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Understanding Numerical Methods: Error Analysis and Function Representation

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rachel
41 diapositives · 559 Views · Oct 12, 2014

This guide provides an introduction to numerical methods, focusing on approximation, errors, and function representations. It explores key concepts such as differentiation, integration, and the sources of error in calculations. Learn about significant digits, rounding errors, and the impact of binary representation on numerical accuracy. The guide also discusses the Taylor series and its relevance in numerical approximations. Perfect for engineering students or anyone interested in mastering numerical methods in computations.

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  1. NCR Register http://numericalmethods.eng.usf.edu

  2. INTRODUCTION, APPROXIMATION AND ERRORS http://numericalmethods.eng.usf.edu

  3. Identify the picture of your instructor http://numericalmethods.eng.usf.edu

  4. 01.01INTRODUCTION http://numericalmethods.eng.usf.edu

  5. To find velocity from acceleration vs time data, the mathematical procedure used is • Differentiation • Integration http://numericalmethods.eng.usf.edu

  6. The form of the exact solution to is • . • . • . • . http://numericalmethods.eng.usf.edu

  7. y 5 a b 2 c x 7 Given the f (x) vs x curve, and the magnitude of the areas as shown, the value of • -2 • 2 • 12 • Cannot be determined http://numericalmethods.eng.usf.edu

  8. A steel cylindrical shaft at room temperature is immersed in a dry-ice/alcohol bath. A layman estimates the reduction in diameter by using while using the value of the thermal expansion coefficient at -108oF. Seeing the graph below, the magnitude of contraction you as a USF educated engineer would calculate would be ______________than the layman’s estimate. • Less • More • Same http://numericalmethods.eng.usf.edu

  9. END http://numericalmethods.eng.usf.edu

  10. 01.02MEASURINGERRORS http://numericalmethods.eng.usf.edu

  11. The number of significant digits in 2.30500 is • 3 • 4 • 5 • 6 http://numericalmethods.eng.usf.edu

  12. The absolute relative approximate error in an iterative process at the end of the tenth iteration is 0.007%. The least number of significant digits correct in the answer is • 2 • 3 • 4 • 5 http://numericalmethods.eng.usf.edu

  13. Three significant digits are expected to be correct after an iterative process. The pre-specified tolerance in this case needs to be less than or equal to • 0.5% • 0.05% • 0.005% • 0.0005% http://numericalmethods.eng.usf.edu

  14. 01.03SOURCES OF ERROR http://numericalmethods.eng.usf.edu

  15. Dreaming http://numericalmethods.eng.usf.edu

  16. The error caused by representing numbers such as 1/3 approximately is called • Round-off error • Truncation error http://numericalmethods.eng.usf.edu

  17. The number 6.749832 with 3 significant digits with rounding is • 6.74 • 6.75 • 6.749 • 6.750 http://numericalmethods.eng.usf.edu

  18. The error caused by using only a few terms of the Maclaurin series to calculate ex results mostly in • Truncation Error • Round off Error http://numericalmethods.eng.usf.edu

  19. The number 6.749832 with 3 significant digits with chopping is • 6.74 • 6.75 • 6.749 • 6.750 http://numericalmethods.eng.usf.edu

  20. END http://numericalmethods.eng.usf.edu

  21. 01.04BINARY REPRESENTATION http://numericalmethods.eng.usf.edu

  22. This is what you have been saying about your TI-30Xa • I don't care what people say The rush is worth the priceI pay I get so high when you're with meBut crash and crave you when you are away • Give me back now my TI89Before I start to drink and whine TI30Xa calculators make me cryIncarnation of of Jason will you ever die • TI30Xa – you make me forget the high maintenance TI89. • I never thought I will fall in love again! http://numericalmethods.eng.usf.edu

  23. (8)10=(?)2 • 1110 • 1011 • 0100 • 1000 http://numericalmethods.eng.usf.edu

  24. The binary representation of (0.3)10 is • (0.01001……...)2 • (0.10100……...)2 • (0.01010……...)2 • (0.01100……...)2 http://numericalmethods.eng.usf.edu

  25. 01.05FLOATING POINT REPRESENTATION http://numericalmethods.eng.usf.edu

  26. Smallest positive number in a 7 bit word where 1st bit is used for sign of number, 2nd bit for sign of exponent, 3 bits for mantissa and 2 bits for exponent • 0.000 • 0.125 • 0.250 • 1.000 http://numericalmethods.eng.usf.edu

  27. Five bits are used for the biased exponent. To convert a biased exponent to an unbiased exponent, you would • add 7 • subtract 7 • add 15 • subtract 15 http://numericalmethods.eng.usf.edu

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

  29. 01.07TAYLOR SERIES http://numericalmethods.eng.usf.edu

  30. Taylor series • if values of h are small • if function and all its derivatives are defined and continuous at x • if function and all its derivatives are defined and continuous in [x,x+h] is only valid http://numericalmethods.eng.usf.edu

  31. END http://numericalmethods.eng.usf.edu

  32. (01011)2 =(?)10 • 7 • 11 • 15 • 22 http://numericalmethods.eng.usf.edu

  33. The machine epsilon in a 7 bit number where 1st bit is used for sign of number, 2nd bit for sign of exponent, 3 bits for mantissa and 2 bits for exponent • 0.125 • 0.25 • 0.5 • 1.0 http://numericalmethods.eng.usf.edu

  34. The number of significant digits in 0.0023406 is • 4 • 5 • 6 • 7 http://numericalmethods.eng.usf.edu

  35. The number of significant digits in 2350 is • 3 • 4 • 5 • 3 or 4 http://numericalmethods.eng.usf.edu

  36. To find velocity from location vs time data of the body, the mathematical procedure used is • Differentiation • Integration http://numericalmethods.eng.usf.edu

  37. Given y= sin(2x), dy/dx at x=3 is • 0.9600 • 0.9945 • 1.920 • 1.989 http://numericalmethods.eng.usf.edu

  38. In a five bit fixed representation, (0.1)10 is represented as (0.00011)2. The true error in this representation most nearly is • 0.00625 • 0.053125 • 0.09375 • 9.5x10-8 http://numericalmethods.eng.usf.edu

  39. What will Maury Povich say when he dies, goes to heaven, and sees God? • Why am I here; I should be with the lawyers. • Is Connie here? • You are the Father. • You are not the father. http://numericalmethods.eng.usf.edu

  40. I walk like a pimp – Jeremy Reed You know it's hard out here for a pimp, When he tryin to get this money for the rent, For the Cadillacs and gas money spent

  41. Largest positive number in a 7 bit number where 1st bit is used for sign of number, 2nd bit for sign of exponent, 3 bits for mantissa and 2 bits for exponent • 1.875 • 4 • 7 • 15 http://numericalmethods.eng.usf.edu