The Look and Feel of Zero

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The Look and Feel of Zero

• Dr. D. E. (Steve) Stevenson
• School of Computing Clemson
• Steve_at_cs.clemson.edu

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Contents

• Expectation
• Wilkinson Numbers
• IEEE 754/854 Floating Point
• Cautionary Tale
• When Zero is not Zero

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Students Viewpoint
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Student Expectations

• The students expect the numbers to work like they
  did in high school and the way the instructors
  pretend on the board.

5
Experiment

• Folk theorem Conversion between rectangular and
  polar coordinates is perfect.
• Constructive reality It aint.
• Well come back in a second

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Wilkinson Numbers
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Wilkinson and Kahan

• James H. Wilkinson (1950s) was the first to
  propose computing numbers be like scientific
  notation.
• IEEE finally standardized floating point
  arithmetic (well sorta) in the mid-1980s after
  much work by William Kahan.
• Ill call the theoretical set Wilkinson Numbers.

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Wilkinson Numbers for (b,p,e)

• The number w 0.0 plus the numbers

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?, ?, and ?

• Let b, p, and e(emin, emax) be integers. All the
  numbers w in the pattern

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Explore
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Answers

• The total range is 0.10010-2 to 0.999102.
• 990 mantissas spread from 0.100 to 0.999
• Logarithmically distributed.

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Error Basics

• For real f(x), denote the approximation by
• Error
• Absolute error
• Relative error

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For Functions,

• For functions, we have
• In, other words,

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Examples to Try
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Condition Numbers

• How hard is it to compute a number given the rule
  f and a slight change in the parameter x?
• The familiar linear algebra concept can be
  expanded.

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Condition Number
This last expression is called the condition
number of y relative to x
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Examples to Try
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The Trick
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Arithmetic
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Beware of Plus/Minus
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Beware the Approximations
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Fun with ?

• The real value truncated to 50 digits is 3.14159
  26535 89793 23846 26433 83279 50288 41971 69399
  37510
• How good an approximation is 22/7.
• Is 355/113 better than 22/7?

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Now, For What Values
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Finally, Polar-Rectangular

• 10,000,000 trials in box 1,000x1,000.
• Random points, convert to polar and invert.
• 1/3 (approx) were correct.
• Of the rest
• Mean is 357e
• Standard Deviation is 210e