1. 10/16/20151 Sources of Error Major: All Engineering Majors Authors: Autar Kaw, Luke Snyder http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates

2. Sources of Error http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu

3. 3 Two sources of numerical error 1)Round off error 2)Truncation error

4. 4 Round-off Error http://numericalmethods.eng.usf.edu

5. 5 Round off Error Caused by representing a number approximately

6. 6 Problems created by round off error 28 Americans were killed on February 25, 1991 by an Iraqi Scud missile in Dhahran, Saudi Arabia. The patriot defense system failed to track and intercept the Scud. Why?

7. 7 Problem with Patriot missile Clock cycle of 1/10 seconds was represented in 24-bit fixed point register created an error of 9.5 x 10 -8 seconds. The battery was on for 100 consecutive hours, thus causing an inaccuracy of

8. 8 Problem (cont.) The shift calculated in the ranging system of the missile was 687 meters. The target was considered to be out of range at a distance greater than 137 meters.

9. 9 Effect of Carrying Significant Digits in Calculations http://numericalmethods.eng.usf.edu

10. 10 Find the contraction in the diameter T a =80 o F; T c =-108 o F; D=12.363” α = a 0 + a 1 T + a 2 T 2

11. 11 Thermal Expansion Coefficient vs Temperature T( o F)α (μin/in/ o F) -3402.45 -3003.07 -2204.08 -1604.72 -805.43 06.00 406.24 806.47

12. 12 Regressing Data in Excel (general format) α = -1E-05T 2 + 0.0062T + 6.0234

13. 13 Observed and Predicted Values T( o F)α (μin/in/ o F) Given α (μin/in/ o F) Predicted -3402.452.76 -3003.073.26 -2204.084.18 -1604.724.78 -805.435.46 06.006.02 406.246.26 806.476.46 α = -1E-05T 2 + 0.0062T + 6.0234

14. 14 Regressing Data in Excel (scientific format) α = -1.2360E-05T 2 + 6.2714E-03T + 6.0234

15. 15 Observed and Predicted Values T( o F)α (μin/in/ o F) Given α (μin/in/ o F) Predicted -3402.452.46 -3003.073.03 -2204.084.05 -1604.724.70 -805.435.44 06.006.02 406.246.25 806.476.45 α = -1.2360E-05T 2 + 6.2714E-03T + 6.0234

16. 16 Observed and Predicted Values T( o F) α (μ in/in/ o F) Given α (μ in/in/ o F) Predicted α (μ in/in/ o F) Predicted -3402.452.462.76 -3003.073.033.26 -2204.084.054.18 -1604.724.704.78 -805.435.445.46 06.006.02 406.246.256.26 806.476.456.46 α = -1.2360E-05T 2 + 6.2714E-03T + 6.0234 α = -1E-05T 2 + 0.0062T + 6.0234

17. 17 THE END

18. 18 Truncation Error http://numericalmethods.eng.usf.edu

19. 19 Truncation error Error caused by truncating or approximating a mathematical procedure.

20. 20 Example of Truncation Error Taking only a few terms of a Maclaurin series to approximate If only 3 terms are used,

21. 21 Another Example of Truncation Error Using a finiteto approximate P Q secant line tangent line Figure 1. Approximate derivative using finite Δx

22. 22 Another Example of Truncation Error Using finite rectangles to approximate an integral.

23. 23 Example 1 —Maclaurin series Calculate the value ofwith an absolute relative approximate error of less than 1%. n 11_____ 22.21.254.545 32.920.7224.658 43.2080.2888.9776 53.29440.08642.6226 63.31510.0207360.62550 6 terms are required. How many are required to get at least 1 significant digit correct in your answer?

24. 24 Example 2 —Differentiation Findforusing and The actual value is Truncation error is then, Can you find the truncation error with

25. 25 Example 3 — Integration Use two rectangles of equal width to approximate the area under the curve for over the interval

26. 26 Integration example (cont.) Choosing a width of 3, we have Actual value is given by Truncation error is then Can you find the truncation error with 4 rectangles?

27. Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit http://numericalmethods.eng.usf.edu/topics/sources_of_err or.html

28. THE END http://numericalmethods.eng.usf.edu