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Mescon logarithms

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Mescon logarithms

  1. 1. Base Agnostic Approximations of Logarithms<br />Josh Woody<br />University of Evansville<br />Presented at MESCON 2011<br />
  2. 2. Overview<br />Motivation<br />Approximation Techniques<br />Applications<br />Conclusions<br />
  3. 3. Motivation<br />Big β€œOh” notation<br />Compares growth of functions<br />Common classes are<br />How does 𝑂(𝑛log𝑛)fit? Compared to 𝑂𝑛1.5 or 𝑂(𝑛)?<br />Other Authors<br />Topic barely addressed in texts<br />Β <br />𝑂1, 𝑂𝑛, 𝑂𝑛log𝑛, 𝑂𝑛2, 𝑂(2𝑛)<br />Β <br />
  4. 4. Approximation Technique 1<br />Integration<br />Integrate the log function<br />𝐹π‘₯= 𝑓π‘₯𝑑π‘₯=Β logπ‘₯𝑑π‘₯=π‘₯π‘™π‘œπ‘”Β π‘₯Β βˆ’π‘₯+𝐢<br />Note that log x is still present, presenting recursion<br />Did not pursue further<br />Β <br />
  5. 5. Approximation Technique 2<br />Derivation<br />Derive the log function<br />𝑓′π‘₯=1π‘₯=π‘₯βˆ’1Β <br />What if we twiddle with the exponent by Β±.01 and integrate?<br />𝑔π‘₯=100π‘₯0.01βˆ’100Β <br />Β <br />
  6. 6. Approximation 2 Results<br />Error at x = 50 is Β±4.2%<br />Error grows with increasing x<br />Can be reduced with more significant figures<br />
  7. 7. Approximation Technique 3<br />Taylor Series<br />Infinite series<br />Reasonable approximation truncates series<br />Argument must be < 1 to converge<br />
  8. 8. Approximation 3 Results<br />Good approximation, even with only 3 terms<br />But approximation only valid for small region<br />
  9. 9. Approximation Technique 4<br />Chebychev Polynomial<br />Infinite Series<br />Approximates β€œminimax” properties<br />Peak error is minimized in some interval<br />Slightly better convergence than Taylor<br />
  10. 10. Approximation 4 Results<br />Centered about 0<br />Can be shifted<br />Really bad approximation outside region of convergence<br />Good approximation inside<br />
  11. 11. Conclusions<br />Infinite series not well suited to task<br />Too much error in portions of number line<br />Derivation attempt is best<br />𝑔π‘₯=100π‘₯0.01βˆ’100Β <br />Β <br />
  12. 12. Applications<br />Suppose two algorithms run in 𝑂(𝑛log𝑛)and 𝑂(𝑛1.5)<br />Which is faster?<br />Since log 𝑛=π‘œπ‘›0.01, the𝑂(𝑛log𝑛 ) algorithm is faster.<br />Β <br />
  13. 13. What base is that?<br />Base in this presentation is always e.<br />Base conversion was insignificant portion of work<br />Change of Base formula always sufficient<br />
  14. 14. The End<br />Slides will be posted on JoshWoody.com tonight<br />Questions, Concerns, or Comments?<br />

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