cis541 07 integration

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Numerical Integration

Roger Crawfis



March 24, 2015 OSU/CSE 541 2

Quadrature

• We talk in terms of Quadrature Rules – 1. The process of making something square. 2.

Mathematics The process of constructing a squareequal in area to a given surface. 3. Astronomy A

configuration in which the position of one celestial

od! is "#$ from another celestial od!% as measured

from a third. – The American &eritage' (ictionar!) *ourth +dition. ,###



March 24, 2015 OSU/CSE 541 3

Outline

• (efinite -ntegrals

• ower and /pper 0ums

– Reimann -ntegration or Reimann 0ums• /niforml!1spaced samples

– Trape2oid Rules

– Romerg -ntegration

– 0impson3s Rules – Adaptive 0impson3s 0cheme

• 4on1uniforml! spaced samples – 5aussian Quadrature *ormulas



March 24, 2015 OSU/CSE 541 4

What does an integral represent6

7asic definition of an integral)

Motivation

∫ =b

adx x f area89 ∫ ∫ =

d

c

b

adxdy x f volume89

∑∫ =

∞→∆=

n

k

k

b

a n x x f dx x f

:

89lim89

sum of height × width

f 9 x8

∆ x

n

ab x

−=∆where



March 24, 2015 OSU/CSE 541 5

• +valuate the integral% without doing the

calculation anal!ticall!.

• 4ecessar! when either)

– -ntegrand is too complicated to integrate anal!ticall!

– -ntegrand is not precisel! defined ! an equation% i.e.%we are given a set of data 9 xi , yi8% i ; :% ,% <% =%n

Motivation

∫ = b

adx x f I 89

dxe x

x x>.#

,

# >.#:

8:cos9,

∫ + ++



March 24, 2015 OSU/CSE 541 6

• -ntegration is a summing process. Thus virtuall!all numerical appro?imations can e represented

!

in which wi are the weights% xi are the sampling

points% and E t is the truncation error • @alid for an! function that is continuous on the

closed and ounded interval of integration.

t

b

ai

n

i

i E x f wdx x f I +== ∫ ∑=

8989:

eimann Integral !heorem



March 24, 2015 OSU/CSE 541 7

• The most common numerical integration formula

is ased on equall! spaced data points.

• (ivide x" , xnB into n intervals 9n≥:8

∫ n x

xdx x f

#

89

#artitioning the Integral

: ,

#

# : :

9 8 9 8 9 8 9 8n

n

n

x x x x

x x x x

f x dx f x f x f x

−

= + + +∫ ∫ ∫ ∫ L



March 24, 2015 OSU/CSE 541 8

$%%er &ums

• Assume that f'x()" ever!where*

• -f within each interval% we could determine

the ma?imum value of the function% then we

have)

• where

( )#

:

:

#

9 8n x n

i i i

i x

f x M x x−

+=

≤ −∑∫

{ }:sup 9 8 )i i i M f x x x x += ≤ ≤ &u%remum + least

u%%er bound



March 24, 2015 OSU/CSE 541 9

$%%er &ums

• 5raphicall!)

x" x x- x. x/



March 24, 2015 OSU/CSE 541 10

0ower &ums

• ikewise% still assuming that f'x()"

ever!where*

• -f within each interval% we could determine

the minimum value of the function% then we

have)

• where

( )#

:

:#

9 8n x n

i i ii x

f x m x x−

+=

≥ −

∑∫

{ }:inf 9 8 )i i im f x x x x += ≤ ≤ Infimum + greatest

lower bound



March 24, 2015 OSU/CSE 541 11

0ower &ums

• 5raphicall!

x" x x- x. x/



March 24, 2015 OSU/CSE 541 12

1iner #artitions

• We now have a ound on the integral of thefunction for some partition 9 x" ,**,xn8)

• As n→∞% one would assume that the sum of theupper ounds and the sum of the lower oundsapproach each other.

• This is the case for most functions% and we callthese Riemann1integrale functions.

( ) ( )#

: :

: :

# #

9 8n x

n n

i i i i i i

i i x

m x x f x M x x− −+ +

= =

− ≤ ≤ −∑ ∑∫



March 24, 2015 OSU/CSE 541 13

2ounding the Integral

• 5raphicall!

x" x x- x. x/



March 24, 2015 OSU/CSE 541 14


• &alving each interval 'much like 0ab()

x" x/ x3 x4 x5



March 24, 2015 OSU/CSE 541 15


• ne more time)

x" x3 x5 x x4



March 24, 2015 OSU/CSE 541 16

Monotonic 1unctions

• 4ote that if a function is monotonicall!

increasing 9or decreasing8% then the lower

sum corresponds to the left partition values%and the upper sum corresponds to the right

partition values.

x" x/ x3 x4 x5



March 24, 2015 OSU/CSE 541 17

0ab and Integration

• Thinking ack to la:% what were the limits

or the integration6

• -s the sin function monotonic on this

interval6

• 0hould the Reiman sum e an upper or

lower sum6



March 24, 2015 OSU/CSE 541 18

#olynomial A%%roximation

• Rather than search for the ma?imum or minimum%

we replace f'x( with a known and simple function.

• Within each interval we appro?imate f 9 x8 ! an mth order pol!nomial.

m

mm xa xa xaa x % ++++= ...89 ,

,:#



March 24, 2015 OSU/CSE 541 19

• The m3s 9order of the pol!nomials8 ma! e the

same or different.

