 # ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 5 Approximations, Errors and The Taylor Series.

## Presentation on theme: "ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 5 Approximations, Errors and The Taylor Series."— Presentation transcript:

ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 5 Approximations, Errors and The Taylor Series

Objectives Distinguish between algorithm errors and roundoff errors. Introduce Taylor Theorem Calculate the numerical error of a finite difference formula for derivatives

Significant Figures

Designate the reliability of a numerical value The significant digits of a number are those that can be used with confidence

Accuracy and Precision Accuracy Precision

Truncation Errors vivi titi t i+1 v i+1 True Slope Approximate Slope Truncation errors due to using approximation in place of exact solution

Roundoff Errors A=  d 2

Roundoff Errors A1=  d 1 2 A2=  d 2 2 d1=1.0d2=1.000001 A1=3.14A2=3.14000628

Error Definition True Value = Approximation + Error Numerical Error E t =true value - approximation Does not account for order of magnitude

Error Definition True Value = Approximation + Error Relative Error = True Error/True Value  t = (True Error/True Value)100%

Approximate Error Numerical Methods: the true value is not known apriori

Approximate Error Numerical Methods: approximate procedures Mclaurin Series Exact

Approximate Error

The Taylor Series Predict value of a function at one point in terms of the function value and its derivatives at another point

Taylor’s Theorem

Error of Order (x i+1 – x i ) n+1

Numerical Differentiation

First Divided Difference

Forward Difference vivi titi t i+1 v i+1 True Slope Approximate Slope

Backward Difference

Central Difference Forward Backward Central

Homework Problems 4.5 and 4.6 Due Wednesday Sept 10

Download ppt "ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 5 Approximations, Errors and The Taylor Series."

Similar presentations