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?