NUMERICAL ERRORStudent Notes

ENGR 351 Numerical Methods for EngineersSouthern Illinois University CarbondaleCollege of EngineeringDr. L.R. Chevalier

Objectives• To understand error terms• Become familiar with notation and

techniques used in this course

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 accuracy

incr

easi

ng p

reci

sion

Accuracy and Precision

Error Definitions• Numerical error - use of approximations

to represent exact mathematical operations and quantities

• true value = approximation + error• error, et=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.

valuetruevalueestimatedvaluetrue

valuetrueerrortrue

t

100e

• True relative percent error

ExampleConsider a problem where the true answer is

7.91712. If you report the value as 7.92, answer the following questions.

1. How many significant figures did you use?2. What is the true error?3. What is the true relative percent error?

Error definitions cont.• May not know the true answer

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

100.

..

100

approxpresentapproxpreviousapproxpresent

ionapproximaterroreapproximat

ae

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

e

ee

ExampleConsider a series expansion to estimate trigonometric functions

xxxxxx .....!7!5!3

sin753

Estimate sin(p/ 2) to three significant figures. Calculate et and ea. STRATEGY

StrategyTerms Results et % ea %

12345

Stop when ea ≤ es

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

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 procedureTo 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

TAYLOR SERIES

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

''' iii

iiiii xxxfxxxfxfxf

Taylor Series Expansion

ii

nni

n

iiiii

xxsizestephwhere

Rhn

xf

hxfhxfhxfxfxf

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

hxfhxfhxfxfxf

ExampleUse zero through fourth order Taylor series expansion to approximate f(1) given f(0) = 1.2 (i.e. h = 1). Calculate et after each step. f x 01 015 0 5 0 25 1 24 3 2. . . . .x x x x

Note:f(1) = 0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1x

f(x)

STRATEGY

Strategy• Estimate the function using only the first term

• Use x = 0 to estimate f(1), which is the y-value when x = 1

• Calculate error, et• Estimate the function using the first and second

term• Calculate the error, et• Progressively add terms

Objectives• To understand error terms• Become familiar with notation and

techniques used in this course