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Numerical Integration
Roger Crawfis
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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. ,###
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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
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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
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• +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,
∫ + ++
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• -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
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• 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
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$%%er &ums
• Assume that f'x()" ever!where*
• -f within each interval% we could determine
the ma?imum value of the function% then we
have)
• where
( )#
:
:
#
9 8n x n
i i i
i x
f x M x x−
+=
≤ −∑∫
{ }:sup 9 8 )i i i M f x x x x += ≤ ≤ &u%remum + least
u%%er bound
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$%%er &ums
• 5raphicall!)
x" x x- x. x/
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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
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0ower &ums
• 5raphicall!
x" x x- x. x/
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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− −+ +
= =
− ≤ ≤ −∑ ∑∫
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2ounding the Integral
• 5raphicall!
x" x x- x. x/
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2ounding the Integral
• &alving each interval 'much like 0ab()
x" x/ x3 x4 x5
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2ounding the Integral
• ne more time)
x" x3 x5 x x4
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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
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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
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#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 ,
,:#
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• 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
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• 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
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!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
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!ra%e7oid ule
• -mprovement6
x" x x- x. x/
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• 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
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,
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
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+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
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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
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• -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
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• 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
:
:
,,
6om%osite !ra%e7oid ule
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( ) ( ) ( )
( )
( ) ( ) ( )
n
b f iha f a f
ab
b f iha f a f h
I
n
i
n
i
,
,
,,
:
:
:
:
+++−=
+++=
∑
∑−
=
−
=
width
Average height
6om%osite !ra%e7oid ule
• Think of this as the width times the average
height*
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• 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
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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 +−+−+=
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( )
( ) ( )
( )
,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
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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
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( ) ( )
( ) ( )
<,
,
<
:, :,
::<:.K #.E #., ,.<N
:,
t
b a h b a
E f f n
− −′′ ′′= =
= − − − =
Exam%le
• The error can thus e estimated as)
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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
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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
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( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )
( )
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,
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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
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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
=
=
= =
= =
≈
=
≈
=
∑∫∫ ∫ ∑ ∫
∑ ∑
∑∑
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Multi+dimensional Integration
• *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
:
,
,
,
,
,
:
: , , , , , :
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Multi+dimensional Integration
• -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.
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Multi+dimensional Integration
• -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 −−− = = =
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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
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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
−
=
= + + + ∑
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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
−
=
− −
= =
−
=
= + + +
= + + + + + + = + + +
∑
∑ ∑
∑
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• 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
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• 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
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• The true integral value can e written
• This is true for an! iteration
ichardson Extra%olation
( ) ( )h E h I I +=
( ) ( ) ( ) ( ),,:: h E h I h E h I I +=+=
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ichardson Extra%olation
• *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
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• This leads to)
• *or integration% we have)
( ) ( ) ,
,
,
:,:
h
hh E h E ≈
( ) ( ) ( ) ( ),,,
,
,
:,: h E h I
hhh E h I +≈+
ichardson Extra%olation
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ichardson Extra%olation
• 0olving for E 9h-8)
• And plugging ack into the estimatedintegral.
( ) ( ) ( )
,
,
,
:
,:,
:h
h
h I h I h E
−
−≈
( ) ( ),, h E h I I +=
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• eads to)
• etting h- ; h, D,
ichardson Extra%olation
( ) ( ) ( ) ( )[ ]:,,,:
, :D
:
h I h I hhh I I −−+≈
( ) ( ) ( )[ ]
( ) ( ):,
:,,,
<
:
<
H
:,
:
h I h I I
h I h I h I I
−≈
−−+≈
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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
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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−≈
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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
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omberg Integration
• *or e?ample% 9 ; :% k ; : leads to
( ):%# #%# :
:%: :
H H :
< < , <
I I h
I I I h
− = ≈ − ÷
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• 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
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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 ; ,
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, 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 ; #
Exact integral is *;."3///.
Exam%le
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Exam%le
, H
segments 9 8 9 8 : #.E #.:N,E
, #.H :.#KEE :.<KNHKKN
H #., :.HEHE :.K,<HKKKN
h O h O h
k
9
9 9;,% k ;,8
k ; ,k ; :
Exact integral is *;."3///.
