Chapter-04

# Chapter-04 - Truncation Errors and the Taylor Series...

This preview shows pages 1–5. Sign up to view the full content.

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

View Full Document

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Truncation Errors and the Taylor Series Chapter 4 • Non-elementary functions such as trigonometric, exponential, and others are expressed in an approximate fashion using Taylor series when their values, derivatives, and integrals are computed. • Any smooth function can be approximated as a polynomial. Taylor series provides a means to predict the value of a function at one point in terms of the function value and its derivatives at another point. x F(x) The approximation to F(x)=-0.1x 4-0.15x 3-0.5x 2-0.25x+1 at x=1, by 0 th , 1 st and 2 nd order Taylor expansion F(x i+1 ) = F(x i ) F(x i+1 )= F(x i )+F’(x i )h F(x i+1 )= F(x i )+F’(x i )h+1/2F’(x i )h 2 True value x i+1 =1 x i =0 h = x i+1- x i Example: To get the cos (x) for small x: If x=0.5 cos (0.5) =1-0.125+0.0026041-0.0000127+ … =0.877582 From the supporting theory, for this series, the error is no greater than the first omitted term. L +- +- = ! 6 ! 4 ! 2 1 cos 6 4 2 x x x x 0000001 . 5 . ! 8 8 = = ∴ ξ φορ ξ • Any smooth function can be approximated as a polynomial. f(x i+1 ) ≈ f(x i ) zero order approximation, only true if x i+1 and x i are very close to each other....
View Full Document

## This note was uploaded on 02/15/2010 for the course CHEN 320 taught by Professor Staff during the Spring '08 term at Texas A&M.

### Page1 / 14

Chapter-04 - Truncation Errors and the Taylor Series...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online