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Sources of Error

Sources of Error. Major: All Engineering Majors Authors: Autar Kaw, Luke Snyder http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates. Sources of Error http://numericalmethods.eng.usf.edu. Two sources of numerical error. Round off error

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Sources of Error

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  1. 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 Errorhttp://numericalmethods.eng.usf.edu

  3. Two sources of numerical error • Round off error • Truncation error

  4. Round-off Errorhttp://numericalmethods.eng.usf.edu

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

  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. 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. 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. Effect of Carrying Significant Digits in Calculationshttp://numericalmethods.eng.usf.edu

  10. Find the contraction in the diameter Ta=80oF; Tc=-108oF; D=12.363” α = a0+ a1T + a2T2

  11. Thermal Expansion Coefficient vs Temperature

  12. Regressing Data in Excel(general format) α = -1E-05T2 + 0.0062T + 6.0234

  13. Observed and Predicted Values α = -1E-05T2 + 0.0062T + 6.0234

  14. Regressing Data in Excel (scientific format) α = -1.2360E-05T2 + 6.2714E-03T + 6.0234

  15. Observed and Predicted Values α = -1.2360E-05T2 + 6.2714E-03T + 6.0234

  16. Observed and Predicted Values α = -1.2360E-05T2 + 6.2714E-03T + 6.0234 α = -1E-05T2 + 0.0062T + 6.0234

  17. THE END

  18. Truncation Errorhttp://numericalmethods.eng.usf.edu

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

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

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

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

  23. Example 1 —Maclaurin series Calculate the value of with an absolute relative approximate error of less than 1%. 6 terms are required. How many are required to get at least 1 significant digit correct in your answer?

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

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

  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_error.html

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

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