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SE301:Numerical MethodsTopic 6
Numerical Integration
SE301_Topic 6 Al-Amer2005 ١
Dr. Samir Al-Amer Term 063
Lecture 17 Introduction to Numerical
Integration
SE301_Topic 6 Al-Amer2005 ٢
DefinitionsUpper and Lower SumsTrapezoid MethodExamples
SE301_Topic 6 Al-Amer2005 ٣
IntegrationIntegrationDefinite Integrals
Definite Integrals are numbers.
21
2
1
0
1
0
2
==∫xxdx
Indefinite Integrals
Indefinite Integrals of a function are functionsthat differ from each other by a constant.
∫ += cxdxx2
2
SE301_Topic 6 Al-Amer2005 ٤
Fundamental Theorem of CalculusFundamental Theorem of Calculus
for solution form closedNo
for tiveantideriva no is There
) f(x)(x) 'F (i.e. f of tiveantideriva is , intervalan on continuous is If
2
2
∫
∫ −=
=
b
a
x
x
b
a
dxe
e
F(a)F(b)f(x)dx
F[a,b]f
SE301_Topic 6 Al-Amer2005 ٥
The Area Under the CurveThe Area Under the Curve
One interpretation of the definite integral is
Integral = area under the curve
f(x)
∫=b
af(x)dxArea
a b
SE301_Topic 6 Al-Amer2005 ٦
Numerical Integration Methods
Numerical integration Methods Covered in this course
Upper and Lower Sums
Newton-Cotes Methods:
Trapezoid Rule
Simpson Rules
Romberg Method
Gauss Quadrature
SE301_Topic 6 Al-Amer2005 ٧
Upper and Lower SumsUpper and Lower SumsThe interval is divided into subintervals
f(x)
a b3210 xxxx
{ }
{ }{ }
( )
( )ii
n
ii
ii
n
ii
iii
iii
n
xxMPfUsumUpper
xxmPfLsumLower
xxxxfMxxxxfm
Define
bxxxxaPPartition
−=
−=
≤≤=≤≤=
=≤≤≤≤==
+
−
=
+
−
=
+
+
∑
∑
1
1
0
1
1
0
1
1
210
),(
),(
:)(max:)(min
...
SE301_Topic 6 Al-Amer2005 ٨
Upper and Lower SumsUpper and Lower Sums
( )
( )
2
2 integral theof Estimate
),(
),(
1
1
0
1
1
0
LUError
UL
xxMPfUsumUpper
xxmPfLsumLower
ii
n
ii
ii
n
ii
−≤
+=
−=
−=
+
−
=
+
−
=
∑
∑
f(x)
3210 xxxxa b
SE301_Topic 6 Al-Amer2005 ٩
ExampleExample
3,2,1,041
1,169,
41,
161
169,
41,
161,0
)intervals equalfour (4
1,43,
42,
41,0
1
3210
3210
1
0
2
==−
====
====
=⎭⎬⎫
⎩⎨⎧=
+
∫
iforxx
MMMM
mmmm
n
PPartition
dxx
ii 143
21
410
SE301_Topic 6 Al-Amer2005 ١٠
ExampleExample( )
( )
81
6414
6430
21
3211
6414
6430
21integraltheofEstimate
64301
169
41
161
41),(
),(
6414
169
41
1610
41),(
),(
1
1
0
1
1
0
=⎟⎠⎞
⎜⎝⎛ −<
=⎟⎠⎞
⎜⎝⎛ +=
=⎥⎦⎤
⎢⎣⎡ +++=
−=
=⎥⎦⎤
⎢⎣⎡ +++=
−=
+
−
=
+
−
=
∑
∑
Error
PfU
xxMPfUsumUpper
PfL
xxmPfLsumLower
ii
n
ii
ii
n
ii
143
21
410
SE301_Topic 6 Al-Amer2005 ١١
Upper and Lower SumsUpper and Lower Sums
• Estimates based on Upper and Lower Sums are easy to obtain for monotonicfunctions (always increasing or always decreasing).
• For non-monotonic functions, finding maximum and minimum of the function can be difficult and other methods can be more attractive.
