error, et=true value - approximation • subscript trepresents the true error • shortcoming....gives no sense of magnitude • normalize by true value to get true relative error

8. Error definitions cont. • True relative percent error

9. Example • Consider a problem where the true answer is 7.91712. If you report the value as 7.92, answer the following questions. • How many significant figures did you use? • What is the true error? • What is the true relative percent error?

10. Error definitions cont. • May not know the true answer apriori • This leads us to develop an iterative approach to numerical methods

11. Error definitions cont. • Usually not concerned with sign, but with tolerance • Want to assure a result is correct to nsignificant figures

12. Example Consider a series expansion to estimate trigonometric functions Estimate sin(p/ 2) to three significant figures. Calculate et and ea. STRATEGY

13. Strategy Stop when ea ≤ es

14. Error Definitions cont. • Round off error - originate from the fact that computers retain only a fixed number of significant figures • Truncation errors - errors that result from using an approximation in place of an exact mathematical procedure

15. Error Definitions cont. • Round off error - originate from the fact that computers retain only a fixed number of significant figures • Truncation errors - errors that result from using an approximation in place of an exact mathematical procedure To gain insight consider the mathematical formulation that is used widely in numerical methods - TAYLOR SERIES

16. TAYLOR SERIES • Provides a means to predict a function value at one point in terms of the function value at and its derivative at another point

17. TAYLOR SERIES Zero order approximation This is good if the function is a constant.

18. Taylor Series Expansion First order approximation slope multiplied by distance Still a straight line but capable of predicting an increase or decrease - LINEAR

19. Taylor Series Expansion Second order approximation - captures some of the curvature

20. Taylor Series Expansion

21. Taylor Series Expansion

22. Example Use zero through fourth order Taylor series expansion to approximate f(1) given f(0) = 1.2 (i.e. h = 1). Calculate et after each step. Note: f(1) = 0.2 STRATEGY

23. Strategy • Estimate the function using only the first term • Use x = 0 to estimate f(1), which is the y-value when x = 1 • Calculate error, et • Estimate the function using the first and second term • Calculate the error, et • Progressively add terms

24. Objectives • To understand error terms • Become familiar with notation and techniques used in this course