Post on 26-Dec-2015
CSE 330 : Numerical Methods
Lecture 16: Numerical Integration - Simpson’s Rule
Dr. S. M. Lutful Kabir
Visiting Professor, BRAC University
& Professor (on leave) IICT, BUET
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Basis of Simpson’s 1/3rd Rule
Trapezoidal rule was based on approximating the integrand by a first
order polynomial, and then integrating the polynomial in the interval of
integration. Simpson’s 1/3rd rule is an extension of Trapezoidal rule
where the integrand is approximated by a second order polynomial.
Hence
b
a
b
a
dx)x(fdx)x(fI 2
Where is a second order polynomial. )x(f2
22102 xaxaa)x(f
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Basis of Simpson’s 1/3rd Rule
Choose
)),a(f,a( ,ba
f,ba
22))b(f,b(an
das the three points of the function to evaluate a0, a1 and a2.
22102 aaaaa)a(f)a(f
2
2102 2222
ba
aba
aaba
fba
f
22102 babaa)b(f)b(f
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Basis of Simpson’s 1/3rd Rule
Solving the previous equations for a0, a1 and a2 give
22
22
02
24
baba
)a(fb)a(abfba
abf)b(abf)b(faa
2212
2433
24
baba
)b(bfba
bf)a(bf)b(afba
af)a(afa
2222
222
baba
)b(fba
f)a(f
a
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Basis of Simpson’s 1/3rd Rule
Then
b
a
dx)x(fI 2
b
a
dxxaxaa 2210
b
a
xa
xaxa
32
3
2
2
10
32
33
2
22
10
aba
aba)ab(a
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Basis of Simpson’s 1/3rd Rule
Substituting values of a0, a1, a 2 give
)b(fba
f)a(fab
dx)x(fb
a 24
62
Since for Simpson’s 1/3rd Rule, the interval [a, b] is brokeninto 2 segments, the segment width
2
abh
)b(fba
f)a(fh
dx)x(fb
a 24
32
Hence
Because the above form has 1/3 in its formula, it is called Simpson’s 1/3rd Rule.
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Example 1
a) Use Simpson’s 1/3rd Rule to find the approximate value of x
The distance covered by a rocket from t=8 to t=30 is given by
30
8
892100140000
1400002000 dtt.
tlnx
b) Find the true error, tE
c) Find the absolute relative true error, t
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Solution
a)
30
8
)( dttfx
)b(fba
f)a(fab
x2
46
)(f)(f)(f 3019486
830
67409017455484426671776
22.).(.
m.7211065
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Solution (cont)b) The exact value of the above integral is
30
8
892100140000
1400002000 dtt.
tlnx
m.3411061True Error
72110653411061 ..Et m.384
c) Absolute relative true error,
%.
..t 100
3411061
72110653411061
%.03960
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Multiple Segment Simpson’s 1/3rd Rule
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Multiple Segment Simpson’s 1/3rd Rule
Just like in multiple segment Trapezoidal Rule, one can subdivide the interval
[a, b] into n segments and apply Simpson’s 1/3rd Rule repeatedly overevery two segments. Note that n needs to be even. Divide interval[a, b] into equal segments, hence the segment width
n
abh
nx
x
b
a
dx)x(fdx)x(f0
where
ax 0 bxn
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Multiple Segment Simpson’s 1/3rd Rule
Apply Simpson’s 1/3rd Rule over each interval,
...)x(f)x(f)x(f
)xx(dx)x(fb
a
6
4 21002
...)x(f)x(f)x(f
)xx(
6
4 43224
f(x)
. . .
