round off and error - chapter 4chapter4rev1

7/29/2019 Round Off and Error - Chapter 4Chapter4Rev1

1/20

Part 1Chapter 4

Roundoff and

Truncation Errors

PowerPoints organized by Dr. Michael R. Gustafson II, Duke University and

Prof. Steve Chapra, Tufts UniversityAll images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.


2/20

Chapter Objectives

Understanding the distinction between accuracyand precision.

Learning how to quantify error.

Learning how error estimates can be used to decidewhen to terminate an iterative calculation.

Understanding how roundoff errors occur because

digital computers have a limited ability to represent

numbers.

Understanding why floating-point numbers have

limits on their range and precision.


3/20

Objectives (cont)

Recognizing that truncation errors occur whenexact mathematical formulations are represented

by approximations.

Knowing how to use the Taylor series to estimatetruncation errors.

Understanding how to write forward, backward, and

centered finite-difference approximations of the first

and second derivatives. Recognizing that efforts to minimize truncation

errors can sometimes increase roundoff errors.


4/20

Accuracy and Precision

Accuracyrefers to how closely a computed or measuredvalue agrees with the true value, whileprecision refers to

how closely individual computed or measured values

agree with each other.

a) inaccurate and imprecise

b) accurate and imprecise

c) inaccurate and precise

d) accurate and precise


5/20

Error Definitions

True error (Et): the difference between thetrue value and the approximation.

Absolute error (|Et|): the absolute difference

between the true value and theapproximation.

True fractional relative error: the true errordivided by the true value.

Relative error (t): the true fractional relativeerror expressed as a percentage.


6/20

Error Definitions (cont)

The previous definitions of error relied on knowinga true value. If that is not the case, approximationscan be made to the error.

The approximate percent relative error can be

given as the approximate error divided by theapproximation, expressed as a percentage - thoughthis presents the challenge of finding theapproximate error!

For iterative processes, the error can beapproximated as the difference in values betweensucessive iterations.


7/20

Using Error Estimates

Often, when performing calculations, we maynot be concerned with the sign of the error

but are interested in whether the absolute

value of the percent relative error is lowerthan a prespecified tolerance s. For such

cases, the computation is repeated until | a

|

round off and error - chapter 4chapter4rev1

Documents