CISE301_Topic6 1

SE301: Numerical Methods

Topic 6 Numerical Differentiation

Lecture 23

KFUPM

Read Chapter 23, Sections 1-2

CISE301_Topic6 2

Lecture 23Numerical

Differentiation

First order derivatives High order derivatives Richardson Extrapolation Examples

CISE301_Topic6 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

CISE301_Topic6 4

Recall

nn

nn

h

hEh

hCEthatsuchCfiniterealhOE

hOhxfhxfhxfxfhxf

hxfhxf

dxdf

similar to rateat zero gapproachin isorder of is E

:,, )(

)(!3

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

:TheoremTaylor

)()(lim

43)3(2)2(

0

CISE301_Topic6 5

Three Formula

them?judge wedo How better? is methodWhich

2)()()(:DifferenceCentral

)()()(:DifferenceBackward

)()()(:DifferenceForward

hhxfhxf

dxxdf

hhxfxf

dxxdf

hxfhxf

dxxdf

CISE301_Topic6 6

The Three Formulas

CISE301_Topic6 7

Forward/Backward Difference Formula

)()()()('

)()()()('

)()(')()( :DifferenceBackward

_______________________________________________________

)()()()('

)()()()('

)()(')()(:DifferenceForward

2

2

2

2

hOh

hxfxfxf

hOhxfxfhxf

hOhxfxfhxf

hOh

xfhxfxf

hOxfhxfhxf

hOhxfxfhxf

CISE301_Topic6 8

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

CISE301_Topic6 9

The Three Formula (Revisited)

answer.better a give toexpected is formula difference Centralaccuracy.in comparable are formulas difference backward andForward

)(2

)()()(:DifferenceCentral

)()()()(:DifferenceBackward

)()()()(:DifferenceForward

2hOh

hxfhxfdxxdf

hOh

hxfxfdxxdf

hOh

xfhxfdxxdf

CISE301_Topic6 10

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

CISE301_Topic6 11

Other Higher Order Formulas

.ordererror theobtainand themprove toTheoremTaylor use can Youpossible. also are)...(),(for formulasOther

)( with FormulasCentral

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

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

)()(2)()(

)3()2(

2

4)4(

3)3(

2)2(

xfxf

hOError

hhxfhxfxfhxfhxfxf

hhxfhxfhxfhxfxf

hhxfxfhxfxf

Example Use forward, backward and centered difference

approximations to estimate the first derivate of:f(x) = –0.1x4 – 0.15x3 – 0.5x2 – 0.25x + 1.2at x = 0.5 using step size h = 0.5 and h = 0.25

Note that the derivate can be obtained directly:f’(x) = –0.4x3 – 0.45x2 – 1.0x – 0.25The true value of f’(0.5) = -0.9125

In this example, the function and its derivate are known. However, in general, only tabulated data might be given.

CISE301_Topic6 12

Solution with Step Size = 0.5 f(0.5) = 0.925, f(0) = 1.2, f(1.0) = 0.2 Forward Divided Difference:

f’(0.5) (0.2 – 0.925)/0.5 = -1.45|t| = |(-0.9125+1.45)/-0.9125| = 58.9%

Backward Divided Difference: f’(0.5) (0.925 – 1.2)/0.5 = -0.55|t| = |(-0.9125+0.55)/-0.9125| = 39.7%

Centered Divided Difference: f’(0.5) (0.2 – 1.2)/1.0 = -1.0|t| = |(-0.9125+1.0)/-0.9125| = 9.6%

CISE301_Topic6 13

Solution with Step Size = 0.25 f(0.5)=0.925, f(0.25)=1.1035, f(0.75)=0.6363 Forward Divided Difference:

f’(0.5) (0.6363 – 0.925)/0.25 = -1.155|t| = |(-0.9125+1.155)/-0.9125| = 26.5%

Backward Divided Difference: f’(0.5) (0.925 – 1.1035)/0.25 = -0.714|t| = |(-0.9125+0.714)/-0.9125| = 21.7%

Centered Divided Difference: f’(0.5) (0.6363 – 1.1035)/0.5 = -0.934|t| = |(-0.9125+0.934)/-0.9125| = 2.4%

CISE301_Topic6 14

Discussion For both the Forward and Backward difference,

the error is O(h) Halving the step size h approximately halves the

error of the Forward and Backward differences The Centered difference approximation is more

accurate than the Forward and Backward differences because the error is O(h2)

Halving the step size h approximately quarters the error of the Centered difference.

CISE301_Topic6 15

CISE301_Topic6 16

Richardson Extrapolation

...)(')(2

)()()(

:)(

formula?better aget Can we

)(2

)()()(':DifferenceCentral

66

44

22

2

hahahaxfhh

hxfhxfh

fixedxandxfHold

hOh

hxfhxfxf

CISE301_Topic6 17

Richardson Extrapolation

)()(31)2/(

34)('

...1615

43)('3)

2(4)(

...222

)(')2

(

...)(')(2

)()()(

:)(

4

66

44

6

6

4

4

2

2

66

44

22

hOhhxf

hahaxfhh

hahahaxfh

hahahaxfhh

hxfhxfh

fixedxandxfHold

Use two derivate estimates to compute a third, more accurate approximation

Richardson Extrapolation Example Use the function:

f(x) = –0.1x4 – 0.15x3 – 0.5x2 – 0.25x + 1.2

Starting with h1 = 0.5 and h2 = 0.25, compute an improved estimate of f’(0.5) using Richardson Extrapolation

Recall the true value of f’(0.5) = -0.9125

CISE301_Topic6 18

Solution The first-derivative estimates can be computed

with centered differences as:

CISE301_Topic6 19

example for thisresult exact theproduces Which

9125.0)1(31)934375.0(

34)(

31)2/(

34)5.0('

:applyingby obtained be can estimate improved The

%4.2 || ,934375.05.0

)25.0()75.0()25.0(

%6.9 || ,0.11

2.12.01

)0()1()5.0(

5.0at 2

)5.0()5.0()(

hhf

ff

ff

xh

hfhfh

t

t

CISE301_Topic6 20

Richardson Extrapolation Table

D(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)

CISE301_Topic6 21

Richardson Extrapolation Table

)1,1(14

1)1,(14

4),(

:

2)0,( :

mnDmnDmnD

Others

hnDColumnFirst

mm

m

n

CISE301_Topic6 22

Example

. derivative theof estimate theas D(2,2)Obtain

10 ion with Extrapolat Richardson Use

60at

:of derivative y thenumericall Evaluatecos

.h

.xxf(x) (x)

CISE301_Topic6 23

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)

2hh)-f(x-h)f(x(h)

CISE301_Topic6 24

ExampleRichardson Table

1.09157)11(151)12(

1516)22(

1.09157)01(31)02(

34)12(

1.09156)00(31)01(

34)1,1(

)1,1(14

1)1,(14

4),(

09115.1 D(2,0)1.08988, D(1,0), 08483.1)0,0(

,D,D,D

,D,D,D

,D,DD

mnDmnDmnD

D

mm

m

CISE301_Topic6 25

ExampleRichardson Table

1.08483

1.08988 1.091561.09115 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 step size h.