Lectures16and17 - The Big-O Order Symbol: Another Way to...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: The Big-O Order Symbol: Another Way to Think of Error To motivate the concept we're about to introduce, consider the following sit- uation: suppose you've shown that the 4 th-order Maclaurin polynomial for a certain function is, say, 1- x 3 + 1 10 x 4 . Suppose also that you go ahead and calculate that the 5 th derivative of this function is zero at zero, so that this same expression is also the 5 th-order Maclaurin polynomial. How would you incorporate this extra piece of information in your work? Well, in the notation we've used in this course we have a fairly e cient way to do this: we could state that f ( x ) 1- x 3 + 1 10 x 4 + R 5 , ( x ) . Even if we don't feel the need to proceed with nding an upper bound on the magnitude of the error term, the fact that we've written R 5 , shows clearly that we know the x 5 term to be zero. What we are about to introduce is essentially an alternative to that notation, which will be easier to work with and is universally recognized. De nition: Given two functions f and g, we say that f is of order g as x x and write f ( x ) = O ( g ( x )) as x x (1) if there exists a constant A (greater than zero) such that | f ( x ) | A | g ( x ) | on some interval around x (although the point x itself may be excluded from the interval, since the idea is to describe the behaviour of f in the limit as we approach x ). For our purposes the function g will be a power of ( x- x ) , and of course if we're dealing with Maclaurin series then it will simply be a power of x . Note: you might be familiar with a similar concept used in computer science, which deals with behaviour as n instead of x x . Examples: a) Since we know that | x 3 | | x 2 | for all x [- 1 , 1] , we can state that x 3 = O ( x 2 ) as x . Since it is also true that | x 3 | | x | on the same interval, we can also state that x 3 = O ( x ) as x . Both statements are equally correct (just look at our de nition; in these examples the constant A is just 1. In fact, we can similarly state that x 3 = O ( x 3 ) as x , and x 3 = O (1) as x (this last statement just says that x 3 is bounded near zero; | x 3 | A for some number A when x is small enough)....
View Full Document

This note was uploaded on 04/05/2010 for the course MATH 119 taught by Professor Harmsworth during the Spring '08 term at Waterloo.

Page1 / 6

Lectures16and17 - The Big-O Order Symbol: Another Way to...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online