Chapter 5 – Numerical Integration.pptx

7/27/2019 Chapter 5 Numerical Integration.pptx

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CHAPTER 5 NUMERICAL INTEGRATION

Prepared by:

Engr. Romano A. Gabrillo

MEngg - MfgE


2/23

INTRODUCTION

Engineers and scientist are frequently facedwith the problem of differentiating or integratingfunctions which are defined in a tabular orgraphical form rather than as explicit functions.

Sometimes there are certain explicit functionswhich are difficult to integrate in terms ofelementary functions. One simple example is:

v = dx/dtwhere v=velocity; x=distance; t=time


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METHODS FOR SOLVING NUMERICAL

INTEGRATION

One can resort to number integration if:

1. F(x) is not known as a closed form but specified

only at discrete points.

2. F(x) is expressed analytically but cannot beintegrated in a closed form. In this chapter the

following methods for numerical integration will

be discussed:

1. Trapezoidal Rule

2. Simpsons Rule


4/23

TRAPEZOIDAL RULE

This is based on linear interpolation formula

of Newtons forward difference using First

order Newtons forward difference

y = y0 + y0T + Error

= y0+ y0T + f () h2 T(T-1)

2

Integrating both sides: (See Figure 1)

1

0

1

0

2''

00 )1(2

)(x

xdTTTh

fTyyhydxI


5/23

FIGURE 1


6/23

Substituting fory0 as (y1 + y0) in theabove equation we get:

)6/1(2

)(

2

2

2

3

3

2

)(

2

''21

0

2

00

1

0

2''2

00

fhTyyhI

TThfTyTyh

ErrorLocalyyh

I

fhyy

yhI

][2

12

)(

2

10

''2

01

0


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HENCE, IF WE INTEGREATE A FUNCTION BY

TRAPEZOIDAL RULE WE GET:

Upper and lower bound for the error can be

found out by substituting upper and lowerbound values for f ().

|)(|12

||

][2

''2

10

fh

errorlocaltheand

yyh

I


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COMPOSITE INTEGRATION FORMULA

(TRAPEZOIDAL)

One way to reduce the error associated with alow order integration formula is to subdivide theinterval of integration a, b into smaller intervalsand to use the trapezoidal rule separately on

each sub interval.

Repeated application of lower order formula ispreferred to the single application of higher

order formula, partly because of the simplicity ofthe low order formulae and partly because ofcomputational difficulties.


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TO GET I BY TRAPEZOIDAL RULE FROM X0XN,WE GET I FOR EACH SUBINTERVAL SUCH AS

(x0 x1), (x1 x2)and add them.


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]........2[2/:

);(12/][2/

);(12/][2/

);(12/][2/

1210

1

''3

1

21

''3

21

10

''3

10

0

1

21

10

nnxxT

nnnnxx

xx

xx

yyyyyhII

Adding

XXfhyyhI

XXfhyyhI

XXfhyyhI

n

nn


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GLOBAL ERROR

)()(12

''

0

2

fxxh

n

The global error accumulated in n steps in

going from x0 to xn is known as global Error nh3


12/23

SIMPSONS RULE

Simpsons Rule is based on quadratic

interpolation function between three points.

The quadratic interpolation function may be

written as:


13/23

In the equation above after2y0 we havewritten two terms because the fourthterm vanishes on integration.

)3)(2)(1(24

)2)(1(6

)1(2

0

4

0

3

0

2

00

TTTTy

TTTy

TTy

yyy

)(

90)0(

6)3/2(

222

)3)(2)(1(

24

)......1(

2

)3)(2)(1(24

).....1(2

44

0

3

0

2

00

2

0

0

4

0

2

00

0

4

0

2

00

2

0

fhyy

yyh

dTTTTTy

TTy

yyhydx

TTTTy

TTy

yyy

x

x


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SUBSTITUTING FOR

210

210

0120

2

010

61

64

61

2/

43/

2

yyyLI

LashforngSubstituti

yyyhI

getweyyyyandyyasy


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COMPOSITE INTEGRATION FORMULA

The interval xn x0 must be divided into even

number of intervals so as to apply Simpsons

Rule


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)()(

180

:

........)(2)....(4)3/(

:

;

90

4)3/(

.. .............

;904)3/(

;90

4)3/(

0

4

6421310

2

5

12

42

5

132

20

5

210

0

2

42

20

IV

n

nnxxS

nn

IV

nnnxx

IV

xx

IV

xx

fxxh

ErrorGlobal

yyyyyyyyhII

Adding

xxfh

yyyhI

xxf

h

yyyhI

xxfh

yyyhI

n

nn


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EXAMPLE NO. 1

Integrate using a) trapezoidal rule b)Simpsons rule and also compute error bounds

Using trapezoidal rule where h is divided by 4parts.

dxx2

1/1

ValueTruexdxxxf 6931471.0log/1)(2

1

2

1

6970238.0

)5714285.06667.08.0(25.01225.0

T

I


18/23

TABLE 1

x f(x) T

1 1 0

1.25 0.8 1

1.5 0.66667 2

1.75 0.5714265 32.0 0.5 4

Actual error

= |true value numerically computed value|

= 0.003876


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BY USING TRAPEZOIDAL RULE

Global Error = h2/12 (xn x0) f()f(x)= 2/x3f() max = 2/13 =2f() min = 2/23 = 0.25

Error Upper Bound= 0.252 (2-1)(2) = 0.010426

12 Error Lower Bound

= 0.252 (2-1)(0.25) = 0.001302

12

0.001302


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BY SIMPSONS RULE

IS = 0.25[1+0.5+4(0.8+0.5714285)+2(0.6667)]3

= 0.6932595

By using Simpsons Rule the Global Error

= h4 (xn x0) fIV ()

180fIV(x) = 24/x5

x=1; fIV() max = 24/15 = 24x=2; fIV() max = 24/25 = 0.75


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UPPER BOUND FOR THE ERROR

= (0.25)4 x (2-1) x 24 = 0.00052

180

LOWER BOUND FOR THE ERROR

= (0.25)4 x (2-1) x 0.75 = 0.0000215

180


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BUT ACTUAL ERROR BYSIMPSONS RULE

= 0.6931471 0.6932595 = 0.000107

0.0000215 < 0.000107 < 0.00052

lower bound error < actual error < upper bounderror

Therefore the final answer is acceptable


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END OF CHAPTER 5

End of Module

Laboratory Experiments starts next week up toend of June

Project submission is also at the end of June Final Exam

June 7, 2013 (Friday Afternoon) (To be confirmed)

Coverage Chapter 4-5

Thanks for listening!

Documents

Chapter 5 – Numerical Integration.pptx