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 ([email protected])Woo Yeon-Moon([email protected])Kang Nam-Hee([email protected])

mailto:[email protected]

2

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

3

11.3.1 Composite Trapezoid Rule11.3.2 Composite Simpson’s Rule

CHOI WonJong ([email protected])

4

11.3.1 Composite Trapezoid Rule


5

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.

6


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

7


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

8


Example 11.9

n=1 n=2 n=3

n=4 n=20 n=100

9


Example 11.9

2

1

1 [log | | ]

1 2[log | 2 | ] [log |1| ] log 0.693147180559951

b baa

dx x Cx

dx C Cx

10

11.3.2 Composite Simpson’s Rule


11


Example 11.10

12


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

13


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

14


Example 11.10

15


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

16


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.

17

11.3.3 Extrapolation Methods for Quadrature

Woo Yeon-Moon([email protected])

18

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

19

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

20

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

21

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

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

22

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

23

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

24

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

25

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

26


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

27


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

28

Matlab function for Romberg Integration

29

11.4 Gaussian Quadrature

Kang Nam-Hee ([email protected])

30

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

31


Gaussian Quadrature Formular (cont.)

n=2

n=3

32


Example 11.13 integral of exp(-x2) Using G.Q

n Xi ci

23

4

±0.557753 0±0.77459±0.861136±0.339981

18/95/90.347850.652145

Table 11.2 parameters of Gaussian quadrature

33

Gaussian-Legendre Polynomials


34

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]

35

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


36

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


37

Example 11.14 (cont) So we can get

Apply Gaussian Quadrature to the integral with n = 2


38

Matlab function for Gaussian Quadrature


39

11.5 MATLAB’s Methods

Woo Yeon-Moon([email protected])

40

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)

Documents

11.4 gaussian quadrature