11. numerical differentiation and integration 11.3 better numerical integration, 11.4 gaussian...
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11. Numerical Differentiation and Integration11.3 Better Numerical Integration, 11.4 Gaussian Quadrature, 11.5 MATLAB’s Methods
Natural Language Processing LabDept. of Computer Science and Engineering, Korea Univertity
CHOI Won-Jong ([email protected])Woo Yeon-Moon([email protected])Kang Nam-Hee([email protected])
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Contents
11.3 BETTER NUMERICAL INTEGRATION 11.3.1 Composite Trapezoid Rule 11.3.2 Composite Simpson’s Rule 11.3.3 Extrapolation Methods for Quadrature
11.4 GAUSSIAN QUADRATURE 11.4.1 Gaussian Quadrature on [-1, 1] 11.4.2 Gaussian Quadrature on [a, b]
11.5 MATLAB’s Methods
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11.3 BETTER NUMERICAL INTEGRATION
Composite integration(복합적분 ) : Applying one of the lower order methods presented in the previous section repeatedly on several sub intervals.
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11.3.1 Composite Trapezoid Rule
If we divide the interval of integration, [a, b], into two or more subintervals and use the trapezoid rule on each subintervals, we obtain the composite trapezoid rule.
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11 1
1 1
( ) ( ) ( ) [ ( ) ( )] [ ( ) ( )]2 2
[ ( ) 2 ( ) ( )] [ ( ) 2 ( ) ( )]2 4
b x b
a a x
h hf x dx f x dx f x dx f a f x f x f b
h b af a f x f b f a f x f b
2
b ah
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11.3.1 Composite Trapezoid Rule
If we divide the interval into n subintervals, we get
1
1
1 1
1 1
( ) ( ) ( )
[ ( ) ( )] [ ( ) ( )]2 2
[ ( ) 2 ( ) 2 ( ) ( )]2
n
b x b
a a x
n
n
f x dx f x dx f x dx
h hf a f x f x f b
b af a f x f x f b
n
b ah
n
MATLAB CODE
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11.3.1 Composite Trapezoid Rule
Example 11.9
n=1 n=2 n=3
n=4 n=20 n=100
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11.3.1 Composite Trapezoid Rule
Example 11.9
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1
1[log | | ]
1 2[log | 2 | ] [log |1| ] log 0.69314718055995
1
b baa
dx x Cx
dx C Cx
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11.3.2 Composite Simpson’s Rule
Example 11.10
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11.3.2 Composite Simpson’s Rule
Applying the same idea of subdivision of intervals to Simpson’s rule and requiring that n be even gives the composite Simpson rule.
[a,b] 를 two subintervals [a,x2], [x2, b] 로 나눈다면 ,
2 ,2 4
b a b ax h
2
2
1 2 2 3
( ) ( ) ( )
[ ( ) 4 ( ) ( )] [ ( ) 4 ( ) ( )]3 3
b x b
a a xf x dx f x dx f x dx
h hf a f x f x f x f x f b
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11.3.2 Composite Simpson’s Rule
In general, for n even, we have h=(b-a)/n, and Simpson’s rule is
b ah
n
1 2 3 4 2 1( ) [ ( ) 4 ( ) 2 ( ) 4 ( ) 2 ( ) 2 ( ) 4 ( ) ( )]3
b
n na
hf x dx f a f x f x f x f x f x f x f b
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11.3.2 Composite Simpson’s Rule
Example 11.10
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11.3.2 Composite Simpson’s Rule
Example 11.11 Length of Elliptical Orbit
2 2 2 2
3( ) cos( ), ( ) sin( )
4
( ') ( ') 0.25 16sin ( ) 9cos ( )b b
a a
x r r y r r
L x y dr r r dr
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11.3.2 Composite Simpson’s Rule
Example 11.11 Length of Elliptical Orbit
2 2 2 2
3( ) cos( ), ( ) sin( )
4
( ') ( ') 0.25 16sin ( ) 9cos ( )b b
a a
x r r y r r
L x y dr r r dr
days 0 10 20 30 40 50 60 70 80 90 100r = [0.00 1.07 1.75 2.27 2.72 3.14 3.56 4.01 4.53 5.22 6.28]
Using Composite Simpson’s Rule and the length between day 0 and 10 (n=20) is 0.88952. (Trapezoid=0.889567, Text=0.8556)Using Composite Simpson’s Rule and the length between day 60 and 70 (n=20) is 0.382108. (Trapezoid=0.382109, Text=0.3702)The former is 2.3279 times faster than the latter.
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Richardson Expolation
Truncation error(절단 오차 )• ( , ) ( , )I I f h E f h
21 1
1
( ) [ ( ) 2 ( ) ... 2 ( ) ( )]2
bj
n jja
hf x dx f a f x f x f b c h
2
1 1[ ( ) 2 ( ) ... 2 ( ) ( )]2 n
hf a f x f x f b ch
41 2 2 3[ ( ) 4 ( ) 2 ( )] [ ( ) 4 ( ) ( )]
3 3
h hf a f x f x f x f x f b ch
사다리꼴
simpson
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Richardson Expolation
To obtain an estimate that is more accurate• using two or more subintervals (h를 줄임 )
- 그러나 , 세부구간의 수가 일정한 범위를 넘어서면 round-off error가 커지게 된다 .
Richardson Extrapolation간격이 다른 2개의 식을 구한 결과를 대수적으로 정리함으로써보다 정확한 값을 산출
계산오차
세부 구간의 수
simpson
trapezoid
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Richardson Extrapolation
Richardson Extrapolation using the trapezoid rule
(if h_2 = ½ h_1)
2 21 1 2 2( ) ( )T TI I h ch I h ch
2 14 ( ) ( )
3
I h I hI
2 12 2
1
2
( ) ( )( )
1
I h I hI I h
h
h
Simpson rules
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Example 11.12 Integral of 1/x
start with one subinterval (h=1)
two subintervals (h=1/2)
to apply Richardson extrapolation
exact value of the integral is ln(2)=0.693147..