• (ifferent choices for m3s lead to different

formulas)

: : ,

:

# #

: ,# # #

9 8 9 8 9 8 ... 9 8n m m m n

nm n mn

x x x x

m m m x x x x

f x dx % x dx % x dx % x dx+ + +

+ −= + + +∫ ∫ ∫ ∫

89<DEs0impsonFcuic<

89:D<s0impsonFquadratic,

89Trape2oidlinear :

+rror *ormulaGol!nomial

H

H

,

hO

hO

hO

m

Newton+6otes 1ormulas



March 24, 2015 OSU/CSE 541 20

• 0implest wa! to appro?imate the area under a

curve – using first order polynomial 9a straight

line8• /sing 4ewton3s form of the interpolating

pol!nomial)

• 4ow% solve for the integral)

( ) ( )∫ ∫ ≈= b

a

b

adx x %dx x f I :

( ) ( ) ( ) ( )

( )a xab

a f b f a f x % −

−−

+=:

!ra%e7oid ule



March 24, 2015 OSU/CSE 541 21

!ra%e7oid ule

( ) ( ) ( )

( )∫

−

−−

+≈b

adxa x

ab

a f b f a f I

( )( ) ( )[ ]b f a f

ab I +

−≈

,

a b

f 9a8 f 9b8

heightaveragewidth ×≈ I

!ra%e7oid ule



March 24, 2015 OSU/CSE 541 22

!ra%e7oid ule

• -mprovement6

x" x x- x. x/



March 24, 2015 OSU/CSE 541 23

• The integration error is)

• Where h 8 b + a and ξ is an unknown point where

a I ξ I b 9intermediate value theorem8

• Jou get e?act integration if the function% f, is

linear 9 f ′′ ; #8

( ) ( ) ,<

:,

89

:,

:h f

abh f E t ξ ξ ′′−

−=′′−=89

<

hO

!ra%e7oid ule Error



March 24, 2015 OSU/CSE 541 24

,

89 xe x f −=-ntegrate from a ; # to b ; ,

( ) ( ) ( )[ ] [ ]

#:E<.:89:

8#98,9,

8#,9,

#H

,

#

,

=+×=

+−=+−≈

=

−

−∫

ee

f f b f a f ab

dxe I x

/se trape2oidal rule)

Exam%le



March 24, 2015 OSU/CSE 541 25

+stimate error) ( ) <

:,

:h f E t ξ ′′−=

Where h 8 b + a and a I ξ I b

(on3t know ξ 1 use average value,

8H,989, x

e x x f −

+−=′′

,>KH.#8,9

,8#9

=′′−=′′

f

f

( ) ( )[ ]>E.#

,

,#

:,

,<

=′′+′′

−=≈ f f

E E at

,#, =−=h

Exam%le



March 24, 2015 OSU/CSE 541 26

Lore intervals% etter result error ∼ O9h-8B

0

1

2

3

4

5

6

7

3 5 7 9 11 13 15

n ; ,

0

1

2

3

4

5

6

7

3 5 7 9 11 13 15

n ; <

0

1

2

3

4

5

6

7

3 5 7 9 11 13 15

n ; H

0

1

2

3

4

5

6

7

3 5 7 9 11 13 15

n ; E



March 24, 2015 OSU/CSE 541 27

• -f we do multiple intervals% we can avoid duplicate

function evaluations and operations)

• /se nM: equall! spaced points.• +ach interval has)

• 7reak up the limits of integration and e?pand.

n

abh

−=

( ) ( ) ( )∫ ∫ ∫ −+

++ +++= b

hb

ha

ha

ha

adx x f dx x f dx x f I ...,

6om%osite !ra%e7oid ule



March 24, 2015 OSU/CSE 541 28

• 0ustituting the trape2oid rule for each integral.

• Results in the Composite Trape2oid *ormula)

( ) ( ) ( )

( )( ) ( )[ ]

( )( ) ( )[ ]

( )( ) ( )[ ]

,

,,

, ,

....,

...a h a h b

a a h b h

a h a a h a h f a f a h f a h f a h

b b h f b h f b

I f x dx f x dx f x dx+ +

+ −

+ − + − −= + + + + + +

− ++ + − +

= + + +∫ ∫ ∫

( ) ( ) ( )

+++= ∑−

=

b f iha f a f h

I n

i

:

:

,,




March 24, 2015 OSU/CSE 541 29

( ) ( ) ( )

( )

( ) ( ) ( )

n

b f iha f a f

ab

b f iha f a f h

I

n

i

n

i

,

,

,,

:

:

:

:

+++−=

+++=

∑

∑−

=

−

=

width

Average height


• Think of this as the width times the average

height*



March 24, 2015 OSU/CSE 541 30

• The error can e estimated as)

• Where% is the average second derivative.

• -f n is douled% h → hD, and E a → E aDH• 4ote% that the error is dependent upon the

width of the area eing integrated.

( ) ( ) f n

ab

f

hab

E a ′′−

=′′−

= ,

<,

:,:,

f ′′

89,

hO

Error



March 24, 2015 OSU/CSE 541 31

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1 1.2

Exam%le

• -ntegrate)

• from

a;#.,to

b;#.E

( ) >H<, ,##E:#N<#:H#,#<.# x x x x x x f +−+−+=



March 24, 2015 OSU/CSE 541 32

( )

( ) ( )

( )

,K.::

,

E:.<,,.<H,.#E.#

,

=

+−=

+−=

b f a f

ab I

Exam%le

• A single application of the Trape2oid rule.

• +rror)( ) ( )<

:,

:ab f E t −′′−= ξ



March 24, 2015 OSU/CSE 541 33

Exam%le

• We don3t know ξ so appro?imate with

average f ′′

( ) <, H###"N,#H<E#,E# x x x x f +−+−=′′

( ) H<, :###<,H#,:"#,E#,# x x x x x f +−+−=′

( )

K.:<:,.#E.#

8,.#98E.#9

,.#E.#

E.#

,.#

−=−

′−′=

−′′=′′ ∫

f f

dx f x f



March 24, 2015 OSU/CSE 541 34

( ) ( )

( ) ( )

<,

,

<

:, :,

::<:.K #.E #., ,.<N

:,

t

b a h b a

E f f n

− −′′ ′′= =

= − − − =

Exam%le

• The error can thus e estimated as)



March 24, 2015 OSU/CSE 541 35

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1 1.2

True value of integral is :,.E,. Trape2oid

rule is ::.,K 1 within appro? error 1 E t is :,O



March 24, 2015 OSU/CSE 541 37

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1 1.2

E t is now ,O



March 24, 2015 OSU/CSE 541 38

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )

( )

NK.:, :,

,,.<H,,.<>:K.<#:E.,,"<.:<<H.N,<:.<K.#

8K89,9

E.#N.#K.#>.#H.#<.#,,.#,.#E.#

=

++++++=

++++++−= f f f f f f f

I

$sing &ix Intervals

• /se intervals 9#.,%#.<8%9#.<%#%H8% etc.

