NUMERICAL ERROR

ENGR 351

Numerical Methods for Engineers

Southern Illinois University Carbondale

College of Engineering

Dr. L.R. Chevalier

Copyright© 1999 by Lizette R. Chevalier

Permission is granted to students at Southern Illinois University at Carbondaleto make one copy of this material for use in the class ENGR 351, NumericalMethods for Engineers. No other permission is granted.

All other rights are reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means,electronic, mechanical, photocopying, recording, or otherwise, withoutthe prior written permission of the copyright owner.

SIMPLE STATISTICS

Arithmetic mean Standard deviation

21

2

21

1

......

n

yys

n

yyy

n

yy

iy

ni

Simple Statistics cont.

100..

1

2

2

y

sVC

n

yys

y

iy

Variance, sy2

Coefficient of variation

Pseudo Code Review for First Programming Assignment

Raw

Dat

a

Cod

e

Out

put D

ata

ASCII• Acronym for American Standard Code for

Information Interchanged• Pronounced ask-key• Also referred to as a text file

– Generate in C++, Fortran, Basic, etc. editors– Generate in Notepad– Can generate in Word, but need to save as a text file

• Executable programs are never stored in ASCII format

Pseudo Code Review for First Programming Assignment

Raw

Dat

a

Cod

e

Out

put D

ata

These are in ASCII format, unless you are using an executable version of your code. In that case, only the data (input and output) are ASCII format.

Pseudo Code for Average

Read an ASCII file that1. Inputs number of students n2. Inputs scores y(n)

1077.839.256.798.288.786.276.382.593.678.2

Data fileinput.dat

DIMENSION YOPEN INPUT.DATREAD NDO I=1,NREAD Y(N)CONTINUEDO J=1,NSUM=SUM+Y(J)CONTINUEAVERAGE = SUM/NPRINT “AV”

Pseudo Code for Average

PROBLEM

Modify the pseudo-code to calculate the variance.

1

2

2

n

yys i

y

Approximation and ErrorsSignificant Figures

• 4 significant figures– 1.845– 0.01845– 0.0001845

• 43,500 ? confidence

• 4.35 x 104 3 significant figures

• 4.350 x 104 4 significant figures

• 4.3500 x 104 5 significant figures

Accuracy and Precision

• Accuracy - how closely a computed or measured value agrees with the true value

• Precision - how closely individual computed or measured values agree with each other– number of significant figures– spread in repeated measurements or

computations

increasing accuracyin

crea

sing

pre

cisi

on

Error Definitions

• Numerical error - use of approximations to represent exact mathematical operations and quantities

• true value = approximation + error– error, t=true value - approximation

– subscript t represents the true error– shortcoming....gives no sense of magnitude– normalize by true value to get true relative error

Error definitions cont.

100valuetrue

errortruet

• True relative percent error• But we may not know the true answer apriori

Error definitions cont.• May not know the true answer apriori

a

approximate error

approximation 100

• This leads us to develop an iterative approach of numerical methods

100.

..

100

approxpresentapproxpreviousapproxpresent

ionapproximaterroreapproximat

a

Error definitions cont.

• Usually not concerned with sign, but with tolerance

• Want to assure a result is correct to n significant figures

%105.0 2 ns

sa

Example

Consider a series expansion to estimate trigonometricfunctions

xxxx

xx .....!7!5!3

sin753

Estimate sin / 2 to three significant figures

Error Definitions cont.• Round off error - originate from the fact

that computers retain only a fixed number of significant figures

• Truncation errors - errors that result from using an approximation in place of an exact mathematical procedure

To gain insight consider the mathematical formulation that is used widely in numerical methods - TAYLOR SERIES

TAYLOR SERIES

• Provides a means to predict a function value at one point in terms of the function value at and its derivative at another point

• Zero order approximation

ii xfxf 1

This is good if the function is a constant.

Taylor Series Expansion

• First order approximation

slope multiplied by distance

Still a straight line but capable of predicting an increase or decrease - LINEAR

iiiii xxxfxfxf 11 '

Taylor Series Expansion

• Second order approximation - captures some of the curvature

2111 !2

''' ii

iiiiii xx

xfxxxfxfxf

Taylor Series Expansion

ii

nni

n

iiiii

xxsizestephwhere

Rhn

xf

hxf

hxf

hxfxfxf

1

......

321

!

!3

'''

!2

'''

Taylor Series Expansion

1

11

1

......

321

!1

!

!3'''

!2''

'

iin

n

n

ii

nni

n

iiiii

xxhn

fR

xxsizestephwhere

Rhn

xf

hxf

hxf

hxfxfxf

Example

Use zero through fourth order Taylor series expansion to approximate f(1) given f(0) = 1.2 (i.e. h = 1)

f x 01 015 0 5 0 25 1 24 3 2. . . . .x x x x

-1

-0.5

0

0.5

1

1.5

0 0.5 1 1.5

x

f(x)

Note:f(1) = 0.2

Functions with infinite number of derivatives

• f(x) = cos x

• f '(x) = -sin x

• f "x) = -cos x

• f "'(x) = sin x

• Evaluate the system where xi = /4 and xi+1

= /3

• h = /3 - /4 = /12

Functions with infinite number of derivatives

• Zero order– f( /3) = cos (/4 ) = 0.707 t = 41.4%

• First order– f( /3) = cos (/4 ) - sin (4 )(/12) t = 4.4%

• Second order– f( /3) = 0.498 t = 0.45%

• By n = 6 t = 2.4 x 10-6 %

Exam Question

How many significant figures are in the following numbers?

A. 3.215

B. 0.00083

C. 2.41 x 10-3

D. 23,000,000

E. 2.3 x 107

TAYLOR SERIES PROBLEM

Use zero- through fourth-order Taylor series expansions to predict f(4) for f(x) = ln x using a base point at x = 2. Compute the percent relative error t for each approximation.