Post on 18-Dec-2015
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 1
EE 3561 : Computational Methods
Unit 6Numerical
Differentiation
Dr. Mujahed AlDhaifallah ( Term 342)
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 2
Lecture 17Numerical Differentiation
First order derivatives High order derivatives Richardson Extrapolation Examples
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 3
Motivation
How 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
• Calculus is the mathematics of change. Because engineers must continuously deal with systems and processes that change, they always need to estimate the value of f '(x) for a given function f(x)..• Standing in the heart of calculus are the mathematical concepts of differentiation and integration:
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 4
Recall Numerical differentiationand integration•The derivative represents the rate of change of a dependent variable with respect to an independent variable.
The difference approximation
If x is allowed to approach zero, the difference becomes a derivative
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 5
Difference Formulas
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Recall
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Three formula
them?judge wedo How better? is method Which
2
)()()(DifferenceCentral
)()()(DifferenceBackward
)()()(DifferenceForward
h
hxfhxf
dx
xdf
h
hxfxf
dx
xdf
h
xfhxf
dx
xdf
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 8
Forward Difference formula
)()()(
)('
)()()()('
)()(')()( :DifferenceBackward
_______________________________________________________
)()()(
)('
)()()()('
)()(')()(DifferenceForward
2
2
2
2
hOh
hxfxfxf
hOhxfxfhxf
hOhxfxfhxf
hOh
xfhxfxf
hOxfhxfhxf
hOhxfxfhxf
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 9
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
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 10
The Three formula (revisited)
2
( ) ( ) ( )Forward Difference ( )
( ) ( ) ( )Backward Difference ( )
( ) ( ) ( )Central Difference ( )
2
Forward and backward difference formulas are comparable in accurcy
Cent
df x f x h f xO h
dx h
df x f x f x hO h
dx h
df x f x h f x hO h
dx h
ral difference formula is expected to give better answer
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 11
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
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 12
Other Higher Order Formulas
ordererror theobtainand themprove toTheoremTaylor use can 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
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 13
HIGH-ACCURACY DIFFERENTIATION FORMULAS• The forward Taylor series expansion is:
• From this, we can write
• Substitute the second derivative approximation into the formula to yield:
• By collecting terms
21
1
''( )( ) ( ) ( )
2
( ) ( ) ''( )( )
2
ii i i
i i ii
f xf x f x f x h h
f x f x f xf x h
h
h
xfxfxfxf
hhxfxfxf
h
xfxfxf
iiii
iii
iii
2
)(3)(4)()(
2
)()(2)()()(
)(
12
212
1
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 14
HIGH-ACCURACY DIFFERENTIATION FORMULAS Inclusion of the 2nd derivative term has
improved the accuracy to O(h2). This is the forward divided difference
formula for the first derivative.
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Forward Formulas
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Backward Formulas
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Centered Formulas
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ExampleEstimate f '(1) for f(x) = ex + x using the centered
formula of O(h4) with h = 0.25.
• Solution
From Table 23.3:
5.15.012
25.125.01
1
75.025.01
5.05.012
12
)()(8)(8)()(
2
1
1
2
2112
hxx
hxx
x
hxx
hxx
h
xfxfxfxfxf
ii
ii
i
ii
ii
iiiii
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 19
• substituting the values results in
717.33
)149.2()867.2(8)740.4(8982.5
)25.0(12
)5.0()75.0(8)25.1(8)5.1()(
ffff
xf i
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 20
Numerical Differentiation Richardson Extrapolation Examples
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 21
Richardson Extrapolation
...)(')(
2
)()()(
)(
formula?better a get weCan
)(2
)()()('DifferenceCentral
66
44
22
2
hahahaxfh
h
hxfhxfh
fixedxandxfHold
hOh
hxfhxfxf
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 22
Richardson Extrapolation
4
66
44
6
6
4
4
2
2
66
44
22
3
)2
(4)()('
...16
15
4
3)('3)
2(4)(
...222
)(')2
(
...)(')(
2
)()()(
)(
hO
hh
xf
hahaxfh
h
ha
ha
haxf
h
hahahaxfh
h
hxfhxfh
fixedxandxfHold
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 23
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)
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 24
Richardson Extrapolation Table
whend terminateissolution The•
)1,1()1,(14
1)1,(),(
2)0,(:
mnDmnDmnDmnD
others
hnDColumnFirst
m
n
smnmna DD 1,,
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 25
Example
.derivative theof
estimate theas D(2,2)Obtain0.1,h
withtion Extrapolan RichardsoUse
.6.0xat
of derivative y thenumericall Evaluatecos
(x)xf(x)
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 26
ExampleFirst Column
f(x h)-f(x-h)(h)
2hf(0.7)-f(0.5)
(0.1) 1.08480.2
f(0.65)-f(0.55)(0.05) 1.0899
0.1f(0.625)-f(0.575)
(0.025) 1.09110.05
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 27
ExampleRichardson Table
1.09157111214
11222
1.09157010214
10212
1.09156000114
10111
11114
11
09115.1D(2,0) 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
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 28
ExampleRichardson Table
1.0848
1.0899 1.09157
1.0911 1.09157 1.09157
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.
EE3561_Unit 6 (c)AL-DHAIFALLAH1435 29
Summary Several formulas are available to
determine first order, second order or higher order derivatives
Richardson Extrapolation provides high accuracy estimates of the first order derivative