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Numerical

Analysis

Lecture 29

Numerical

Numerical

Analysis

Analysis

Lecture 29

Lecture 29

Chapter 7

Numerical

Differentiation

and

Integration

Chapter 7

Numerical

Differentiation

and

Integration

NEWTON-COTES

INTEGRATION FORMULAE

THE TRAPEZOIDAL RULE

( COMPOSITE FORM )

SIMPSON’S RULES

( COMPOSITE FORM )

ROMBERG’S INTEGRATIONDOUBLE INTEGRATION

DIFFERENTIATION USINGDIFFERENCE OPREATORS:Applications:Remember Using forward differenceoperator

∆

, the shift operator , the

backward difference operator and theaverage difference operator , we obtainedsevral formulae and we worked out thefollowing example :

1.001.163.

13.9641.

101.

x

( )

f

x

0.00.20.40.60.81.

Solution

Since x = 0 and

0.2 appear at and nearbeginning of the table, it isappropriate to use formulaebased on forward differencesto find the derivatives. Thedifference table for the givendata is depicted below:

Example

Find

and

from the table

(2.2)

y



( 2 .2 )

y





x

y

( x )

Solution:Since x=2.2 occurs at the end of the table, it is appropriate to usebackward difference formulaefor derivatives. The backwarddifference table for the givendata is shown below:

Using backward difference formulae for

and

we have

(

)

y

x



y

x

2

3

4

1

2

3

4

n

n

n

n

n

y

y

y

y

y

h











 

















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Therefore, Also

1

(2.2)

2

3

5(1.8043)

4

y











 









 

2

3

4

2

1

11

12

n

n

n

n

y

y

y

y

h





 



 













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Case IV:

Derivation of two

and three point formulae:

Retaining only the first term in equation :

2

3

4

0

0

0

0

0

0

1

2

3

4

y

y

y

Dy

y

y

h





































1

(

)

(

)

i

i

i

i

i

i

y

y

y

y

h

h

y x

h

y x

h







 









we can get another usefulform for the first derivative as

1

(

)

(

)

i

i

i

i

i

i

y

y

y

y

h

h

y x

y x

h

h







 









Adding the last two equations, we have

(

)

(

)

2

i

i

i

y x

h

y x

h

y

h







 