SE301_14 (c)AL-AMER2003 1

SE031Numerical MethodsTopic 6

Numerical Differentiation

Dr. Samir Al-Amer ( Term 053)

Modified by Dr. Baroudi(072)Read chapter 23.1 and 23.2

SE301_14 (c)AL-AMER2003 2

Numerical Differentiation First order derivatives High order derivatives Richardson Extrapolation Examples

SE301_14 (c)AL-AMER2003 3

MotivationHow do you evaluate the

derivative of a tabulated function.

How do we determine the velocity and acceleration from tabulated measurements.

Time(second)

Displacement

(meters)

0 30.1

5 48.2

10 50.0

15 40.2

SE301_14 (c)AL-AMER2003 4

Recall

nn

nn

h

hEh

hCEthatsuchCfiniterealhOE

hOhxfhxf

hxfxfhxf

TheoremTaylor

h

xfhxf

dx

df

similar to rateat zero gapproachin isorder of is E

, )(

)(!3

)(

!2

)()(')()(

:

)()(lim

43)3(2)2(

0

SE301_14 (c)AL-AMER2003 5

Three formula

them?judge wedo How better? is methodWhich

2

)()()(DifferenceCentral

)()()(DifferenceBackward

)()()(DifferenceForward

h

hxfhxf

dx

xdf

h

hxfxf

dx

xdf

h

xfhxf

dx

xdf

SE301_14 (c)AL-AMER2003 6

Forward Difference formula

)()()(

)('

)()()()('

)()(')()( :DifferenceBackward

_______________________________________________________

)()()(

)('

)()()()('

)()(')()(DifferenceForward

2

2

2

2

hOh

hxfxfxf

hOhxfxfhxf

hOhxfxfhxf

hOh

xfhxfxf

hOxfhxfhxf

hOhxfxfhxf

SE301_14 (c)AL-AMER2003 7

Central Difference formula

)(2

)()()('

...!3

)(2)('2)()(

...!4

)(

!3

)(

!2

)()(')()(

...!4

)(

!3

)(

!2

)()(')()(

DifferenceCentral

2

3)3(

4)4(3)3(2)2(

4)4(3)3(2)2(

hOh

hxfhxfxf

hxfhxfhxfhxf

hxfhxfhxfhxfxfhxf

hxfhxfhxfhxfxfhxf

SE301_14 (c)AL-AMER2003 8

The Three formula (revisited)

answer better give toexpected is formula difference Central

accurcyin comparable are formulas difference backward andForward

)(2

)()()(DifferenceCentral

)()()()(

DifferenceBackward

)()()()(

DifferenceForward

2hOh

hxfhxf

dx

xdf

hOh

hxfxf

dx

xdf

hOh

xfhxf

dx

xdf

SE301_14 (c)AL-AMER2003 9

Higher Order Formulas

12

)(

)()()(2)(

)(

...!4

)(2

!2

)(2)(2)()(

...!4

)(

!3

)(

!2

)()(')()(

...!4

)(

!3

)(

!2

)()(')()(

2)4(

22

)2(

4)4(2)2(

4)4(3)3(2)2(

4)4(3)3(2)2(

hfError

hOh

hxfxfhxfxf

hxfhxfxfhxfhxf

hxfhxfhxfhxfxfhxf

hxfhxfhxfhxfxfhxf

SE301_14 (c)AL-AMER2003 10

High Accuracy Differentiation Formulas (1) High-accuracy divided-difference formulas can be generated

by including additional terms from the Taylor series expansion.

Consider the forward finite-divided difference

)(2

)(3)(4)()(

)(2

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

)()()(2)(

)(

)(2

)()()()(

2

)()()()(

212

22

121

212

21

21

hOh

xfxfxfxf

hOhh

xfxfxf

h

xfxfxf

hOh

xfxfxfxf

hOhxf

h

xfxfxf

hxf

hxfxfxf

iiii

iiiiii

iiii

iiii

iiii

SE301_14 (c)AL-AMER2003 11

High Accuracy Differentiation Formulas (2) Consider the backward finite-divided difference

Consider the central finite-divided difference

)(2

)(3)(4)()(

)()()(

)(

212

1

hOh

xfxfxfxf

hOh

xfxfxf

iiii

iii

)(12

)()(8)(8)()(

)(2

)()()(

42112

211

hOh

xfxfxfxfxf

hOh

xfxfxf

iiiii

iii

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Example

Central and Backward Forward, :caseeach for error true theFind

.250 of size step a and difference-divided finitite using 50At

2.125.05.015.01.0

of derivativefirst theEstimate234

.h.x

xxxxf(x)

SE301_14 (c)AL-AMER2003 13

Other Higher Order Formulas

ordererror obtain theand themprove toTheoremTaylor usecan You

possible. also are)...(),(for formulasOther

)2()(4)(6)(4)2()(

2

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

)()(2)()(

)3()2(

4)4(

3)3(

2)2(

xfxf

h

hxfhxfxfhxfhxfxf

h

hxfhxfhxfhxfxf

h

hxfxfhxfxf

SE301_14 (c)AL-AMER2003 14

Richardson Extrapolation

xfaa

hahahah

hxfhxfxf

xfh

xfh

h

hxfhxfxf v

and on depend ,..., Constants

...2

)()()('

...)(!5

)(!32

)()()('

DifferenceCentral

42

66

44

22

4'''

2

SE301_14 (c)AL-AMER2003 15

Richardson Extrapolation

4

66

44

6

6

4

4

2

2

66

44

22

)2

(4)(3

1)('

get werearrange, and 3-by divide

...16

15

4

3)('3)

2(4)(

...222

)(')2

(

...)(')(

2

)()()(

define ,)(

hOh

hxf

hahaxfh

h

ha

ha

haxf

h

hahahaxfh

h

hxfhxfh

fixedxandxfHold

SE301_14 (c)AL-AMER2003 16

Richardson Extrapolation Table

)1,1()1,(14

1)1,(),(

2)0,(:

mnDmnDmnDmnD

others

hnDColumnFirst

m

n

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Richardson Extrapolation TableD(0,0)=Φ(h)

D(1,0)=Φ(h/2) D(1,1)

D(2,0)=Φ(h/4) D(2,1) D(2,2)

D(3,0)=Φ(h/8) D(3,1) D(3,2) D(3,3)

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Example

.derivative theof

estimate theas D(2,2)Obtain 0.1,h

ion with Extrapolat Richardson Use

.6.0at x

of derivative y thenumericall Evaluatecos

(x)xf(x)

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ExampleFirst Column

1.091150.05

f(0.575)-)625.f(0(0.025)

1.089880.1

f(0.55)-)65.f(0(0.05)

1.084830.2

f(0.5)-7).f(0(0.1)

2h

h)-f(x-h)f(x(h)

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ExampleRichardson Table

1.09148111214

11222

1.09146010214

10212

1.09114000114

10111

11114

11

09115.1D(2,1) 0.10988,D(1,0) , 08483.1)0,0(

2

),D(),D(),D(),D(

),D(),D(),D(),D(

),D(),D(),D(),D(

),m-D(n)D(n,m-)D(n,m-D(n,m)

D

m

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ExampleRichardson Table

1.08483

1.08988 1.09114

1.09115 1.09146 1.09148

This is the best estimate of the derivative of the function

All entries of the Richardson table are estimates of the derivative of the function. The first column are estimates using the central difference formula with different h.

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Summary