– 9n ; K% h ; #.:8

True value of integral is :,.E,



March 24, 2015 OSU/CSE 541 39

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1 1.2

E t is now #.>O



March 24, 2015 OSU/CSE 541 40

Multi+dimensional Integration

• Consider a two1dimensional case.

( ): : :

## # #

:

# #

# #

# #

% %

%

%

%

n

ii

n

i

i

n n

i 9

i 9

n n

i 9

i 9

i f x y dxdy A f y dy

n

i A f y dy

n

i 9 A A f n n

i 9 A A f

n n

=

=

= =

= =

≈

=

≈

=

∑∫∫ ∫ ∑ ∫

∑ ∑

∑∑



March 24, 2015 OSU/CSE 541 41


• *or the Trape2oid Rule% this leads to

weights in the following pattern): , , , , , :

, H H H H H ,

, H H H H H ,

, H H H H H ,

, H H H H H ,

, H H H H H ,

: , , , , , :

{ } { }

[ ] { }

[ ]

[ ] [ ]

,

: #% #%

, :% % , :% ::

, P:% : :% % :H

H :% % , :% % ,

i9

i n 9 n

i n 9 n A

i n 9 nn

i n 9 n

∈ ∈ ∈ − ∈ −= ∈ − ∈ −

∈ − ∈ −

K

K

K K

:

,

,

,

,

,

:

: , , , , , :



March 24, 2015 OSU/CSE 541 42


• -f we use the weights from the Trape2oid

rule% the error is still O9h,8.

• &owever% there are now n, function

evaluations.

– +quall!1spaced samples on a square region.



March 24, 2015 OSU/CSE 541 43


• -n general% given k dimensions% we have N8 nk function evaluations)

• -f the dimension is high% this leads to a

significant amount of additional work ingoing from h→ hD,. – Rememer this for Lonte1Carlo -ntegration.

( ) ( ) ( ),,

, , k k k O h O n O n O N −−− = = =



March 24, 2015 OSU/CSE 541 44

educing the Error

• To improve the estimate of the integral% we

can either)

– Add more intervals

– /se a higher order pol!nomial

– /se Richardson +?trapolation to e?amine the

limit as h→#.• Called Romerg -ntegration



March 24, 2015 OSU/CSE 541 45

Adding More Intervals

• -f we have an estimate for one value of h%

do we need to recompute ever!thing for a

value of hD,6

( ) ( ) ( ):

:

,,

n

h

i

h I f a f a ih f b

−

=

= + + + ∑



March 24, 2015 OSU/CSE 541 46

Adding More Intervals

• This is called the Recursive Trapezoid

Rule in the ook.

• We have n→ ,n and h→ hD,.

( )

, :

:,

: :

: #

:

#

9 8 , 9 8H ,

9 8 , , 9 8H ,

,, H ,

n

h

i

n n

i i

nh

i

h h I f a f a i f b

h h f a f a ih f a ih f b

I h h f a ih

−

=

− −

= =

−

=

= + + +

= + + + + + + = + + +

∑

∑ ∑

∑



March 24, 2015 OSU/CSE 541 47

• 5iven two numerical estimates otained

using different h3s% compute higher1order

estimate• 0tarting with a step si2e h,% the e?act value is

• 0uppose we reduce step si2e to h-

8989 ::

nhOh A A +=

8989 ,,

nhOh A A +=

ecall ichardson Extra%olation



March 24, 2015 OSU/CSE 541 48

• Lultipl!ing the ,nd eqn ! 9h,Dh-8n and

sutracting from the :st eqn)

• The new error term is generall! O9h,n:,8 or

O9h,n:-8.

:

8989

,

:

:,

,

:

−

−

=n

n

h

h

h Ah Ah

h

A

ichardson Extra%olation



March 24, 2015 OSU/CSE 541 49

• The true integral value can e written

• This is true for an! iteration


( ) ( )h E h I I +=

( ) ( ) ( ) ( ),,:: h E h I h E h I I +=+=



March 24, 2015 OSU/CSE 541 50


• *or e?ample) /sing 9n ; ,8

• where c is a constant

• Therefore)( )

( ) ,

,

,

:

,

:

h

h

h E

h E ≈

f ch E ′′≈

,

error ; O9h-

8B

order of error in

tra%e7oidal rule



March 24, 2015 OSU/CSE 541 51

• This leads to)

• *or integration% we have)

( ) ( ) ,

,

,

:,:

h

hh E h E ≈

( ) ( ) ( ) ( ),,,

,

,

:,: h E h I

hhh E h I +≈+




March 24, 2015 OSU/CSE 541 52


• 0olving for E 9h-8)

• And plugging ack into the estimatedintegral.

( ) ( ) ( )

,

,

,

:

,:,

:h

h

h I h I h E

−

−≈

( ) ( ),, h E h I I +=



March 24, 2015 OSU/CSE 541 53

• eads to)

• etting h- ; h, D,


( ) ( ) ( ) ( )[ ]:,,,:

, :D

:

h I h I hhh I I −−+≈

( ) ( ) ( )[ ]

( ) ( ):,

:,,,

<

:

<

H

:,

:

h I h I I

h I h I h I I

−≈

−−+≈



March 24, 2015 OSU/CSE 541 54

• We comined two O9h-8 estimates to get an

O9h.8 estimate.

• Can also comine two O9h.8 estimates to

get an O9h; 8 estimate.

( ) ( )l m h I h I I :>

:

:>

:K

−≈

omberg Integration



March 24, 2015 OSU/CSE 541 55

omberg Integration

• 5reater weight is placed on the more

accurate estimate.

• Weighting coefficients sum to unit!

– i.e% 9:K1:8D:>;:

• Can continue% ! comining two O9h; 8

estimates to get an O9h<8 estimate.

