11.4 gaussian quadrature

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 (wjchoi@nlp.korea.ac.kr)Woo Yeon-Moon(wooym@nlp.korea.ac.kr)Kang Nam-Hee(nhkang@nlp.korea.ac.kr)

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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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11.3.1 Composite Trapezoid Rule11.3.2 Composite Simpson’s Rule

CHOI WonJong (wjchoi@nlp.korea.ac.kr)

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11.3.1 Composite Trapezoid Rule

CHOI WonJong (wjchoi@nlp.korea.ac.kr)

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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.

1

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

2b 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 a f a f x f x f bn

b ahn

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 0.693147180559951

b baa

dx x Cx

dx C Cx

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11.3.2 Composite Simpson’s Rule

CHOI WonJong (wjchoi@nlp.korea.ac.kr)

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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 ahn

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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11.3.3 Extrapolation Methods for Quadrature

Woo Yeon-Moon(wooym@nlp.korea.ac.kr)

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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 nh f a f x f x f b ch

41 2 2 3[ ( ) 4 ( ) 2 ( )] [ ( ) 4 ( ) ( )]

3 3h 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 hhh

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..

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01

1 1 1 1 3[ (1) (2)] [ ] 0.752 2 1 2 4

dx I f fx

11 1 1 2 1 17[ (1) 2 (1.5) (2)] [ ] 0.70834 4 1 1.5 2 24

I f f f

1[4 ( ) ( )]3 2

hA A A h

1 01[4 ] [4(0.7083) 0.7500] / 3 0.69443

I 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 0.05560.7083 0.6944 0.0139

(2)

EE

E E

2( ) ( )E h O h

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Romberg Integration Approximate an Error

Trapezoid 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 hhh

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

2degree 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 11

1( ) [ ( ) 2 ( ) ... 2 ( ) ( )]2

b

na

hf x dx dx f a f x f x f bx

Ⅰ Ⅱ

h=1 0.75000.69440.69330.6943

h=1/2 0.7083h=1/4 0.6970h=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

1 dxx

0.7500

0.6944

0.6932

0.6931

0.6931

0.6931

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 Gaussian Quadrature

Kang Nam-Hee (nhkang@nlp.korea.ac.kr)

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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

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4

±0.557753 0±0.77459±0.861136±0.339981

18/95/90.347850.652145

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

Woo Yeon-Moon(wooym@nlp.korea.ac.kr)

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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)