Dr. Nirav Vyas numerical method 3.pdf

8/20/2019 Dr. Nirav Vyas numerical method 3.pdf

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Numerical Methods - Numerical

Integration

N. B. Vyas

Department of Mathematics,Atmiya Institute of Tech. and Science, Rajkot (Guj.)[email protected]

N. B. Vyas Numerical Methods - Numerical Integration


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Numerical Integration

Let I =

ba

y dx where y = f (x) takes the values y0, y1, . . . , yn for

x0, x1, . . . , xn



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Let I =

ba


x0, x1, . . . , xn

Let us divide the interval (a, b) into n sub-intervals of width h sothat x0 = a, x1 = a + h = x0 + h, x2 = x0 + 2h, . . .,xn = x0 + nh = b then



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Let I =

ba


x0, x1, . . . , xn


I = ba

y dx = x0+nhx0

f (x) dx



5/18


Let I =

ba


x0, x1, . . . , xn


I = ba

y dx = x0+nhx0

f (x) dx

Trapezoidal rule:

b=x0+nh

a=x0

f (x)dx = h

2 [(y0 + yn) + 2 (y1 + y2 + .... + yn)]; h = b − a

n

If the number of strips is increased; that is, h is decreased, thenthe accuracy of the approximation is increased.



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Simpson’s 1

3rd rule:


N I


7/18


Simpson’s 1

3rd rule:

x0+nh x0

f (x)dx = h3 [(y0 + yn) + 4(y1 + y3 + ....)

+2(y3 + y4 + ....)]; h = b−an


N I


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Simpson’s 1

3rd rule:

x0+nh x0

f (x)dx = h3 [(y0 + yn) + 4(y1 + y3 + ....)

+2(y3 + y4 + ....)]; h = b−an

while applying this rule, the given interval must be divided intoeven number of equal sub-intervals. i.e. n must be even.


N I


9/18


Simpson’s 1

3rd rule:

x0+nh x0

f (x)dx = h3 [(y0 + yn) + 4(y1 + y3 + ....)

+2(y3 + y4 + ....)]; h = b−an


Simpson’s 3

8th rule:


N I


10/18


Simpson’s 1

3rd rule:

x0+nh x0

f (x)dx = h3 [(y0 + yn) + 4(y1 + y3 + ....)

+2(y3 + y4 + ....)]; h = b−an


Simpson’s 3

8th rule:

x0+nh x0 f (x)dx =

3h

8 [(y0 + yn) + 3(y1 + y2 + y4 + y5 + ....)+2(y3 + y6 + ....)]; h =

b−an




11/18


Simpson’s 1

3rd rule:

x0+nh x0

f (x)dx = h3 [(y0 + yn) + 4(y1 + y3 + ....)

+2(y3 + y4 + ....)]; h = b−an


Simpson’s 3

8th rule:

x0+nh x0 f (x)dx =

3h

8 [(y0 + yn) + 3(y1 + y2 + y4 + y5 + ....)+2(y3 + y6 + ....)]; h =

b−an

while applying this rule, the number of sub-intervals should betaken as a multiple of 3 i.e. n must be multiple of 3




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Gaussian Integration Formula:

1 −1

f (t)dt =

n

i=1

wif (ti)

It should be noted here that, t = ±1 is obtained by setting

x = 1

2 [(b + a) + t (b − a)]




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Gaussian Integration Formula: The following table gives thevalues for n = 2, 3, 4, 5


Example


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Example

Ex. Evaluate1

0

e−x2

dx by using Gaussion integration formula for

n = 3.

Sol. Here, we have to first convert the given integral from 0 to 1 into

an integral from −1 to 1. x = 12 [(b + a) + t (b − a)], a = 0 and

b = 1

∴ x = t + 1

2 ⇒ dx =

dt

2

∴

1

0 exp(−x

2

)dx =

1

2

1

−1 exp

−

1

4 (t + 1)

2dt


Error


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Error

Error in Quadrature Formula:

If y p is a polynomial representing the function y = f (x) in theinterval [x0, xn] the error in the quadrature formula is given by

E =

xn x0

f (x) =

xn x0

y pdx


Error


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Error

Error in Trapezoidal rule:

|error| ≤ (b − a)h2

12|f (M )|

where f (M ) = max|f 0(x)|, |f

1(x)|, ..., |f

n−1(x)|

∴ error is of order h2

total error = dh3

12 y0 + y

1 + ... + y

n−1


Error


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Error

Error in Simpson’s 1

3rd rule:

|error| ≤ (b − a) h4

180|f 4(M )|

where f 4(M ) = max|y40|, |y

42|, ..., |y

4n−2|


total error = h5

90 y40 + y

42 + ... + y

4n−2


Error


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Error

Error in Simpson’s 3

8th rule:

|error| ≤ (b − a)h4

80|f 4(M )|

where f 4(M ) = max|y40|, |y

43|, ..., |y

4n−3|


total error = 3h5

80 y40 + y

43 + ... + y

4n−3


Dr. Nirav Vyas numerical method 3.pdf

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