( ) ( )l m h I h I I K<

:

K<

KH−≈



March 24, 2015 OSU/CSE 541 56

• 5eneral pattern is called Romerg -ntegration

– 9 ) level of sudivision% 9M: has more intervals.

– k ) level of integration% k ; : is original trape2oid

estimate O9h-8B% k ; , is improved O9h.8B% etc.

( )% :%

% : % % :%

H :

H : H :

k

9 k 9 k

9 k 9 k 9 k 9 k k k

I I

I I I I −

+ −

−

≈ = + −− −

omberg Integration



March 24, 2015 OSU/CSE 541 57

omberg Integration

• *or e?ample% 9 ; :% k ; : leads to

( ):%# #%# :

:%: :

H H :

< < , <

I I h

I I I h

− = ≈ − ÷



March 24, 2015 OSU/CSE 541 58

• Consider the function)

• -ntegrate from a ; # to b ; #.E

• /sing the trape2oidal rule !ields the

following results)

>H<, H##"##KN>,##,>,.#89 x x x x x x f +−+−+=

Exam%le



March 24, 2015 OSU/CSE 541 59

intervals -ntegral

: #.E #.:N,E

, #.H :.#KEE

H #., :.HEHE

h

Exam%le

• Trape2oid Rules)

Exact integral is *;."3///.

<KNHKKN.:8:N,E.#9<

:8#KEE.:9

<

H=−= I

9 9;:% k ;:8

k

9

k ; # k ; :

9 ; #

9 ; :

9 ; ,



March 24, 2015 OSU/CSE 541 60

, Hsegments 9 8 9 8

: #.E #.:N,E

, #.H :.#KEE :.<KNHKKN

H #., :.HEHE

h O h O h

k

9

K,<HKKKN.:8#KEE.:9<

:8HEHE.:9

<

H=−= I 9 9;,% k ;:8

k ; :k ; #


Exam%le



March 24, 2015 OSU/CSE 541 61

Exam%le

, H

segments 9 8 9 8 : #.E #.:N,E


H #., :.HEHE :.K,<HKKKN

h O h O h

k

9

9 9;,% k ;,8

k ; ,k ; :


KH#><<<H.:8<KNHKKN.:9:>

:8K,<HKKKN.:9

:>

:K=−= I



March 24, 2015 OSU/CSE 541 62

, H Ksegments 9 8 9 8 9 8

: #.E #.:N,E


H #., :.

h O h O h O h

HEHE :.K,<HKKKN :.KH#><<<H

k

9

k ; <k ; ,k ; :

Exam%le



March 24, 2015 OSU/CSE 541 63

, H K Esegments 9 8 9 8 9 8 9 8

: #.E #.:N,E


H

h O h O h O h O h

#., :.HEHE :.K,<HKKKN :.KH#><<<H

E #.: 66 66 66 66

668KH#><<<H.:9K<

:9668

K<

KH=−= I 9 9;<% k ;<8

k

9

k ; <

Exam%le

• 7etter and etter results can e otained !

continuing this



March 24, 2015 OSU/CSE 541 64

omberg Integration

• -s this that significant6

• Consider the cost of computing the

Trape2oid Rule for :### data points. – Refinement would lead to ,### data points.

• -mplies an additional :##< operations using theRecursive Trape2oid Rule.

• 4ot to mention the :### 9e?pensive8 function evals.

– Romerg -ntegration cost)• Three additional operations – no function evals



March 24, 2015 OSU/CSE 541 65

=igher+Order #olynomials

• Recall)

89<DEs0impsonFcuic<89:D<s0impsonFquadratic,

89Trape2oidlinear :

+rror *ormulaGol!nomial

H

H

,

hOhO

hO

m

: : ,

:

# #

: ,

# # #

9 8 9 8 9 8 ... 9 8n m m m n

n

m n mn

x x x x

m m m x x x x

f x dx % x dx % x dx % x dx+ + +

+ −

= + + +

∫ ∫ ∫ ∫



March 24, 2015 OSU/CSE 541 66

• -f we use a ,nd order pol!nomial 9need <

points or , intervals8)

– agrange form.

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( ) dx x f

x x x x

x x x x

x f x x x x

x x x x x f

x x x x

x x x x I

x

x

−−

−−+

−−

−−+

−−

−−= ∫

,

:,#,

:#

:

,:#:

,##

,#:#

,:,

#

+=

,,#

: x x x

&im%son>s ?/ ule



March 24, 2015 OSU/CSE 541 67

&im%son>s ?/ ule

• Requiring equall!1spaced intervals)

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

,

#

# # # ## :

# #

,

, ,

,

,

x

x

x x h x x h x x x x h I f x f xh h h h

x x x x h f x dx

h h

− − − − − − −= + − − −

− − −+

∫



March 24, 2015 OSU/CSE 541 68

• -ntegrate and simplif!)

0

2

4

6

8

10

12

3 5 7 9 11 13 15

( ) ( ) ( )[ ],:# H

<

x f x f x f h

I ++≈,

abh

−=

&im%son>s ?/ ule

Quadratic

#olynomial



March 24, 2015 OSU/CSE 541 69

• -f we use a 8 x" and b 8 x-% and

x, 8 9b:a8D,

&im%son>s ?/ ule

( ) ( ) ( ) ( )

K

H ,:# x f x f x f ab I

++−≈

widthaverage height



March 24, 2015 OSU/CSE 541 70

• +rror for 0impson3s :D< rule

⇒-ntegrates a cuic e?actl!) ( ) ( ) #H =ξ f

( )

( )

( ) ( )

( )ξ ξ H

>

H>

,EE#"# f

ab

f

h

E t −

−=−=

,

abh

−=

89>

hO

&im%son>s ?/ ule



March 24, 2015 OSU/CSE 541 71

6om%osite &im%son>s ?/ ule

• As with Trape2oidal rule% can use multiple

applications of 0impson3s :D< rule.

• 4eed even numer of intervals

– An odd numer of points are required.



March 24, 2015 OSU/CSE 541 72


• +?ample) " points% H intervals

-5

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5



March 24, 2015 OSU/CSE 541 73

• As in composite trape2oid% reak integral up

into nD, su1integrals)

• 0ustitute 0impson3s :D< rule for each

integral and collect terms.


