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January 22,2015 Thursday Repl class Dr Azizan 1 FCM2043 Computational Methods Week 2(4), Lecture 6 -Finite Difference Approximations of Higher Derivatives- Lecturer: Dr Azizan [email protected]

Computational Method - Higher Derivatives

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Page 1: Computational Method - Higher Derivatives

January 22,2015 Thursday Repl class

Dr Azizan 1

FCM2043Computational Methods

Week 2(4), Lecture 6-Finite Difference Approximations

of Higher Derivatives-Lecturer: Dr Azizan

[email protected]

Page 2: Computational Method - Higher Derivatives

January 22,2015 Thursday Repl class

Dr Azizan 2

Lesson outcome

• At the end of this session, you should be able to use a centered difference approximation of O(h2) to estimate the second derivative of a function.

Page 3: Computational Method - Higher Derivatives

January 22,2015 Thursday Repl class

Dr Azizan 3

);19(fromitgsubtractinand2by)12(gmultiplyinand

)12......(..........!3

)(!2

)()(')()(

expansion;seriesTaylorForwardtheRecall

)19.....(!3

)2(!2

)2)(()2)((')()(

:)(oftermsin)(forexpansionseriesTaylorforwardawritewethis,doTos.derivativehigherofestimation

numericalderivetousedbecanexpansionseriesTaylor

3)3(

21

3)3(2

2

2

hxfhxfxfxfxf

hfhxfhxfxfxf

xfxf

iiiii

iiii

ii

Page 4: Computational Method - Higher Derivatives

January 22,2015 Thursday Repl class

Dr Azizan 4

....12

)(7)()(

)20....().........()()()(2)(

)()()(2)()()()(2)(

)()()(2)(;derivativesecondtheafterseriesthetruncateNow

...12

)(7)()()()(2)(

2.....!3

)(2!2

)(2)('2)(2)(2

)19(......!3

)2(!2

)2)(()2)((')()(

2)3(

212

2212

2212

22

12

4)4(

3)3(212

3)3(

21

3)3(2

2

hxfhxfhOwhere

xfhOh

xfxfxf

xfh

RxfxfxfhxfRxfxfxf

Rhxfxfxfxf

Rhxfhxfhxfxfxfxf

Rhxfhxfhxfxfxf

Rhfhxfhxfxfxf

ii

iiii

iiii

iiii

iiii

ni

iiiii

nii

iii

ni

iii

'second forward finite difference'

Page 5: Computational Method - Higher Derivatives

January 22,2015 Thursday Repl class

Dr Azizan 5

Exercise• Perform manipulations to obtain a 'second

backward finite difference'

......12

)(7)()(where

)21)......(()()(2)()(

2)4(

)3(

221

hxfhxfhO

hOh

xfxfxfxf

ii

iiii

Page 6: Computational Method - Higher Derivatives

January 22,2015 Thursday Repl class

Dr Azizan 6

Exercise• Perform manipulations to obtain a 'second

centered finite difference'

)23.....(

)()()()(

)(

asexpressedbecandifferencefinitecenteredsecondtheely,Alternativ

......360

)(12

)()(where

)21)......(()()(2)()(

11

4)6(

2)4(

2

22

11

hh

xfxfh

xfxf

xf

hxfhxfhO

hOh

xfxfxfxf

iiii

i

ii

iiii

Page 7: Computational Method - Higher Derivatives

January 22,2015 Thursday Repl class

Dr Azizan 7

Example

expansion.seriesTaylortheoftermremaindertheofbasisthe

onresultsyourInterpret.derivativesecondtheofvaluetruethewithestimatesyourCompare

.125.0and25.0sizesstepusing2at887625)(

functiontheofderivativesecondtheestimateto)(ofionapproximatdifferencecenteredaUse

23

2

hxxxxxf

hO

Page 8: Computational Method - Higher Derivatives

January 22,2015 Thursday Repl class

Dr Azizan 8

Solution

28812)2(150)2(is2atderivativesecondtheofvaluetrueThe

12150)(71275)('

887625)(2

23

fx

xxfxxxf

xxxxf

Page 9: Computational Method - Higher Derivatives

January 22,2015 Thursday Repl class

Dr Azizan 9

Solution

.288)25.0(

)75.1()2(2)25.2()2(

)()(2)()(

)25.2()(25.2102)2()(2

859383975175125.0Using

2

211

11

11

ffff

hxfxfxfxf

fxfxfxfx

.).f()x f( .xh

iiii

ii

ii

i-i-

Page 10: Computational Method - Higher Derivatives

January 22,2015 Thursday Repl class

Dr Azizan 10

Solution

288)125.0(

)82617.68)102(26738.139)125.0(

)875.1()2(2)125.2()2(

)()(2)()(

125.0Using

2

2

211

ffff

hxfxfxfxf

h

iiii

Page 11: Computational Method - Higher Derivatives

January 22,2015 Thursday Repl class

Dr Azizan 11

Conclusion• Both results are exact because the errors are a

function of 4th and higher derivatives which are zero for a 3rd order polynomial function.

Page 12: Computational Method - Higher Derivatives

January 22,2015 Thursday Repl class

Dr Azizan 12

Homework

ions.approximatdifferencefinitecenteredandbackwardforward,theusing

functiontheofsderivativesecondandfirsttheFind.25.0with]2,2[intervalthe

on42)(functiontheConsider 3

hxxxf