SE301_Topic 6 Al-Amer2005 ١٢
Newton-Cotes MethodsIn Newton-Cote Methods, the function is approximated by a polynomial of order nComputing the integral of a polynomial is easy.
( )
1)(...
2)()()(
...)(1122
10
10
+−
++−
+−≈
+++≈
++
∫
∫∫
nabaabaabadxxf
dxxaxaadxxfnn
n
b
a
b
a
nn
b
a
SE301_Topic 6 Al-Amer2005 ١٣
Newton-Cotes MethodsTrapezoid Method (First Order Polynomial are used)
Simpson 1/3 Rule (Second Order Polynomial are used),
( )
( )∫∫
∫∫
++≈
+≈
b
a
b
a
b
a
b
a
dxxaxaadxxf
dxxaadxxf
2210
10
)(
)(
SE301_Topic 6 Al-Amer2005 ١٤
Trapezoid MethodTrapezoid Method
)()()()( axab
afbfaf −−−
+
f(x)
ba ( )2
)()(
2)()(
)()()(
)()()()(
)(
2
afbfab
xab
afbf
xab
afbfaaf
dxaxab
afbfafI
dxxfI
b
a
b
a
b
a
b
a
+−=
−−
+
⎟⎠⎞
⎜⎝⎛
−−
−=
⎟⎠⎞
⎜⎝⎛ −
−−
+≈
=
∫
∫
SE301_Topic 6 Al-Amer2005 ١٥
Trapezoid MethodDerivation-One interval
( )
( )2
)()(
)()(2
)()()()()(
2)()()()()(
)()()()()(
)()()()()(
22
2
afbfab
abab
afbfabab
afbfaaf
xab
afbfxab
afbfaaf
dxxab
afbfab
afbfaafI
dxaxab
afbfafdxxfI
b
a
b
a
b
a
b
a
b
a
+−=
−−−
+−⎟⎠⎞
⎜⎝⎛
−−
−=
−−
+⎟⎠⎞
⎜⎝⎛
−−
−=
⎟⎠⎞
⎜⎝⎛
−−
+−−
−≈
⎟⎠⎞
⎜⎝⎛ −
−−
+≈=
∫
∫∫
SE301_Topic 6 Al-Amer2005 ١٦
Trapezoid MethodTrapezoid Method
f(x)
)(bf
( ))()(2
bfafabArea +−
=
)(af
ba
SE301_Topic 6 Al-Amer2005 ١٧
Trapezoid MethodTrapezoid MethodMultiple Application RuleMultiple Application Rule
f(x)
s trapezoid theof
areas theof sum)(
... segments into dpartitione
is b][a, intervalThe
210
=
=≤≤≤≤=
∫b
a
n
dxxf
bxxxxan
( )1212
2)()( xxxfxfArea −
+=
x3210 xxxx
a b
SE301_Topic 6 Al-Amer2005 ١٨
Trapezoid MethodTrapezoid MethodGeneral Formula and special caseGeneral Formula and special case
( )( ))()(21)(
... equal)y necessarilot segments(nn into divided is interval theIf
11
1
0
210
iiii
n
i
b
a
n
xfxfxxdxxf
bxxxxa
+−≈
=≤≤≤≤=
++
−
=∑∫
[ ] ⎥⎦
⎤⎢⎣
⎡++≈
=−
∑∫−
=
+
1
10
1
)()()(21)(
allfor points) base spacedEqualiy ( CaseSpecial
n
iin
b
a
ii
xfxfxfhdxxf
ihxx
SE301_Topic 6 Al-Amer2005 ١٩
Example
Given a tabulated values of the velocity of an object.
Obtain an estimate of the distance traveled in the interval [0,3].