( ) ( ) ( )∫ ∫ ∫ −+++=

n

n

x

x

x

x

x

xdx x f dx x f dx x f I

,

H

,

,

#

...

( )

( ) ( ) ( ) ( ): ,

#

:%<%> ,%H%K

H ,

<

n n

i 9 n

i 9

f x f x f x f x

I b an

− −

= =

+ + += −

∑ ∑

n:: data points% an odd numer



March 24, 2015 OSU/CSE 541 74

• dd coefficients receive a weight of H% even

receive a weight of ,.

• (oesn3t seem ver! fair% does it6

coefficients on

numerator : H :

: H :

:H :

i ; #

i ; n




March 24, 2015 OSU/CSE 541 75

• The error can e estimated !)

• -f n is douled% h → hD, and E a → E aD:K

( ) 8H9H

8H9>

:E#:E#

f hab

f nh

E a−

==

8H9 f

is the average Hth derivative

89 HhO

Error Estimate



March 24, 2015 OSU/CSE 541 76

• -ntegrate from a ; # to b ; ,.

• /se 0impson3s :D< rule)

( ) ( )

( ) ( )

,,

# : ,#

# : H

: H 9 8

<

: # H 9:8 ,<

: 9 H 8 #.E,""H

<

x I e dx h f x f x f x

f f f

e e e

−

− −

= ≈ + +

= + +

= + + =

∫

,

89 xe x f −=

, :,

# :,

,:# ===+====−= b xba xa xabh

Exam%le



March 24, 2015 OSU/CSE 541 77

• +rror estimate)

• Where h 8 b + a and a I ξ I b

• (on3t know ξ – use average value

( ) ( )ξ H>

"# f

h E t −=

( )

( )

( ) ( )

( ) ( )

( )H H H

> > # : ,H: :

"# "# <t a

f x f x f x E E f + + ≈ = − = −

Exam%le



March 24, 2015 OSU/CSE 541 78

• et3s look at the pol!nomial again)

– *rom a ; # to b ; #.E

>H<, H##"##KN>,##,>,.#89 x x x x x x f +−+−+=

E.# H.#,

# H.#,

,:# ===+

====−

= b xba

xa xab

h

( ) ( )[ ]

( ) ( )[ ]

<KNHKKKN.:

E.#8H.#9H#<

8H.#9

89H

<

: 89 ,:#

,

#

=

++=

++≈=

∫ f f f

x f x f x f hdx x f I


Another Exam%le



March 24, 2015 OSU/CSE 541 79

• Actual +rror) 9using the known e?act value8

• +stimate error) 9if the e?act value is not

availale8

• Where a I ξ I b*

( ) ( )ξ H>

"# f

h E t −=

,N<#KKKK.#<KNHKKKN.:1:.KH#><<<H == E :KO

Error



March 24, 2015 OSU/CSE 541 80

• Compute the fourth1derivative

• Latches actual error prett! well.

Error

x x f HE###,:K##898H9+−=

( ) ( ) ( ) ( ) ,N<#KKKN.#H.#"#

H.#

"#

H.# H>

:

H>

=−=−=≈ f x f E E at

middle %oint



March 24, 2015 OSU/CSE 541 81

• -f we use H segments instead of :)

– x ; #.# #., #.H #.K #.EB#.,

b ah

n

−= =

,<,.#8E.#9 HKH.<8K.#9

H>K.,8H.#9 ,EE.:8,.#9 ,.#8#9

==

===

f f

f f f

( )

( ) ( ) ( ) ( )

( )

K,<HKKN.:

:,

,<,.#8H>K.,9,8HKH.<,EE.:9H,.#E.#

8H89<9

8E.#98K.#9H8H.#9,8,.#9H8#9#E.#

<

,H:

>%<%:

,

K%H%,

#

=

++++=

++++−=

+++−=

∑ ∑−

=

−

=

f f f f f

n

x f x f x f x f

ab I

n

n

i

n

9

9i


Exam%le 6ontinued



March 24, 2015 OSU/CSE 541 82

• Actual +rror) 9using the known e?act value8

• +stimate error) 9if the e?act value is not

availale8

#:N#KKKN.#K,<HKKN.:1:.KH#><<<H == E :O

( ) ( ) ( ) ( ) ##E>.#H.#

"#

,.#

"#

,.# H>

,

H>

−=−=−=≈ f x f E E at

middle %oint

Error



March 24, 2015 OSU/CSE 541 83

Error

• Actual is twice the estimated% wh!6

• Recall)

x x f HE###,:K##898H9 +−=

[ ] { }9 H8 9 H8

#%#.E

9 H8

ma? 9 8 9#8 ,:K##

9#.H8 ,H##

x f x f

f

∈

= = −

=



March 24, 2015 OSU/CSE 541 84

Error

• Rather than estimate% we can ound the

asolute value of the error)

• *ive times the actual% ut provides a safer

error metric.

( ) ( ) ( ) ( )> >

H H#., #.,# #.#NKE

"# "#a E f f ξ = − ≤ =



March 24, 2015 OSU/CSE 541 85

&im%on>s ?/ ule

• 0impson3s :D< rule uses a ,nd order pol!nomial

– need < points or , intervals

– This implies we need an even numer of intervals.• What if !ou don3t have an even numer of

intervals6 Two choices)

:. /se 0impson3s :D< on all the segments e?cept the last

9or first8 one% and use trape2oidal rule on the one left. – Gitfall 1 larger error on the segment using trape2oid

,. /se 0impson3s <DE rule.



March 24, 2015 OSU/CSE 541 86

• 0impson3s <DE rule uses a third order pol!nomial

– need < intervals 9H data points8

<

<

,

,:#< 8989 xa xa xaa x % x f +++=≈

( ) ( )∫ ∫ ≈=

<

#

<

#<

x

x

x

x dx x %dx x f I

&im%son>s /?< ule



March 24, 2015 OSU/CSE 541 87

&im%son>s /?< ule

• (etermine a3s with agrange pol!nomial

• *or evenl! spaced points

<

ab

h

−

=

( ) ( ) ( ) ( )[ ]<,:# <<E

< x f x f x x f h I +++=



March 24, 2015 OSU/CSE 541 88

• 0ame order as :D< Rule.

