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Number Representation and Accuracy in Computing Systems

Learn about number representation, floating-point accuracy, significant digits, and the impact on precision and error in computing systems. Explore Taylor's theorem for mathematical approximation. Understand rounding, chopping, and machine number limitations.

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Number Representation and Accuracy in Computing Systems

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  1. Lecture 2Number Representation and accuracy Number Representation Normalized Floating Point Representation Significant Digits Accuracy and Precision Rounding and Chopping Reading assignment: Chapter 2 (c)Al-Dhaifallah 1435

  2. Representing Real Numbers • You are familiar with the decimal system • Decimal System Base =10 , Digits(0,1,…9) • Standard Representations (c)Al-Dhaifallah 1435

  3. Normalized Floating Point Representation • Normalized Floating Point Representation • No integral part, • AdvantageEfficient in representing very small or very large numbers (c)Al-Dhaifallah 1435

  4. Binary System • Binary System Base=2, Digits{0,1} (c)Al-Dhaifallah 1435

  5. 7-Bit Representation(sign: 1 bit, Mantissa 3bits,exponent 3 bits) (c)Al-Dhaifallah 1435

  6. Fact • Number that have finite expansion in one numbering system may have an infinite expansion in another numbering system • You can never represent 0.1 exactly in any computer (c)Al-Dhaifallah 1435

  7. Representation • Hypothetical Machine (real computers use ≥ 23 bit mantissa) • Example: If a machine has 5 bits representation distributed as follows Mantissa 2 bits exponent 2 bit sign 1 bit Possible machine numbers (0.25=00001) (0.375= 01111) (1.5=00111) (c)Al-Dhaifallah 1435

  8. Representation Gap near zero (c)Al-Dhaifallah 1435

  9. Remarks • Numbers that can be exactly represented are called machine numbers • Difference between machine numbers is not uniform. So, sum of machine numbers is not necessarily a machine number 0.25 + .375 =0.625 (not a machine number) (c)Al-Dhaifallah 1435

  10. Significant Digits • Significant digits are those digits that can be used with confidence. 0 1 2 3 4 Length of green rectangle = 3.45 significant (c)Al-Dhaifallah 1435

  11. Loss of Significance • Mathematical operations may lead to reducing the number of significant digits 0.123466 E+02 6 significant digits ─ 0.123445 E+02 6 significant digits ────────────── 0.000021E+02 2 significant digits 0. 210000E-02 Subtracting nearly equal numbers causes loss of significance (c)Al-Dhaifallah 1435

  12. Accuracy and Precision • Accuracy is related to closeness to the true value • Precision is related to the closeness to other estimated values (c)Al-Dhaifallah 1435

  13. Accuracy and Precision Better Precision Accuracy is related to closeness to the true value Precision is related to the closeness to other estimated values Better accuracy (c)Al-Dhaifallah 1435

  14. Rounding and Chopping • Rounding: Replace the number by the nearest machine number • Chopping: Throw all extra digits True 1.1681 0 1 2 Rounding (1.2) Chopping (1.1) (c)Al-Dhaifallah 1435

  15. Error DefinitionsTrue Error can be computed if the true value is known (c)Al-Dhaifallah 1435

  16. Error DefinitionsEstimated error Used when the true value is not known (c)Al-Dhaifallah 1435

  17. Notation We say the estimate is correct to n decimal digits if We say the estimate is correct to n decimal digits rounded if (c)Al-Dhaifallah 1435

  18. Summary • Number Representation Number that have finite expansion in one numbering system may have an infinite expansion in another numbering system. • Normalized Floating Point Representation • Efficient in representing very small or very large numbers • Difference between machine numbers is not uniform • Representation error depends on the number of bits used in the mantissa. (c)Al-Dhaifallah 1435

  19. Summary • Rounding Chopping • Error Definitions: • Absolute true error • True Percent relative error • Estimated absolute error • Estimated percent relative error (c)Al-Dhaifallah 1435

  20. Lecture 3Taylor Theorem Motivation Taylor Theorem Examples Reading assignment: Chapter 4 (c)Al-Dhaifallah 1435

  21. Motivation • We can easily compute expressions like b a 0.6 (c)Al-Dhaifallah 1435

  22. Taylor Series (c)Al-Dhaifallah 1435

  23. Taylor SeriesExample 1 (c)Al-Dhaifallah 1435

  24. Taylor SeriesExample 1 (c)Al-Dhaifallah 1435

  25. Taylor SeriesExample 2 (c)Al-Dhaifallah 1435

  26. (c)Al-Dhaifallah 1435

  27. Convergence of Taylor Series(Observations, Example 1) • The Taylor series converges fast (few terms are needed) when x is near the point of expansion. If |x-c| is large then more terms are needed to get good approximation. (c)Al-Dhaifallah 1435

  28. Taylor SeriesExample 3 (c)Al-Dhaifallah 1435

  29. Example 3remarks • Can we apply Taylor series for x>1?? • How many terms are needed to get good approximation??? These questions will be answered using Taylor Theorem (c)Al-Dhaifallah 1435

  30. Taylor Theorem (n+1) termsTruncated Taylor Series Reminder (c)Al-Dhaifallah 1435

  31. Taylor Theorem (c)Al-Dhaifallah 1435

  32. Error Term (c)Al-Dhaifallah 1435

  33. Example 4(The Approximation Error) (c)Al-Dhaifallah 1435

  34. Example 5 (c)Al-Dhaifallah 1435

  35. Example 5 (c)Al-Dhaifallah 1435

  36. Example 5Error term (c)Al-Dhaifallah 1435

  37. Alternative form of Taylor Theorem (c)Al-Dhaifallah 1435

  38. Taylor TheoremAlternative forms (c)Al-Dhaifallah 1435

  39. Derivative Mean-Value Theorem (c)Al-Dhaifallah 1435

  40. Alternating Series Theorem Alternating Series is special case of Taylor Series. (c)Al-Dhaifallah 1435

  41. Alternating SeriesExample 6 (c)Al-Dhaifallah 1435

  42. Remark • In this course all angles are assumed to be in radian unless you are told otherwise (c)Al-Dhaifallah 1435

  43. Maclurine series Find Maclurine Maclurine series expansion of cos (x) Maclurine series is a special case of Taylor series with the center of expansion c = 0 (c)Al-Dhaifallah 1435

  44. Taylor SeriesExample 7 (c)Al-Dhaifallah 1435

  45. Taylor SeriesExample 8 (c)Al-Dhaifallah 1435

  46. Taylor SeriesExample 8 (c)Al-Dhaifallah 1435

  47. Summary • Taylor series expansion is very important in most numerical methods applications approximation remainder • Remainder can be used to • estimate approximation error or • estimate the number of terms to achieve desirable accuracy (c)Al-Dhaifallah 1435

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