ce27. notes. numerical differentiation

8/9/2019 Ce27. Notes. Numerical Differentiation

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Numerical Differentiation

eatingeatingeatingeating


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Introduction Some numerical algorithms require

evaluation of the derivative of a function.

When the function is difficult todifferentiate analytically or when thefunction is given empirically as in table of

values, approximate calculation of thederivative is necessary.


3/28

Approximating the derivativeThe definition of the derivative is, the limit as

h0 of

practically suggests the program to pick asmall value of h.

Example. Compute the derivative off(x)=ex

atx=1.

( )( ) ( )

h

xfhxfxf

+' )1(


4/28


5/28

Approximating the derivative When h is large, the difference quotient is

the slope of the secant line joining twopoints that are not close enough to eachother.

When h is too small, the differencequotient can exhibit the problem of loss of

significance due to the subtraction ofquantities that are nearly equal.


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Approximating the derivative To determine the best approximation

before the terms start to move away fromthe limit is to computeDk until

In the example,

11 + NNNN DDDD

00012.00007.0 4556 =>= DDDD


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Sources of error The truncation error t comes from higher

terms in the Taylor series expansion:

Round-off error r must be expected sincereal numbers dont have exact

representation in binary.

( ) ( )L

+= "

2

1' hf

h

xfhxff


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Central-difference formulas Assume thatfC3[a,b] and thatx-h,x, x+h

[a,b]. Then

where, there exists a number c=c(x) [a,b]such that

( )( ) ( )

th

hxfhxfxf +

+

2'

( )( ) ( )232

6h

cfht O==

)2(


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Centered-formula of order O(h4

)

Assume thatfC5[a,b] and thatx-2h,x-h, x

x+h, x+2h [a,b]. Then

where( )( ) ( )454

30h

cfht O==

( )( ) ( ) ( ) ( )

t

h

hxfhxfhxfhxfxf +

++++

12

2882'

)3(


10/28

ExampleLetf(x)=cos(x). Calculate approximations for

f(0.8) using Eqs. (2) and (3) with stepsizes h=0.1, 0.01, 0.001, 0.0001 andcompare with true valuef(0.8) = -sin(0.8).

Carry out nine decimal places in

calculations.


11/28

ExampleEq. (2)

Eq. (3)

( )

( ) ( ) ( ) ( )

h

hxfhxfhxfhxf

xf 12

2882

'

++++

( )( ) ( )

h

hxhxf

xf 2'

+


12/28

Differentiating an

Interpolating Polynomial The Lagrange quadratic polynomial P2(x)

that passes throughx=0.7,0.8, and 0.9.

Differentiating

( ) 22 348063157.0159260044.0046875165.1 xxxP =

( ) 716161095.08.0'2 =P


13/28

Differentiating an

Interpolating Polynomial The Lagrange quadric polynomial P4(x)

that passes throughx=0.6,0.7,0.8, 0.9 and1.0.

Differentiating

( )2

4 523291341.000963839.0998452927.0 xxxP +=43

028981100.0026521229.0 xx ++

( ) 717353703.08.0'4 =P


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Richardsons ExtrapolationLet

andD0(h) andD0(2h) denote approximations

tof(x0) that are obtained using Eq. (2)with step sizes h and 2h.

( ) ( )khxfxff kk +== 0

( ) ( )( ) ( ) 2

00

2

00

42''

ChhDxfChhDxf+

+

( )4


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Richardsons ExtrapolationManipulating these two equations, we obtain

which is the central difference formulaEq.(3).

This method of obtaining a formula forf(x0)of higher order from a formula of lowerorder is called extrapolation.

( ) ( ) ( )h

ffffhDhDxf

12

88

3

24' 211200

0

++=


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Richardsons ExtrapolationSuppose that two approximations of order

O(h2k) forf(x0) areDk-1(h) andDk-1(2h) andthat they satisfy

Then an improved approximation is

( ) ( )( ) ( ) L

L

+++

+++

++

+

22

2

12

110

22

2

2

110

442''

kkkk

k

kk

k

hchchDxfhchchDxf

( ) ( ) ( ) ( ) ( ) ( )221122014

24'

+++

=+=

k

k

kk

kk

k hhDhD

hhDxf OO


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ExampleLet

f(x)=cos(x).

Use Eq. (4) with h=0.01 to show how the

linear combination (4D0(h) -D0(2h))/3 canbe used to obtain the approximation to

f(0.8).

Carry nine decimal places in allcalculations.


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ExampleUsing Eq. (4), we obtain

The linear combination becomes

( ) ( ) ( )

( )( ) ( )

717308275.0

04.0

78.082.02

717344150.002.0

79.081.0

0

0

=

=

ffhD

ffhD

( ) ( ) ( ) 717356108.03

248.0' 00 = hDhDf


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Higher Derivatives Taylor series can be used to obtain

central-difference formulas for higherderivatives.

The popular choices are those of order

O(h2) and O(h4).


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Higher DerivativesTo derivef(x) of order O(h2), we write

adding

( ) ( ) ( )( ) ( )( )

( ) ( ) ( ) ( ) ( )( ) L

L

++=

++++=+

62"'

62

"'

332

332

xfhxfhxfhxfhxf

xfhxfhxfhxfhxf

( ) ( ) ( )( ) ( )( )

L+++=++24

2

2

"22

442xfhxfh

xfhxfhxf


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Higher DerivativesSolving forf(x) yields

If truncated at the fourth derivative

( ) ( ) ( ) ( )2

2"

h

hxfxfhxfxf

++=

( )

( )( )

( )( )

( )( ) LL

!22

!62

!42

2226442

kxfhxfhxfh

kk

( )( )( )

12

2"

42

2

101 cfh

h

fffxf

+=


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Central difference formulas

of Order O(h2)

( )

( )

( )( )

( )( )4

210120

4

3

21120

3

2

1010

11

0

464 2

2

2"

2'

h

fffffxf

h

ffffxf

h

fffxf

h

ffxf

++

+

+


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Central difference formulas

of Order O(h4)

( )

( )

( )( )

( )( )4

32101230

4

3

3211230

3

2

210120

2112

0

61239563912

8

81313812

163016"

12

88

'

hfffffffxf

h

ffffffxf

h

fffffxf

h

ffff

xf

+++

+++

++

++


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ExampleLetf(x)=cos(x). Use the formula

with h=0.1, 0.01, and 0.001 to findapproximations to f(0.8).

Compare with the true value of f(0.8) =

-cos(0.8).

( )2

101 2"h

fffxf

+=


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Example The calculation for h=0.01 is

with an error of -0.000016709 The other calculations are summarized:

( ) ( ) ( ) ( ) 696690000.00001.0

79.080.0281.08.0" =

+

ffff


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Exercise 1Letf(x)=sin(x), wherex is measured in

radians.(a) Calculate approximations tof(0.8)using Eq.(3) with h=0.1, 0.01, andcompare withf( 0.8)=cos(0.8).

(b) Use extrapolation formula to compute

the approximation tof( 0.8)


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Exercise 2Use numerical differentiation formula

and h=0.1 to approximatef(1) for thefunctions

(a)f(x) =x2 (b)f(x) =x4

( )2

210120

12

163016"

h

fffffxf

++

ce27. notes. numerical differentiation

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