– Lore function evaluations.

– -nterval width% h% is smaller.

• -ntegrates a cuic e?actl!)

⇒ ( ) ( ) #H =ξ f

( ) ( )H><

E#t E h f ξ = − 89 HhO

Error



March 24, 2015 OSU/CSE 541 89

6om%arison

• 0impson3s :D< rule and 0impson3s <DE rule have thesame order of error – O9h.8

– trape2oidal rule has an error of O9h-8

• 0impson3s :D< rule requires even number ofsegments.

• 0impson3s <DE rule requires multiples of threesegments.

• 7oth 0impson3s methods require evenly spaced data points



March 24, 2015 OSU/CSE 541 90

• n ; :# points ⇒ " intervals

– *irst K intervals 1 0impson3s :D<

– ast < intervals 1 0impson3s <DE

0impson3s :D<

0impson3s <DE

Mixing !echni@ues



March 24, 2015 OSU/CSE 541 91

• We can e?amine even higher1order

pol!nomials.

– 0impson3s :D< 1 ,nd order agrange 9< pts8 – 0impson3s <DE 1 <rd order agrange 9H pts8

• /suall! do not go higher.

• /se multiple segments. – 7ut onl! where needed.

Newton+6otes 1ormulas



March 24, 2015 OSU/CSE 541 92

Ada%tive &im%son>s &cheme

• Recall 0impson3s :D< Rule)

• Where initiall!% we have a; x" and b; x-.

• 0udividing the integral into two)

( ) ( ) ( )[ ],:# H

<

x f x f x f h

I ++≈

( ) ( ) ( ) ( ) ( ): , <H , HK

h I f a f x f x f x f b≈ + + + +



March 24, 2015 OSU/CSE 541 93


• We want to keep sudividing% until we

reach a desired error tolerance% ε.

• Lathematicall!)

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

:

: , <

H<

H , HK

b

a

b

a

h f x dx f a f x f b

h f x dx f a f x f x f x f b

ε

ε

− + + ≤

− + + + + ≤

∫

∫



March 24, 2015 OSU/CSE 541 94


• This will e satisfied if)

• The left and the right are within one1half of theerror.

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

: ,

, <

,

H %K ,

H %K ,

,

c

a

b

c

h f x dx f a f x f x and

h f x dx f x f x f b where

a bc x

ε

ε

− + + ≤

− + + ≤

+= =

∫

∫



March 24, 2015 OSU/CSE 541 95


• ka!% now we have two separate intervals

to integrate.

• What if one can e solved accuratel! withan h;:#1<% ut the other requires man!%

man! more intervals% h;:#1K6



March 24, 2015 OSU/CSE 541 96


• Adaptive 0impson3s method provides a

divide and conquer scheme until the

appropriate error is satisfied ever!where.• @er! popular method in practice.

• Grolem)

– We do not know the e?act value% and hence donot know the error.



March 24, 2015 OSU/CSE 541 97


• &ow do we know whether to continue to

sudivide or terminate6

( ) ( ) ( )

( ) ( ) ( )

( ) ( )>

H

% % %

% H %

K ,:

%"# ,

b

a

I f x dx & a b E a b where

b a a b& a b f a f f b and

b a E a b f

≡ = +

− + = + +

− = −

∫



March 24, 2015 OSU/CSE 541 98


• The first iteration can then e defined as)

• 0usequent sudivision can e defined as)

( )

( ) ( )

,

% %& & a c & c b= +

( ) ( )

( ) ( ) ( ) ( )

: :

: :

%

% % %

I & E where

& & a b E E a b

= +

= =



March 24, 2015 OSU/CSE 541 99


• 4ow% since

• We can solve for E 9,8 in terms of E 9:8.

( ) ( ) ( )

( ) ( )

> >

, H H

>H :

H

: D , : D ,

"# , "# ,

: : :

, "# , :K

h h E f f

h f E

= − −

= − =

( ) ( ) ( ),

% % E E a c E c b= +



March 24, 2015 OSU/CSE 541 100


• *inall!% using the identit!)

• We have)

• Glugging into our definition)

( ) ( ) ( ) ( ): : , , I & E & E = + = +

( ) ( ) ( ) ( ) ( ), : : , ,:>& & E E E − = − =

( ) ( ) ( ) ( ) ( )( ), , , , ::

:> I & E & & & = + = + −



March 24, 2015 OSU/CSE 541 101


• ur error criteria is thus)

• 0implif!ing leads to the termination

formula)

( ) ( ) ( )

( )

, , ::

:> I & & & ε − = − ≤

( ) ( )

( ), :

:>& & ε − ≤



March 24, 2015 OSU/CSE 541 102


• What happens graphicall!)



March 24, 2015 OSU/CSE 541 103

:, :>& subd v& i ideε − ≥ →



March 24, 2015 OSU/CSE 541 104



March 24, 2015 OSU/CSE 541 105

:, :>

,

subdivide& & ε

− ≥ →



March 24, 2015 OSU/CSE 541 106

( ), :,

:

:>

& I & & = + −

:, :>H

& o& d neε

− ≤ →



March 24, 2015 OSU/CSE 541 107

:, :>H

subdivide& & ε

− ≥ →



March 24, 2015 OSU/CSE 541 108

( ), :,

:

:>

& I & & = + −



March 24, 2015 OSU/CSE 541 109

( ), :,

:

:>& I & & = + −



March 24, 2015 OSU/CSE 541 110

I8I left : I right

I right

8I left

: I right



March 24, 2015 OSU/CSE 541 111



March 24, 2015 OSU/CSE 541 112



March 24, 2015 OSU/CSE 541 113


• We graduall! capture the difficult spots.



