Base Agnostic Approximations of Logarithms Josh Woody University of Evansville Presented at MESCON 2011
Jun 26, 2015
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?