Mescon logarithms
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Mescon logarithms

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Mescon logarithms Mescon logarithms Presentation Transcript

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