KH#><<<H.:8<KNHKKN.:9:>
:8K,<HKKKN.:9
:>
:K=−= I
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, H Ksegments 9 8 9 8 9 8
: #.E #.:N,E
, #.H :.#KEE :.<KNHKKN
H #., :.
h O h O h O h
HEHE :.K,<HKKKN :.KH#><<<H
k
9
k ; <k ; ,k ; :
Exam%le
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, H K Esegments 9 8 9 8 9 8 9 8
: #.E #.:N,E
, #.H :.#KEE :.<KNHKKN
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
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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
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=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+ + +
+ −
= + + +
∫ ∫ ∫ ∫
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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
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&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
− − − − − − −= + − − −
− − −+
∫
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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
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• -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
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• +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
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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.
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6om%osite &im%son>s ?/ ule
• +?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
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• As in composite trape2oid% reak integral up
into nD, su1integrals)
• 0ustitute 0impson3s :D< rule for each
integral and collect terms.
6om%osite &im%son>s ?/ ule
( ) ( ) ( )∫ ∫ ∫ −+++=
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
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• 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
6om%osite &im%son>s ?/ ule
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• 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
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• -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
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• +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
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• 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
Exact integral is *;."3///.
Another Exam%le
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• 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
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• 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
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• -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
Exact integral is *;."3///.
Exam%le 6ontinued
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• 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
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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
∈
= = −
=
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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 = − ≤ =
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&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.
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• 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
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&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 +++=
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• 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
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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
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• n ; :# points ⇒ " intervals
– *irst K intervals 1 0impson3s :D<
– ast < intervals 1 0impson3s <DE
0impson3s :D<
0impson3s <DE
Mixing !echni@ues
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• 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
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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≈ + + + +
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Ada%tive &im%son>s &cheme
• 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
− + + ≤
− + + + + ≤
∫
∫
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Ada%tive &im%son>s &cheme
• 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
− + + ≤
− + + ≤
+= =
∫
∫
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Ada%tive &im%son>s &cheme
• 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
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Ada%tive &im%son>s &cheme
• 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.
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Ada%tive &im%son>s &cheme
• &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
≡ = +
− + = + +
− = −
∫
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Ada%tive &im%son>s &cheme
• 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
= +
= =
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Ada%tive &im%son>s &cheme
• 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= +
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March 24, 2015 OSU/CSE 541 100
Ada%tive &im%son>s &cheme
• *inall!% using the identit!)
• We have)
• Glugging into our definition)
( ) ( ) ( ) ( ): : , , I & E & E = + = +
( ) ( ) ( ) ( ) ( ), : : , ,:>& & E E E − = − =
( ) ( ) ( ) ( ) ( )( ), , , , ::
:> I & E & & & = + = + −
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Ada%tive &im%son>s &cheme
• ur error criteria is thus)
• 0implif!ing leads to the termination
formula)
( ) ( ) ( )
( )
, , ::
:> I & & & ε − = − ≤
( ) ( )
( ), :
:>& & ε − ≤
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Ada%tive &im%son>s &cheme
• What happens graphicall!)
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:, :>& subd v& i ideε − ≥ →
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March 24, 2015 OSU/CSE 541 104
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:, :>
,
subdivide& & ε
− ≥ →
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( ), :,
:
:>
& I & & = + −
:, :>H
& o& d neε
− ≤ →
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:, :>H
subdivide& & ε
− ≥ →
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March 24, 2015 OSU/CSE 541 108
( ), :,
:
:>
& I & & = + −
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( ), :,
:
:>& I & & = + −
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I8I left : I right
I right
8I left
: I right
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March 24, 2015 OSU/CSE 541 111
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March 24, 2015 OSU/CSE 541 112
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Ada%tive &im%son>s &cheme
• We graduall! capture the difficult spots.
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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;
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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
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• -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
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• 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
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• 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
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• 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
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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
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• -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
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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
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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
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uassian Quadrature
• This !ields the two point Gauss!egendre
formula
+
−≈
<:
<: f f I
i Q d t
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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
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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
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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
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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
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• 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%
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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 cdx x
xc xc xc x f c x f c x f cdx x
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
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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 ++≈
=igher+order aussian Quadrature
=i h d i Q d t
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∫ − ++−≈:
: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
=igher+order aussian Quadrature
Exam%le
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>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
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( )
++
−≈ >
<
"
>#"
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
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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 ,,::
:
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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
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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
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aussian Quadrature
• Grolems)
– -f we add more data points% like douling the
numer of sample points.