Time (s) 0.0 1.0 2.0 3.0
Velocity (m/s) 0.0 10 12 14
Distance = integral of the velocity
∫=3
0)(Distance dttV
SE301_Topic 6 Al-Amer2005 ٢٠
Example 1Example 1
Time (s) 0.0 1.0 2.0 3.0
Velocity (m/s)
0.0 10 12 14
{ }3,2,1,0are points Baselssubinterva3 into
divided is interval The
( )
29)140(2112)(101 Distance
)()(21)(
1
0
1
1
1
=⎥⎦⎤
⎢⎣⎡ +++=
⎥⎦
⎤⎢⎣
⎡++=
=−=
∑−
=
+
n
n
ii
ii
xfxfxfhT
xxhMethodTrapezoid
SE301_Topic 6 Al-Amer2005 ٢١
Estimating the Error Estimating the Error For Trapezoid methodFor Trapezoid method
?accurcy digit decimal 5 to
)sin( compute toneeded
are intervals spacedequally many How
0∫π
dxx
SE301_Topic 6 Al-Amer2005 ٢٢
Error in estimating the integralError in estimating the integralTheoremTheorem
)(''max12
)(12
then)( eapproximat
toused is Method Trapezoid If :Theorem) (width intervals Equal
on continuous is)('' :Assumption
],[
2
''2
a
xfhabError
[a,b]wherefhabError
dxxf
h[a,b]xf
bax
b
∈
−≤
∈−
−=
=
∫ξξ
SE301_Topic 6 Al-Amer2005 ٢٣
ExampleExample
52
52
],[
2
5
0
106
1021
121)(''
)sin()('');cos()(';0;
)(''max12
1021 error ,)sin(
−
−
∈
−
×≤⇒
×≤≤⇒≤
−====
−≤
×≤∫
π
ππ
π
h
hErrorxf
xxfxxfab
xfhabError
thatsohfinddxx
bax
SE301_Topic 6 Al-Amer2005 ٢٤
ExampleExample
x 1.0 1.5 2.0 2.5 3.0
f(x) 2.1 3.2 3.4 2.8 2.7
( )( )
( )⎥⎦
⎤⎢⎣
⎡++=
−=
+−=
∑
∑
∫
−
=
+
++
−
=
)()(21)(),(
, allfor :
)()(21),(
)(Compute tomethod TrapezoidUse
0
1
1
1
11
1
0
3
1
n
n
ii
ii
iiii
n
i
xfxfxfhPfT
ixxhCaseSpecial
xfxfxxPfTTrapezoid
dxxf
SE301_Topic 6 Al-Amer2005 ٢٥
ExampleExample
x 1.0 1.5 2.0 2.5 3.0
f(x) 2.1 3.2 3.4 2.8 2.7
( )
( )
9.5
7.21.2218.24.32.35.0
)()(21)()( 0
1
1
3
1
=
⎥⎦⎤
⎢⎣⎡ ++++=
⎥⎥⎦
⎤
⎢⎢⎣
⎡++≈ ∑∫
−
=n
n
ii xfxfxfhdxxf
SE301: Numerical Method
Lecture 18Recursive Trapezoid Method
SE301_Topic 6 Al-Amer2005 ٢٦
Recursive formula is used
SE301_Topic 6 Al-Amer2005 ٢٧
Recursive Trapezoid MethodRecursive Trapezoid Method
f(x)
( ))()(2
)0,0(R
intervaloneonbasedEstimate
bfafababh
+−
=
−=
haa +
SE301_Topic 6 Al-Amer2005 ٢٨
Recursive Trapezoid MethodRecursive Trapezoid Method
f(x)
hahaa 2++
( )
[ ])()0,0(21)0,1(
)()(21)(
2)0,1(R
2
intervals 2on based Estimate
hafhRR
bfafhafab
abh
++=
⎥⎦⎤
⎢⎣⎡ +++
−=
−=
Based on previous estimate
Based on new point
SE301_Topic 6 Al-Amer2005 ٢٩
Recursive Trapezoid MethodRecursive Trapezoid Method
f(x)
[
( )
[ ])3()()0,1(21)0,2(
)()(21
)3()2()(4
)0,2(R
4
hafhafhRR
bfaf
hafhafhafab
abh
++++=
⎥⎦⎤++
+++++−
=
−=
hahaa 42 ++Based on previous estimate
Based on new points
SE301_Topic 6 Al-Amer2005 ٣٠
Recursive Trapezoid MethodRecursive Trapezoid MethodFormulasFormulas
[ ]
( )
n
k
abh
hkafhnRnR
bfafab
n
2
)12()0,1(21)0,(
)()(2
)0,0(R
)1(2
1
−=
⎥⎦
⎤⎢⎣
⎡−++−=
+−
=
∑−
=
SE301_Topic 6 Al-Amer2005 ٣١
Recursive Trapezoid MethodRecursive Trapezoid Method
[ ]
( )
( )
( )
( )⎥⎦
⎤⎢⎣
⎡−++−=
−=
⎥⎦
⎤⎢⎣
⎡−++=
−=
⎥⎦
⎤⎢⎣
⎡−++=
−=
⎥⎦
⎤⎢⎣
⎡−++=
−=
+−
=−=
∑
∑
∑
∑
−
=
=
=
=
)1(
2
2
1
2
13
2
12
1
1
)12()0,1(21)0,(,
2
..................
