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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Ohβ notation<br />Compares growth of functions<br />Common classes are<br />How does π(πlogπ)fit? Compared to ππ1.5 or π(π)?<br />Other Authors<br />Topic barely addressed in texts<br />Β <br />π1,Β ππ,Β ππlogπ,Β ππ2,Β π(2π)<br />Β <br />
4. 4. Approximation Technique 1<br />Integration<br />Integrate the log function<br />πΉπ₯=Β ππ₯ππ₯=Β logπ₯ππ₯=π₯πππΒ π₯Β βπ₯+πΆ<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 />πβ²π₯=1π₯=π₯β1Β <br />What if we twiddle with the exponent by Β±.01 and integrate?<br />ππ₯=100π₯0.01β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minimaxβ 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 />ππ₯=100π₯0.01β100Β <br />Β <br />
12. 12. Applications<br />Suppose two algorithms run in π(πlogπ)and π(π1.5)<br />Which is faster?<br />Since logΒ π=ππ0.01, theπ(πlogπΒ ) 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 />