March 24, 2015 OSU/CSE 541 114

Ada%tive &im%son>s 6ode

• 0imple Recursive Grogramstatic const int m_nMaximum_Divisions = 1000;Real IntegrationSimpson( const Real (*f) (Real x) const Real start const Real en! const Real

error_tolerance int "level )

#level $= 1;Real % = (en! & start);Real mi!point = (start $ en!) ' 0;Real f_start = f(start);Real f_en! = f(en!);Real f_mi! = f(mi!point );oneevel = %*( f_start $ +0*f_mi! $ f_en!) ' ,0;Real leftMi!point = (start$ mi!point ) ' 0;Real rig%tMi!point = (en!$ mi!point ) ' 0Real f_mi!eft = f(leftMi!point );Real f_mi!Rig%t = f(rig%tMi!point );t-oevel = %*(f_start $ +0* f_mi!eft $ 0* f_mi! $ +0* f_mi!Rig%t $ f_en!) ' 10;if( level .= m_nMax_Divisions ) '' /erminate t%e process converging too slo-

return t-oevel;



March 24, 2015 OSU/CSE 541 115

Ada%tive &im%son>s 6ode

if( asf( t-oevel & oneevel) 120*error_tolerance) '' Desire! solution reac%e!return t-oevel $ (t-oevel3oneevel) ' 120;

''

'' 4t%er-ise split t%e interval in t-o an! recursivel5 evaluate eac% %alf''leftIntegral = IntegrationSimpson( f start mi!point error_tolerance'0 level );rig%tIntegral = IntegrationSimpson( f mi!point en! error_tolerance'0 level );return leftIntegral $ rig%tIntegral;

6

d



March 24, 2015 OSU/CSE 541 116

• -dea is that if we evaluate the function at certain

points% and sum with certain weights% we will get a

more accurate integral

• +valuation points and weights are pre1computed

and taulated

• 7asic form)

∫ ∑− =

≈=:

: :

8989n

i

ii x f cdx x f I

ci ) weighting factors

xi ) sampling points selected optimall!

uassian Quadrature

NewBB

Q d



March 24, 2015 OSU/CSE 541 117

• 4ote that the interval is etween –: and :

• *or other intervals% a change of variales is used to

transfer the prolem so that it utili2es the interval1:% :B

• This is a linear transform% such that for t ∈a%B)

• We have for x∈1:%:B)

∫ b

adt t f 89

,

89 ab xabt

++−=

ab

abt x

−−−

=,

uassian Quadrature

Q d



March 24, 2015 OSU/CSE 541 118

• As t 8 a ⇒ x ; 1:

• As t 8 b ⇒ x ; :

dxab

dt ,

89 −=

++−=

,

8989

ab xab f t f

dxab xab

f ab

dt t f b

a ∫ ∫ −

++−−

=:

: ,

89

,

8989

uassian Quadrature

Q d



March 24, 2015 OSU/CSE 541 119

• 7asic form of 5aussian quadrature)

• *or n;,% we have)

• This leads to H unknowns) c% c-% x% and x-

– two unknown weights 9c% c-8

– two unknown sampling points 9 x% x-8

∫ ∑−

=

≈=:

:

:

8989n

i

ii x f cdx x f I

( ) ( ),,:: x f c x f c I +≈

uassian Quadrature

i Q d



March 24, 2015 OSU/CSE 541 120

uassian Quadrature

• What we need now% are four known values

for the equation.

• -f we had these% we could then attempt tosolve for the four unknowns.

• et3s make it work for pol!nomials

i Q d



March 24, 2015 OSU/CSE 541 121

• -n particular% let3s look at these simple pol!nomials) – Constant

• f'x(8

– inear • f'x(8x

– Quadratic

• f'x(8x-

– Cuic• f'x(8x/

uassian Quadrature

i Q d



March 24, 2015 OSU/CSE 541 122

uassian Quadrature

• Recalling the formula)

– Constant

• f'x(8

– inear

• f'x(8x

– Quadratic

• f'x(8x-

– Cuic

• f'x(8x/

( ) ( )

( ) ( )

( ) ( )

( ) ( ) <

,,

<

::,,::

:

:

<

,

,,

,

::,,::

:

:

,

,,::,,::

:

:

,:,,::

:

:

#<

,

#

,:

xc xc x f c x f cdx x

xc xc x f c x f cdx x

xc xc x f c x f c xdx

cc x f c x f cdx

+=+==

+=+==

+=+==

+=+==

∫

∫

∫ ∫

−

−

−

−

( ) ( ),,:: x f c x f c I +≈

i Q d



March 24, 2015 OSU/CSE 541 123

• We can now solve for our unknowns) – Note, this is not an easy %roblem and will not be

covered in this class*

>NN.#<:

>NN.#<

:

:

,

:

,:

==

−=−=

==

x

x

cc

uassian Quadrature

i Q d t



March 24, 2015 OSU/CSE 541 124

uassian Quadrature

• This !ields the two point Gauss!egendre

formula

+

−≈

<:

<: f f I

i Q d t



March 24, 2015 OSU/CSE 541 125

uassian Quadrature

• This is exact for all pol!nomials up to and

including degree < 9cuics8.

( )

( )

( )

: : : : :< , < ,

: : : : :

< < , ,

:

<

:

<

: : : : : :: :

< < < < < <

< ,

ax bx cx d dx a x dx b x dx c xdx d dx

a b c d

ax bx cx d

− − − − −

−

+ + + = + + +

− − − = + + + + + + +

=

∫ ∫ ∫ ∫ ∫

+ + +

i Q d t



March 24, 2015 OSU/CSE 541 126

x

f 9 x8

1: :

f 9#.>NN8 f 91#.>NN8

1#.>NN #.>NN

∫ − +−≈:

:8>NN.#98>NN.#989 f f dx x f

uassian Quadrature

E l



March 24, 2015 OSU/CSE 541 127

Exam%le

>H<, H##"##KN>,##,>,.#89 x x x x x x f +−+−+=

dxab xab

f ab

dt t f b

a ∫ ∫ −

++−−

=:

: ,

89

,

8989

( )dt t f

dt t

f dx x f I

∫

∫ ∫

−

−

+=

++−−==

:

:

:

:

E.#

#

H.#H.#H.#

,

#E.#8#E.#9

,

8#E.#989

• -ntegrate f'x( from a ; # to b ; #.E

• Transform from #% #.EB to 1:% :B

E l



March 24, 2015 OSU/CSE 541 128

• 0olving

• And sustituting for the ,1point formula)

Exact integral is 1."#$%333#

( )

dt t t t

t t

dt t f I

∫

∫

−

−

−+−−−+−−−+=

+=

:

: >H<

,

:

:

8H.#H.#9H##8H.#H.#9"##8H.#H.#9KN>

8H.#H.#9,##8H.#H.#9,>,.#H.#

H.#H.#H.#

<D:±=t

E,,>NNNE.:<#>E<N,<.:>:KNH#>>.# =+≈ I

( )dt t f I ∫ −= :

:H.#

Exam%le

=i h d i Q d t



March 24, 2015 OSU/CSE 541 129

• Recall the asic form)

• et3s look at n;<.

