numerical integration continued --- simpson’s rules - we can add more segments or - we can use a...
Post on 21-Dec-2015
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Numerical integration continued ---
Simpson’s rules
- We can add more segments OR
- We can use a higher order polynomial
0
2
4
6
8
10
12
3 5 7 9 11 13 15
Simpson’s 1/3 rule
• use a second order interpolating polynomial
2
0
2
02
x
x
x
xdxxfdxxfI
If we use Lagrange form
dxxf
xxxx
xxxx
xfxxxx
xxxxxf
xxxx
xxxxI
x
x
21202
10
12101
200
2010
212
0
2
201
xxx
Integrate and do some algebra
210 43
xfxfxfh
I
0
2
4
6
8
10
12
3 5 7 9 11 13 15
If we use a=x0 and b=x2, and x1=(b+a)/2
6
4 210 xfxfxfabI
width Average height
Error for Simpson’s 1/3 rule
45
2880f
abE t
As with Trapezoidal rule, can use multiple applications of Simpson’s 1/3 rule
• need even number of segments, odd number of points
-5
0
5
10
15
20
25
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
9 points, 4 segments
n
xfxfxfxf
abIn
n
i
n
jji
3
241
5,3,1
2
6,4,20
As in multiple trapzoid, break integral up
n
n
x
x
x
x
x
xdxxfdxxfdxxfI
2
4
2
2
0
...
Substitute Simpson’s 1/3 rule for each integral and collect terms
Example: Numerically integrate from 0 to 1 using 1) single trapezoid, 2) multiple trapezoid, 3) single Simpson’s 1/3 and 4) multiple Simpson’s 1/3x y
0 -6.8700.125 1.2530.25 2.0540.375 0.6540.5 -0.167
0.625 0.4040.75 1.5720.875 1.301
1 -3.314 -8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
0 0.2 0.4 0.6 0.8 1 1.2
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1 1.2
True, analytic value of I is 0.4749
09.52
31.387.601
2
bfafabI
Really quite bad
%172,14749.0
09.54749.0
tE
1) Single trapezoidal rule
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1 1.2
2) Multiple trapezoidal rule
081.0
1875.075.0...25.0125.0202
25.0
22
1
1
ffffff
bfihafafh
In
i
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1 1.2%834749.0
081.04749.0
tE
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1 1.2
81.1
15.0403
5.0
43 210
fff
xfxfxfh
I
3) Single Simpson’s 1/3 rule
%4814749.0
081.04749.0
tE
-8
-6
-4
-2
0
2
4
0 0.2 0.4 0.6 0.8 1 1.2
4) Multiple Simpson’s 1/3 rule
n
xfxfxfxf
abIn
n
i
n
jji
3
241
5,3,1
2
6,4,20
4659.08*3
175.05.025.0*2875.0625.0375.0125.0*4001
3
241
5,3,1
2
6,4,20
fffffffffn
xfxfxfxf
abIn
n
i
n
jji
%24749.0
4650.04749.0
tE
Simpson’s 1/3 rule is limited to
• applications with equally-spaced data
• even number of segments
• odd number of points
Simpson’s 3/8 rule used when there are
• odd number of segments
• even number of points
Simpson’s 3/8 rule uses a third order Lagrange polynomial
b
a
b
adxxfdxxfI 3
Four equally spaced points, separated by
3210 338
3xfxfxxfhI
3
abh
or
8
33 3210 xfxfxfxfabI
Can do multiple segment application of Simpson’s 3/8 rule.
Can also mix and match Simpson’s 1/3 and 3/8 to fill up segments
Example:
-15
-10
-5
0
5
10
15
20
25
30
35
0 2 4 6 8 10 12 14
x y1 3.6832 5.5173 1.2224 -5.035 -9.546 -9.577 -3.848 7.0319 20.0210 29.2311 25.3712 -4.64
12 points, 11 segmentsEach 3/8 rule application takes 3 segmentsEach 1/3 rule application takes 2 segments
Neither 2 nor 3 go into 11
But 3 3’s and a 2 do.
-15
-10
-5
0
5
10
15
20
25
30
35
0 2 4 6 8 10 12 14
1/3 rule
3/8 rule
3/8 rule
3/8 rule
17.648
33
8
338
33
6
4
111098811
876558
543225
21002
2
0
11
8
8
5
5
2
xfxfxfxfxx
xfxfxfxfxx
xfxfxfxfxx
xfxfxfxx
dxxfdxxfdxxfdxxfIx
x
x
x
x
x
x
x
Higher order Newton-Cotes closed formulas
Simpson’s 1/3 - 2nd order Lagrange
Simpson’s 3/8 - 3rd order Lagrange
we can keep going
but don’t usually - Simpson is accurate enough when applied in multiple segments
Integration with unequal segments
If all unequal, stuck with multiple trapezoid rule application
If you can find some sets of equal segments, use Simpson’s rules