Post on 26-Jun-2015
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Base Agnostic Approximations of Logarithms
Josh WoodyUniversity of Evansville
Presented at MESCON 2011
Overview
• Motivation• Approximation Techniques• Applications• Conclusions
Motivation
• Big “Oh” notation– Compares growth of functions– Common classes are– How does fit? Compared to or ?
• Other Authors– Topic barely addressed in texts
𝑂 (1 ) ,𝑂 (𝑛) ,𝑂 (𝑛 log𝑛) ,𝑂 (𝑛2 ) ,𝑂 (2𝑛)
Approximation Technique 1
• Integration– Integrate the log function
– Note that log x is still present, presenting recursion
– Did not pursue further
Approximation Technique 2
• Derivation– Derive the log function
–What if we twiddle with the exponent by ±.01 and integrate?
Approximation 2 Results• Error at x = 50
is ±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 “minimax” 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 best𝑔 (𝑥 )=100 𝑥0.01−100
Applications
• Suppose two algorithms run in and
• Which is faster?• Since , the algorithm is faster.
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?