• We now have K unknowns) c,% c-% c/% x,% x-% and x/

– three unknown weights 9c,% c- % c/8

– three unknown sampling points 9 x,% x- % x/8

∫ ∑−

=

≈=:

:

:

8989n

i

ii x f cdx x f I

( ) ( ) ( )<<,,:: x f c x f c x f c I ++≈

=igher+order aussian Quadrature

/se K equations 1 constant% linear% quadratic% cuic%



March 24, 2015 OSU/CSE 541 130

/se K equations constant% linear% quadratic% cuic%

Hth order and >th order to find those unknowns

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) >

<<

>

,,

>

::<<,,::

:

:

>

H<<

H,,

H::<<,,::

:

:

H

<

<<

<

,,

<

::<<,,::

:

:

<

,

<<

,

,,

,

::<<,,::

:

:

,

<<,,::<<,,::

:

:

<,:<<,,::

:

:

#

>,

#

<,

#

,:

xc xc xc x f c x f c x f cdx x




xc xc xc x f c x f c x f c xdx

ccc x f c x f c x f cdx

++=++==

++=++==

++=++==

++=++==

++=++==

++=++==

∫ ∫

∫

∫

∫ ∫

−

−

−

−

−

−

=i h d i Q d t



March 24, 2015 OSU/CSE 541 131

• Can solve these equations 9or have some one

smarter than us% like 5uass solve them8.

• Groduces the three point 5auss1egendre formula

– +?act for pol!nomials up to and including degree >

9ecause using >th degree pol!nomial8

NNH>"KK".#>D< # NNH>"KK".#>D<

"D> "DE "D>

,,:

<,:

===−=−=

===

x x x

ccc

( ) ( ) ( )<<,,:: x f c x f c x f c I ++≈


=i h d i Q d t



March 24, 2015 OSU/CSE 541 132

∫ − ++−≈:

:8NN>.#9

"

>8#.#9

"

E8NN>.#9

"

>89 f f f dx x f

1: :

x

f 9#.NN>8 f 91#.NN>8

1#.NN> #.NN>

f 9#8


Exam%le



March 24, 2015 OSU/CSE 541 133

>H<, H##"##KN>,##,>,.#89 x x x x x x f +−+−+=

-ntegrate from a ; # to b ; #.E

Transform from #% #.EB to 1:% :B

( )

dt

t t t

t t

dx x f I

∫

∫

−

−+−−−+

−−−+=

=

:

: >H<

,

E.#

#

8H.#H.#9H##8H.#H.#9"##8H.#H.#9KN>

8H.#H.#9,##8H.#H.#9,>,.#

Exam%le

replace 1#.H with M#.H

Exam%le



March 24, 2015 OSU/CSE 541 134

( )

++

−≈ >

<

"

>#"

E

>

<

"

> f f f I

KH#><<<H.:

HE>"EN>"".#EN<,HHHHH.#,E:<#:,"#.#

=++≈ I

0ustitute into the transform equation and get

Exact integral is 1."#$%333#

Exam%le

• /sing the <1point 5auss1egendre formula)

aussian Quadrature



March 24, 2015 OSU/CSE 541 135

Can develop higher order 5auss1egendre forms

using

( ) ( ) ( )nn x f c x f c x f c I +++≈ ...,,::

@alues for c3s and x3s are taulated

/se the same transformation to map interval onto1:% :B

aussian Quadrature

( ) ( ) ( )xfcxfcxfcdxxfI +++≈= ∫ ...89 ,,::

:



March 24, 2015 OSU/CSE 541 136

:N:<,H>.#

<K#NK:K.#

HKN":<".#

HKN":<".#

<K#NK:K.#

:N:<,H>.#

>##.,<K",KEE

#>#.HNEK,EKN

E"#.>KEEEEEE

#>#.HNEK,EKN

>##.,<K",KEE

>:#.<HNE>HEH

H"#.K>,:H>:>

H"#.K>,:H>:>

>:#.<HNE>HEH

>K#.>>>>>>>>

E"#.EEEEEEEE

>K#.>>>>>>>>

#.:

#.:

"<,HK">:H#

KK:,#"<EK#

,<EK:":EK#

,<EK:":EK#KK:,#"<EK#

"<,HK">:H#

"#K:N"EH>"#

><EHK"<:#:#

###########><EHK"<:#:#

"#K:N"EH>"#

EK::<K<::K#

<<""E:#H<K#<<""E:#H<K#

EK::<K<::K#

NNH>"KKK",############

NNH>"KKK",#

>NN<>#,K",#

>NN<>#,K",#

*

*

*

**

*

*

*

**

*

*

**

*

**

*

*

*

−−

−

−

−

−

−−−

, < H > Kn

ci

xi

( ) ( ) ( )nn x f c x f c x f cdx x f I +++∫ − ...89 ,,:::

aussian Quadrature



March 24, 2015 OSU/CSE 541 137

aussian Quadrature

• Requires function evaluations at non1uniforml! spaced points within theintegration interval – not appropriate for cases where the function is

unknown

– not suited for dealing with taulated data that

appear in man! engineering prolems• -f the function is known% its efficienc! can

e a decided advantage

aussian Quadrature



aussian Quadrature

• Grolems)

– -f we add more data points% like douling the

numer of sample points.

cis541 07 integration

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