5. Integration

2. Quadrature as Box Counting

3. Algorithm: Trapezoid Rule

4. Algorithm: Simpson’s Rule

5. Integration Error

6. Algorithm: Gaussian Quadrature

7. Empirical Error Estimate

8. Experimentation

9. Higher Order Rules

5.2. Quadrature as Box Counting

Riemann: /

01

limb b a h

ih

ia

f x d x h f x

Numerical:

1

b N

i iia

f x d x w f x

w = weight

• Keeping N finite can still give exact results, e.g., polynomials.

• Aim: accurate result for small N.

• No universal “best” algorithm.

Tips

• Remove singularities first.

• By putting them at endpoints of sub-intervals

• By change of variable.

1 0 1

1 1 0

x f x d x x f x d x x f x d x

1 11/3 3

0 0

3x d x y d y 1/3y x

21 1

2 20 0

12

1 2

f yf xd x d y

x y

2 1y x

• Speed up or slow down (by change of variable or step size)

in slowly or rapidly varying region.

Algorithms for Evenly Spaced Points

Evenly spaced points :

1

b N

i iia

f x d x w f

1ix a i h b a

hN

1, ,i N

Name Degree wi

Trapezoid 1

Simpson’s 2

3/8 3

Milne 4

11 ,1

2h

11 , 4 ,1

3h

31 , 3 , 3 ,1

8h

114 , 64 , 24 , 64 ,14

45h

i if f x

8.3. Algorithm: Trapezoid Rule

1

1

1

2

i

i

x

i i

x

f x d x h f f

0 1 1 2 2 1 1

1

2

b

N N N N

a

f x d x h f f f f f f f f

0 1 2 2 1

1 1

2 2N N Nh f f f f f f

1

b N

i iia

f x d x w f x

1 1,1 , ,1 ,

2 2iw h

1 0 0

0

1

2

h

f x d x f f h f h

f x ax b

0 1

1

2h f f

0f b

1f a h b

1 0

0

f fa

hb f

1 0

0

f ff x x f

h

5.4. Algorithm: Simpson’s Rule

2f x ax bx c

21

0

21

f ah bh c

f c

f ah bh c

21 1 0

1 1

0

1

21

2

ah f f f

bh f f

c f

322

3

h

h

f x d x a h c h

1 1 0 0

1 22

3 3h f f f f

1 0 1

14

3h f f f

2

1 2

14

3

i

i

x

i i i

x

f x d x h f f f

1

b N

i iia

f x d x w f x

11 , 4 , 2 , 4 , 2 , , 4 , 2 , 4 ,1

3iw h

N = odd int

11

2N N

int1

11 4 1

3

N

ii

w h N

1N h

0 1 2 2 3 4 4 3 2 2 1

14 4 4 4

3

b

N N N N N N

a

f x d x h f f f f f f f f f f f f

0 1 2 3 4 3 2 14 2 4 2 4 2 43 N N N N

hf f f f f f f f f

2

1 2

14

3

i

i

x

i i i

x

f x d x h f f f

b a

5.5. Integration Error

42 3 40 0 0 0 0

1 1 1

2 3! 4!f x f f x f x f x f x

Expand f at middle of interval x [ h, h ] :

43 50 0 0

2 22

2 3 4! 5

h

h

f x d x f h f h f h

Error, Trapezoid :

int

1

2

b ah

N

3int 0tE O N f h

3

02int

1O f b a

N

Error,Simpson :

4 5int 0sE O N f h

5404

int

1O f b a

N

n data points in each interval

n = 2

n = 4

Relative error :,

,t s

t s

E

f

1

nn

n

fb a

N f

1

,

nn

t s n

fb a

N f n = 2,4 for t , s

Round-off error is random : ro mN m machine precision ~ 107 ( single prec) ~ 1015 ( double prec)

,tot t s ro ,

1 10

2tot

t s m

dn

d N N

Min tot : ,t s m

Set scale to

1nf

f 1b a

Trapezoid :

1

nn

mn

fb a N

N f

2

2 11

n nn

m

fN b a

f

2/51510N

2

2 1nmN

tot mN 1

12 1n

m

610 4/515 1210 10tot

Simpson :

2/91510N 10/310 2154 8/915 40/3 1410 10 4.6 10tot

Conclusions

• Simpson’s rule is better than trapezoid.

• It’s possible to get tot m .

• Best result is obtained for N ~ 1000 instead of .

5.6. Algorithm: Gaussian Quadrature

( ) ( )b

a

I d x f x w x1

( )n

k kk

S A f x

( )

( ) '( )

b

kk ka

xA dx

x x x

where is the nth degree member of a complete set of orthogonal polynomials, and { xk } are its roots.

I = S if f is an 2n-1 degree polynomial. Proof :

Integral Polynomial weight Limits

Legendre 1 ( 1, 1 )

Hermite exp( x2 ) ( , )

Laguerre exp( x ) ( , )

Chebyshev I ( 1x2 )1/2 ( 1, 1 )

1

1

( )d x f x

2

( )xd x e f x

( )xd x e f x

1

21

1( )

1d x f x

x

5.6.1. Mapping Integration Points

An integral b

a

F y d y can be transformed into 1

1

f x d x

y x by the linear transform

a

b

Thus

1

21

2

b a

b a

1

2

1

2 y b a x b a

f x b a F y

1

2d y b a d x

1

2y b a x b a

An integral 0

F y d y

can be transformed into 1

1

f x d x

1y

x

by the linear transform

02

0

Thus 2

21

2 1

1 xy

x

f x F yx

21

d y d xx

1 1

1 2y

x

1

2 1

x

x

0

Similarly, one get the following transforms

Interval W y

[ a , b ]

[ 0 , ]

[ , ]

[ b , ]

[ 0 , b ]

1

interval 1

F y d y f x d x

1

2W b a

21

Wx

1

2b a x b a

1

2 1

x

x

0

2

22

2 1

1

xW

x

2

2

1

x

x

21

Wx

2

2

1

2

bW

x

1

2 1

xb

x

1 11

2

xb

x

f x W F y x

5.7. Empirical Error Estimate (Ex.5.1)

Relative error, , as a function of N

for the trapezoid, Simpson, & Gaussian methods, in the calculation of

Answer

11

0

1tI e d t e

numerical exact

exact

5.8. Experimentation

Evaluate

2

0

sin 1000 x d x

and 2

0

sin 100x

x d x

What’s wrong ?

5.9. Higher Order Rules

2 4b

a

A h f x d x h h

Let A(h) be the numerical evaluation of an integral with leading error h2, i.e.,

2 41 1

2 4 16

b

a

hA f x d x h h

44 1 1

3 2 3 4

b

a

hA A h f x d x h Romberg’s

extrapolation

see Burden, § 4.5