x0 x2 xn-
2
xn
x
.....dx)x(fdx)x(fdx)x(fx
x
x
x
b
a
4
2
2
0
n
n
n
n
x
x
x
x
dx)x(fdx)x(f....2
2
4
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Multiple Segment Simpson’s 1/3rd Rule
...)x(f)x(f)x(f
)xx(... nnnnn
6
4 23442
6
4 122
)x(f)x(f)x(f)xx( nnn
nn
Since
hxx ii 22 n...,,,i 42
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Multiple Segment Simpson’s 1/3rd Rule
Then
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)()(4)(2)( 210 xfxfxfhdxxf
b
a
...)x(f)x(f)x(f
h
6
42 432
6
)()(4)(2 234 nnn xfxfxfh
64
2 12 )x(f)x(f)x(fh nnn
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Multiple Segment Simpson’s 1/3rd Rule
b
a
dx)x(f ...)x(f...)x(f)x(f)x(fh
n 1310 43
)}]()(...)()(2... 242 nn xfxfxfxf
)()(2)(4)(3
2
2
1
10 n
n
evenii
i
n
oddii
i xfxfxfxfh
)()(2)(4)(3
2
2
1
10 n
n
evenii
i
n
oddii
i xfxfxfxfn
ab
a) Use four segment Simpson’s 1/3rd Rule to find the approximate value of x.
b) Find the true error, for part (a).c) Find the absolute relative true error, for part (a).
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Example 2Use 4-segment Simpson’s 1/3rd Rule to approximate the distance
tE
covered by a rocket from t= 8 to t=30 as given by
30
8
8.92100140000
140000ln2000 dtt
tx
a
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SolutionUsing n segment Simpson’s 1/3rd Rule,
4
830 h 5.5
So )8()( 0 ftf
)5.58()( 1 ftf )5.13(f
a)
)5.55.13()( 2 ftf )19(f
)5.519()( 3 ftf )5.24(f
)( 4tf )30(f
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Solution (cont.)
)()(2)(4)(3
2
2
1
10 n
n
evenii
i
n
oddii
i tftftftfn
abx
2
2
3
1)30()(2)(4)8(
)4(3
830
evenii
i
oddii
i ftftff
)30()(2)(4)(4)8(12
22231 ftftftff
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Solution (cont.)
)30()19(2)5.24(4)5.13(4)8(6
11fffff
6740.901)7455.484(2)0501.676(4)2469.320(42667.1776
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m64.11061
cont.
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Solution (cont.)
In this case, the true error is
64.1106134.11061 tE
b)
m30.0
The absolute relative true error
%10034.11061
64.1106134.11061
t
c)
%0027.0
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Solution (cont.)
Table 1: Values of Simpson’s 1/3rd Rule for Example 2 with multiple segments
n Approximate Value
Et |Єt |
2468
10
11065.7211061.6411061.4011061.3511061.34
4.380.300.060.010.00
0.0396%0.0027%0.0005%0.0001%0.0000%
Simpson’s 3/8 Rule of Integration
In a similar fashion, Simpson 3/8 rule for integration can be derived by approximating the given function f(x) with the 3rd order (cubic) polynomial f3(x)
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3
2
1
0
32
33
22103
,,,1
)(
a
a
a
a
xxx
xaxaxaaxf
Using Lagrange interpolation, the cubic polynomial function that passes through 4 data points can be explicitly given as
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Simpson’s 3/8 Rule of Integration
3231303
2102
321202
310
1312101
3200
302010
3213
xfxxxxxx
xxxxxxxf
xxxxxx
xxxxxx
xfxxxxxx
xxxxxxxf
xxxxxx
xxxxxxxf
3210 338
3xfxfxfxf
hI
Following the same procedure of Simpson’s 1/3 rule, the expression of the Integral results,
Multi Segments for Simpson’s 3/8 Rule
Similarly the expression for the multi segment Simpson’s Rule can be derived as follows:
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nnnn xfxfxfxf
xfxfxfxfxfxfxfxfhI
123
65433210
33.....
3333
8
3
n
n
ii
n
ii
n
ii xfxfxfxfxf
h 3
,..9,6,3
1
,..8,5,2
2
,..7,4,10 233
8
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Exercise Find the distance covered by the rocket in between
t=8sec and t=30 sec using the expression of previous examples. Apply Simpson’s 3/8 rule and use six segments for finding the distance.
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Thanks
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