LECF03-Numerical-Differentiation-and-Integration.ppt

Numerical-Differentiation-and-Integration

1. NUMERICAL
DIFFERENTIATION AND
INTEGRATION
MAC

2. Overview
•Newton-Cotes Integration
Formulas
•Trapezoidal rule
•Simpson’s Rules

3. Specific Study Objectives
•Understand the derivation of the Newton-Cotes
formulas
•Recognize that the trapezoidal and Simpson’s 1/3
and 3/8 rules represent the areas of 1st, 2nd, and
3rd order polynomials
•Be able to choose the “best” among these formulas
for any particular problem

4. Newton-Cotes Integration
•Common numerical integration scheme
•Based on the strategy of replacing a complicated
function or tabulated data with some
approximating function that is easy to integrate

5. Newton-Cotes Integration
•Common numerical integration scheme
•Based on the strategy of replacing a
complicated function or tabulated data with
some approximating function that is easy to
integrate
 

Ifxdxfxdx
fxaax ax
a
b
n
a
b
n n
n
 

 
0 1....

6. Newton-Cotes Integration
•Common numerical integration scheme
•Based on the strategy of replacing a complicated
function or tabulated data with some
approximating function that is easy to integrate
 

Ifxdxfxdx
fxaax ax
a
b
n
a
b
n n
n
 

 
0 1....
f
n(x) is an nth order
polynomial

7. The approximation of an integral by the area under
- a first order polynomial
- a second order polynomial
We can also approximated the integral by using a
series of polynomials applied piece wise.
0
1
2
3
4
5
0 5 10
x
f
(
x
)
0
1
2
3
4
5
0 5 10
x
f
(
x
)

8. 0
1
2
3
4
5
012345678910
x
f
(
x
)
An approximation of an integral by the area under straight line segments.

9. Newton-Cotes Formulas
•Closed form - data is at the beginning
and end of the limits of integration
•Open form - integration limits extend
beyond the range of data.
0
1
2
3
4
5
012345678910
x
f
(
x
)
0
1
2
3
4
5
012345678910
x
f
(
x
)

10. TRAPEZOIDAL RULE
•First of the Newton-Cotes closed integration formulas
•Corresponds to the case where the polynomial is a first
order
 

Ifxdxfxdx
fxaax
a
b
a
b
n
 

 1
0 1

11.  

Ifxdxfxdx
fxaax
a
b
a
b
n
 

 1
0 1
A straight line can be represented as:


fxfa
fbfa
ba
xa
1 




12.  



Ifxdxfxdx
fa
fbfa
ba
xadx
a
b
a
b
a
b
 
 



 

1
Integrate this equation. Results in the trapezoidal rule.


Iba
fafb


2

14. 

Iba
fafb


2
The concept is the same but the trapezoid is on its side.
h
e
i
g
h
t
base
base
width
h
e
i
g
h
t
h
e
i
g
h
t

16. Error of the Trapezoidal Rule
E f ba
wherea b
t
 

1
12
3
''

This indicates that is the function being integrated is
linear, the trapezoidal rule will be exact.
Otherwise, for section with second and higher order
derivatives (that is with curvature) error can occur.
A reasonable estimate of x is the average value of
b and a

17. EXAMPLE 01: TRAPEZOIDAL RULE
Use trapezoidal rule to integrateUse trapezoidal rule to integrate

18. EXAMPLE 01: TRAPEZOIDAL RULE

19. Multiple Application of the
Trapezoidal Rule
•Improve the accuracy by dividing
the integration interval into a
number of smaller segments
•Apply the method to each
segment
•Resulting equations are called
multiple-application or composite
integration formulas

20. Multiple Application of the
Trapezoidal Rule
The average height represents a
weighted average
of the function values
Note that the interior points are given
twice the weight of the two end points

21. Example 02: Multiple-Application
Trapezoidal Rule

22. SIMPSON’S 1/3 RULE
•Corresponds to the case where the function is a second order
polynomial
 

Ifxdxfxdx
fxaaxax
a
b
a
b
n
 

 2
0 1 2
2

23. SIMPSON’S 1/3 RULE

24. Example 03: Single Application of
Simpson’s 1/3 Rule
Estimated error:

25. Multiple Application of Simpson’s 1/3
Rule

26. Example 05: Multiple-Application
Version of Simpson’s 1/3 Rule
Use multiple application of Simpson’s rule with n = 4 to
estimate the integral of
from a = 0 to b = 0.8.

27. Simpson’s 3/8 Rule

28. Example 05: Simpson’s 3/8 Rule
Use multiple application of Simpson’s rule with n = 4 to
estimate the integral of
from a = 0 to b = 0.8.

29. Example 05: Simpson’s 3/8 Rule

30. End