C04-Defferentiation and Integration

8/3/2019 C04-Defferentiation and Integration

1/20

5/5/2012 by Adam Pamudji R 1

Numerical Differentiation


2/20


First Order Derivative

First Order Taylor Series for x closed to x0

001

0

0001

)()('

)(')(

xx

xfxPxf

xxxfxfxP


3/20


graphical explanation

0

010

0001

)()('

)(')(

xx

xfxPxf

xxxfxfxP

x0 x>x0x


4/20


h

xfhxfxf

)()(' 000

Ifx = x0+ h, then for forward difference we have

Ifx = x0- h, then for backward difference we have

h

hxfxfh

xfhxfxf

00000)()()('


5/20


First Order Derivative

Second Order Taylor Series for x closed to x0

h

hxfhxfxf

xfhhxfhxf

hxf

hxfxfhxf

hxfhxfxfhxf

xxxf

xxxfxfxP

2)('

0)('20

2

)(''

)(')(

2

)('')(')(

!2

)(''

)(')(

00

0

000

2

0

000

2

0

000

2

00

0002


6/20


graphical explanation

h

hxfhxfxf

2

)()(' 000

x0

x0-h

central difference

)(' 0xf

)(0

xf

)(0

hxf )( 0 hxf

x0+h


7/20


Reading Assignment

Please read page 146 to 150 section 4.1for another approach

Find about:

The three point formulas

The five point formulas


8/20


Second Order Derivative

Second Order Taylor Series for x closed to x0

2

000

0

2

0000

2

0000

2

0000

)(2)(''

)(''0)(2

2

)('')(')(

2

)('')(')(

h

hxfxfhxfxf

hxfxfhxfhxf

hxfhxfxfhxf

hxfhxfxfhxf


9/20


Differentiation from interpolation

function The use of polynomials in approximation

problems allow us to find the derivative

and integral from that functions. Read page 145 and 146


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Differentiation and Integration

dxxfdxxP

xfxP

xfxP

n

n

n

''


11/20


Differentiation

xLdxdxf

xLxfdx

d

xP

in

n

i

i

n

iinin

,

0

0,'


12/20


Differentiation

n

ijj

n

iljl

l

ln

ijj

ji

in

n

ijj ji

j

in

xx

xx

xL

dx

d

xx

xxxL

0 0

0

,

0

,

1


13/20


Numerical Integration

We can use theinterpolation function

to approximate anintegral

dxxPdxxf

xPxf

n

n

dxxLxf

dxxLxfdxxP

b

a

in

n

i

i

b

a

n

i

ini

bx

ax

n

,

0

0

,


14/20



For the first and second Lagrangepolynomials with equally spaced nodes

Trapezoidal rule

Simpsons rule

Read page 163 to 165

Learn about closed and open Newton-Cotes formulas (p 168 to p 173)


15/20


Trapezoidal rule

For function f(x)between x0=aandx1=b

h = x1x0

)(''12

)()(2

)(3

10f

hxfxf

hdxxf

b

a

a=x0 b=x1

f

P1

x

y


16/20


Simpson rule

)(

90

)()(4)(3

)(

)4(5

210

fh

xfxfxfh

dxxfb

a

For function f(x)between x0=aand

x2=b

a=x0 b=x2

f

P1

x

y

x1


17/20


Closed and Open Newton-Cotes

Closed N-C Trapezoidal rule

Simpsons rule Simpsons three-eight rule

Open N-C

Midpoint rules: n=0 n=1

n=2

a=x-1 b=x1

f

P1

x

y

x0


18/20


Examples

See the excel file
http://../computation/INTEG00.XLShttp://../computation/INTEG00.XLS


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Gaussian Quadrature

For a given known function

Optimizing accuracy by selecting best positionof nodes.

By using a standard tabulated coeff and node

positions, it is necessary to transform linearlythe function to an interval [-1,1]


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Gauss Quadrature

Read page 198 to 204

Example file
http://../computation/integrasi01.xlshttp://../computation/integrasi01.xls

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