)12()0,2(21)0,3(,
2
)12()0,1(21)0,2(,
2
)12()0,0(21)0,1(,
2
)()(2
)0,0(R,
n
kn
k
k
k
hkafhnRnRabh
hkafhRRabh
hkafhRRabh
hkafhRRabh
bfafababh
SE301_Topic 6 Al-Amer2005 ٣٢
Advantages of Recursive TrapezoidRecursive Trapezoid:
Gives the same answer as the standard Trapezoid method. Make use of the available information to reduce computation time.Useful if the number of iterations is not known in advance.
SE301:Numerical Methods19. Romberg Method
SE301_Topic 6 Al-Amer2005 ٣٣
MotivationDerivation of Romberg Method
Romberg MethodExample
When to stop?
SE301_Topic 6 Al-Amer2005 ٣٤
Motivation for Romberg MethodMotivation for Romberg MethodTrapezoid formula with an interval h gives error of the order O(h2)We can combine two Trapezoid estimates with intervals 2h and h to get a better estimate.
SE301_Topic 6 Al-Amer2005 ٣٥
Romberg MethodRomberg Method
dx f(x) ofion approximat theimprove tocombined are
4h,8h,...2h,h,sizeofintervalswith methodTrapezoid using Estimatesb
a∫
First column is obtained using Trapezoid Method
R(0,0)
R(1,0) R(1,1)
R(2,0) R(2,1) R(2,2)
R(3,0) R(3,1) R(3,2) R(3,3)The other elements are obtained using the Romberg Method
SE301_Topic 6 Al-Amer2005 ٣٦
First Column First Column Recursive Trapezoid MethodRecursive Trapezoid Method
[ ]
( )
n
k
abh
hkafhnRnR
bfafab
n
2
)12()0,1(21)0,(
)()(2
)0,0(R
)1(2
1
−=
⎥⎦
⎤⎢⎣
⎡−++−=
+−
=
∑−
=
SE301_Topic 6 Al-Amer2005 ٣٧
Derivation of Romberg MethodDerivation of Romberg Method
[ ] ...)0,1()0,(31)0,()(
2*41
)2(...641
161
41)0,()(
R(n,0)by obtained is estimate accurate More
)1(...)0,1()(
2 method Trapezoid)()0,1()(
66
44
66
44
22
66
44
22
12
+++−−+=
−
++++=
++++−=
−=+−=
∫
∫
∫
∫ −
hbhbnRnRnRdxxf
giveseqeq
eqhahahanRdxxf
eqhahahanRdxxf
abhwithhOnRdxxf
b
a
b
a
b
a
n
b
a
SE301_Topic 6 Al-Amer2005 ٣٨
Romberg MethodRomberg Method
[ ]
( )
[ ]1,1
)1,1()1,(14
1)1,(),(
)12()0,1(21)0,(
,2
)()(2
)0,0(R
)1(2
1
≥≥
−−−−−
+−=
⎥⎦
⎤⎢⎣
⎡−++−=
−=
+−
=
∑−
=
mnfor
mnRmnRmnRmnR
hkafhnRnR
abh
bfafab
m
k
n
n
R(0,0)
R(1,0) R(1,1)
R(2,0) R(2,1) R(2,2)
R(3,0) R(3,1) R(3,2) R(3,3)
SE301_Topic 6 Al-Amer2005 ٣٩
Property of Romberg MethodProperty of Romberg MethodR(0,0)
