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# Mescon logarithms

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### Transcript

• 1. Base Agnostic Approximations of Logarithms
Josh Woody
University of Evansville
Presented at MESCON 2011
• 2. Overview
Motivation
Approximation Techniques
Applications
Conclusions
• 3. Motivation
Big βOhβ notation
Compares growth of functions
Common classes are
How does π(πlogπ)fit? Compared to ππ1.5 or π(π)?
Other Authors
Β
π1,Β ππ,Β ππlogπ,Β ππ2,Β π(2π)
Β
• 4. Approximation Technique 1
Integration
Integrate the log function
πΉπ₯=Β ππ₯ππ₯=Β logπ₯ππ₯=π₯πππΒ π₯Β βπ₯+πΆ
Note that log x is still present, presenting recursion
Did not pursue further
Β
• 5. Approximation Technique 2
Derivation
Derive the log function
πβ²π₯=1π₯=π₯β1Β
What if we twiddle with the exponent by Β±.01 and integrate?
ππ₯=100π₯0.01β100Β
Β
• 6. Approximation 2 Results
Error at x = 50 is Β±4.2%
Error grows with increasing x
Can be reduced with more significant figures
• 7. Approximation Technique 3
Taylor Series
Infinite series
Reasonable approximation truncates series
Argument must be < 1 to converge
• 8. Approximation 3 Results
Good approximation, even with only 3 terms
But approximation only valid for small region
• 9. Approximation Technique 4
Chebychev Polynomial
Infinite Series
Approximates βminimaxβ properties
Peak error is minimized in some interval
Slightly better convergence than Taylor
• 10. Approximation 4 Results
Can be shifted
Really bad approximation outside region of convergence
Good approximation inside
• 11. Conclusions
Infinite series not well suited to task
Too much error in portions of number line
Derivation attempt is best
ππ₯=100π₯0.01β100Β
Β
• 12. Applications
Suppose two algorithms run in π(πlogπ)and π(π1.5)
Which is faster?
Since logΒ π=ππ0.01, theπ(πlogπΒ ) algorithm is faster.
Β
• 13. What base is that?
Base in this presentation is always e.
Base conversion was insignificant portion of work
Change of Base formula always sufficient
• 14. The End
Slides will be posted on JoshWoody.com tonight