2
0
1
1 1 1 1 3[ (1) (2)] [ ] 0.75
2 2 1 2 4
dxI f f
x
1
1 1 1 2 1 17[ (1) 2 (1.5) (2)] [ ] 0.7083
4 4 1 1.5 2 24I f f f
1[4 ( ) ( )]
3 2
hA A A h
1 0
1[4 ] [4(0.7083) 0.7500]/ 3 0.6944
3I I I
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Example 11.12 Integral of 1/x
Form a table of the approximations
0.6944 ≠0.693147
Ⅰ Ⅱ
h=1 0.75000.6944
h=1/2 0.7083
0
1
20 1
0.75 0.6944 0.0556
0.7083 0.6944 0.0139
(2)
E
E
E E
2( ) ( )E h O h
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Romberg Integration
Approximate an ErrorTrapezoid rules : Richardson extrapolation :
continued ( using simpson rules)4 4
1 1 2 2( ) ( )S SI I h ch I h ch
2 116 ( ) ( )
15
I h I hI
2 12 4
1
2
( ) ( )( )
1
I h I hI I h
h
h
2( )O h4( )O h
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Romberg Integration
Improving the result by Richardson extrapolation
Romberg integration : iterative procedure using Richardson extrapolation
k means the improving level(= )
2 4 6 8 101 2 3 4 5E c h c h c h c h c h
4 ( / 2) ( )( )
4 1
k
k
I h I hI h
2
degree of the error
1st 2nd 3rd
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Example 11.12 Integral of 1/x using Romberg Integration
Trapezoid rule
For k=0, I_0 = 0.75 For k=1, I_1 = 0.7083 For k=2, I_2 = 0.6941
To apply Richardson extrapolation
2
1 1
1
1( ) [ ( ) 2 ( ) ... 2 ( ) ( )]
2
b
n
a
hf x dx dx f a f x f x f b
x
Ⅰ Ⅱ
h=1 0.75000.69440.69330.6943
h=1/2 0.7083
h=1/4 0.6970
h=1/8 0.6941
1( ) 4 ( ) ( )
3 2
hA h A A h
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Example 11.12 Integral of 1/x using Romberg Integration
second level of extrapolation
1( ) 16 ( ) ( )
15 2
hC h B B h
Ⅰ Ⅱ Ⅲ
h=1 0.75000.69440.6933
h=1/2 0.7083 [16(0.6933)-0.6944]/15
h=1/4 0.6970
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Example 11.12 Integral of 1/x using Romberg Integration
five levels of extrapolation to find values for 2
1
1dx
x0.750
00.694
40.693
20.693
10.693
10.693
1
0.7083
0.6933
0.6931
0.6931
0.6931
0.6970
0.6932
0.6931
0.6931
0.6941
0.6931
0.6931
0.6934
0.6931
0.6932
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Matlab function for Romberg Integration
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11.4.1 Gaussian Quadrature on [-1,1]
Gaussian Quadrature Formular
Get the definite integration of f(x) on [-1,1] using linear combinations of coefficient ck and evaluated function value f(xk) at the point xk
Appropriate values of the points xk and ck depend on the choice of n
By choosing the quadrature point x1 ,… xn as the n zeros of the nth-degree Gauss-Legendre polynomial, and by using the appropriate coefficients, the integration formular is exact for polynomials of degree up to 2n-1
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11.4.1 Gaussian Quadrature on [-1,1]
Gaussian Quadrature Formular (cont.)
n=2
n=3
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11.4.1 Gaussian Quadrature on [-1,1]
Example 11.13 integral of exp(-x2) Using G.Q
n Xi ci
2
3
4
±0.557753
0
±0.77459
±0.861136
±0.339981
1
8/9
5/9
0.34785
0.652145
Table 11.2 parameters of Gaussian quadrature
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Gaussian-Legendre Polynomials
11.4.1 Gaussian Quadrature on [-1,1]
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Extends Gaussian Quadrature for f(t) on [a, b] by Transformation f(t) on [a, b] to f(x) on [-1,1]
For the given integral
change interval of t by using next formular
so the interval
11.4.2 Gaussian Quadrature on [a,b]
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Extends Gaussian Quadrature for f(t) on [a, b] (cont.) f(t) rewrite for variable x
remark the factor (b-a)/2 (∵td convert to dx)
Apply f(x) to the integral
11.4.2 Gaussian Quadrature on [a,b]
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Example 11.14 integral of exp(-x2) on [0,2] using G.Q with n = 2
Consider again the integral
Transform f(t) on [0,2] to f(x) on [-1,1] using next formular
11.4.2 Gaussian Quadrature on [a,b]
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Example 11.14 (cont) So we can get
Apply Gaussian Quadrature to the integral with n = 2
11.4.2 Gaussian Quadrature on [a,b]
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Matlab function for Gaussian Quadrature
11.4.2 Gaussian Quadrature on [a,b]
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11.5 MATLAB’s Methods p=polyfit(x,y,n) – find the coefficients of the p
olynomial of degree n polyder(p) - calculates the derivative of polynom
ials diff(x) - x = [1 2 3 4 5];
y = diff(x)y = 1 1 1 1
traps(x,y) Q=quad(‘f’,xmin,xmax) (simpson rules) Q=quad8(‘f’,xmin,xmax) (Newton-Cotes eight-panel
rule)