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Understanding Sources of Error in Engineering Calculations

This article discusses round-off and truncation errors in numerical methods, highlighting examples and their impact on precision. It emphasizes the importance of accurate calculations in engineering. Presented by A. Ranjbar.

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Understanding Sources of Error in Engineering Calculations

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  1. Sources of Errorمنابع خطا Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker Presented by: A. Ranjbar http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for Amol Undergraduate students دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  2. Sources of Errorمنابع خطاhttp://numericalmethods.eng.usf.edu دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  3. Two sources of numerical errorدو منبع خطا • Round off error: خطای روند کردن • Truncation error: خطای برش دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  4. Round-off Errorخطای روند کردنhttp://numericalmethods.eng.usf.edu دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  5. Round off Errorخطای روند کردن • Caused by representing a number approximately تقریب عدد: دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  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? دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  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 دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  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. دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  9. Effect of Carrying Significant Digits in Calculationsاثر کار کردن با بیت های با معنی در محاسباتhttp://numericalmethods.eng.usf.edu دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  10. Find the contraction in the diameterمحاسبه انبساط قطر Ta=80oF; Tc=-108oF; D=12.363” α = a0+ a1T + a2T2 دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  11. Thermal Expansion Coefficient vs Temperatureضریب انبساط حرارتی بر حسب دما دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  12. Regressing Data in Excel(general format)برازش در Excel α = -1E-05T^2 + 0.0062T + 6.0234 دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  13. Observed and Predicted Valuesمقادیر مشاهده شده و پیش بینی α = -1E-05T^2 + 0.0062T + 6.0234 دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  14. Regressing Data in Excel (scientific format) α = -1.2360E-05T^2 + 6.2714E-03T + 6.0234 دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  15. Observed and Predicted Values α = -1.2360E-05T^2 + 6.2714E-03T + 6.0234 دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  16. Observed and Predicted Values SCI: α = -1.2360E-05T2 + 6.2714E-03T + 6.0234 Excel : α = -1E-05T2 + 0.0062T + 6.0234 دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  17. دقت بیشتر به صحت نتایج کمک میکند. THE END دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  18. Truncation Errorخطای برش عددhttp://numericalmethods.eng.usf.edu دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  19. Truncation error • Error caused by truncating or approximating a mathematical procedure. توسط برش یا تقریب یک روال ریاضی بدست می آید. دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  20. Example of Truncation Errorمثال: در نظر گرفتن ترمهای کمتر Taking only a few terms of a Maclaurin series to approximate If only 3 terms are used, دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  21. secant line P tangent line Q Figure 1. Approximate derivative using finite Δx Another Example of Truncation Errorتقریب دلتا برای مشتق Using a finite to approximate دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  22. Another Example of Truncation Errorتقریب انتگرال با مستطیل پایین Using finite rectangles to approximate an integral. دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  23. Example 1 —Maclaurin series Calculate the value of with an absolute relative approximate error of less than 1%. تعداد 6 ترم نیاز است تا با دقت 1 درصد قابل نمایش باشد. دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 6 terms are required. How many are required to get at least 1 significant digit correct in your answer?

  24. Example 2 —Differentiationخطای محاسبه دیفرانسیل: آیا میتوان خطای قطع عدد به ازا را بدست آورد؟ Find for using and The actual value is دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 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 دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  26. Integration example (cont.) Choosing a width of 3, we have Actual value is given by دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل 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 دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

  28. THE END http://numericalmethods.eng.usf.edu دکتر ابوالفضل رنجبر نوعی 89-90-2-آمل- آمل

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