1. Lecture 2 Number Representation and accuracy
Normalized Floating Point Representation
Significant Digits
Accuracy and Precision
Rounding and Chopping
Reading assignment: Chapter 2
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2. Representing Real Numbers
You are familiar with the decimal system
Decimal System Base =10 , Digits(0,1,…9)
Standard Representations
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3. Normalized Floating Point Representation
No integral part,
Advantage Efficient in representing very small or very large numbers
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4. Binary System Binary System Base=2, Digits{0,1} EE 3561_Unit_1
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5. 7-Bit Representation (sign: 1 bit, Mantissa 3bits,exponent 3 bits)
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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
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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)
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8. Representation
Gap near zero
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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
= (not a machine number)
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10. Significant Digits
Significant digits are those digits that can be used with confidence.
Length of green rectangle = 3.45
significant
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11. Loss of Significance
Mathematical operations may lead to reducing the number of significant digits
E significant digits
─ E significant digits
──────────────
E significant digits
E-02
Subtracting nearly equal numbers causes loss of significance
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12. Accuracy and Precision
Accuracy is related to closeness to the true value
Precision is related to the closeness to other estimated values
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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
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14. Rounding and Chopping Rounding (1.2) Chopping (1.1)
Rounding: Replace the number by the nearest
machine number
Chopping: Throw all extra digits
True
Rounding (1.2) Chopping (1.1)
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15. Error Definitions True Error
can be computed if the true value is known
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16. Error Definitions Estimated error
Used when the true value is not known
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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
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18. Summary Number that have finite expansion in one numbering system may
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.
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19. Summary Rounding Chopping Error Definitions: Absolute true error
True Percent relative error
Estimated absolute error
Estimated percent relative error
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20. Lecture 3 Taylor Theorem
Motivation
Taylor Theorem
Examples
Reading assignment: Chapter 4
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21. Motivation We can easily compute expressions like b a 0.6
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22. Taylor Series
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23. Taylor Series Example 1
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24. Taylor Series Example 1
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25. Taylor Series Example 2
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26. EE 3561_Unit_1
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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.
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28. Taylor Series Example 3
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29. Example 3 remarks 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
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30. Taylor Theorem (n+1) terms Truncated Taylor Series Reminder
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31. Taylor Theorem
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32. Error Term
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33. Example 4 (The Approximation Error)
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34. Example 5
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35. Example 5
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36. Example 5 Error term
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37. Alternative form of Taylor Theorem
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38. Taylor Theorem Alternative forms
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39. Derivative Mean-Value Theorem
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40. Alternating Series Theorem
Alternating Series is special case of Taylor Series.
This means that the error is less or equal to the magnitude of the first omitted term in the alternating series.
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41. Alternating Series Example 6
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42. Remark
In this course all angles are assumed to be in radian unless you are told otherwise
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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
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44. Taylor Series Example 7
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45. Taylor Series Example 8
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46. Taylor Series Example 8
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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
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