R(1,0) R(1,1)
R(2,0) R(2,1) R(2,2)
R(3,0) R(3,1) R(3,2) R(3,3))(),()(
Theorem
22 ++=∫ mb
a
hOmnRdxxf
)()()()( 8642 hOhOhOhOError Level
SE301_Topic 6 Al-Amer2005 ٤٠
Example 1Example 1
[ ] [ ]
[ ]
[ ]31
21
83
31
83)0,0()0,1(
141)0,1()1,1(
1,1
)1,1()1,(14
1)1,(),(
83
41
21
21
21))(()0,0(
21)0,1(,
21
5.01021)()(
2)0,0(,1
Compute
1
1
02
=⎥⎦⎤
⎢⎣⎡ −+=−
−+=
≥≥
−−−−−
+−=
=⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=++==
=+=+−
==
∫
RRRR
mnfor
mnRmnRmnRmnR
hafhRRh
bfafabRh
dxx
m
0.5
3/8 1/3
SE301_Topic 6 Al-Amer2005 ٤١
0.5
3/8 1/3
11/32 1/3 1/3
Example 1 cont.Example 1 cont.
[ ]
[ ]
[ ]31
31
31
151
31)1,1()1,2(
141)1,2()2,2(
31
83
3211
31
3211)0,1()0,2(
141)0,1()1,2(
)1,1()1,(14
1)1,(),(
3211
169
161
41
83
21))3()(()0,1(
21)0,2(,
41
2
1
=⎥⎦⎤
⎢⎣⎡ −+=−
−+=
=⎥⎦⎤
⎢⎣⎡ −+=−
−+=
−−−−−
+−=
=⎟⎠⎞
⎜⎝⎛ ++⎟
⎠⎞
⎜⎝⎛=++++==
RRRR
RRRR
mnRmnRmnRmnR
hafhafhRRh
m
SE301_Topic 6 Al-Amer2005 ٤٢
When do we stop?When do we stop?
R(4,4)at STOP examplefor steps ofnumber given aafter
or )1,1()1,( ε≤−−−− mnRmnR
ifSTOP
SE301:Numerical Methods20. Gauss Quadrature
SE301_Topic 6 Al-Amer2005 ٤٣
MotivationGeneral integration formula
Read 22.3-22.3
SE301_Topic 6 Al-Amer2005 ٤٤
MotivationMotivation
( )
⎩⎨⎧
=−=
=
=
⎥⎦
⎤⎢⎣
⎡++=
∑∫
∑∫
=
−
=
nandihnih
cwhere
xfcdxxf
xfxfxfhdxxf
i
n
iii
b
a
n
ini
b
a
05.01,...,2,1
)()(
as expressed becan It
)()(21)()(
Method Trapezoid
0
1
10
SE301_Topic 6 Al-Amer2005 ٤٥
General Integration FormulaGeneral Integration Formula
integral theofion approximat good gives formula that thesoselect wedo How
:Problem::
)()(0
ii
ii
n
iii
b
a
xandc
NodesxWeightsc
xfcdxxf ∑∫=
=
SE301_Topic 6 Al-Amer2005 ٤٦
Lagrange InterpolationLagrange Interpolation
∫∑∫
∫ ∑∫∫
∫∫
=≈⇒
⎟⎠
⎞⎜⎝
⎛=≈
≈
=
=
b
a ii
n
iii
b
a
b
a i
n
ii
b
a n
b
a
n
n
b
a n
b
a
dxxcwherexfcdxxf
dxxfxdxxPdxxf
xxxxPwhere
dxxPdxxf
)()()(
)()()()(
,...,, nodes at thef(x) einterpolat that polynomial a is )(
)()(
0
0
10
l
l
SE301_Topic 6 Al-Amer2005 ٤٧
QuestionQuestion
What is the best way to choose the nodes and the weights?
SE301_Topic 6 Al-Amer2005 ٤٨
Theorem Theorem
12norder of spolynomial allfor exact be willformula The
)()()(
then)( of zeros theare,...,,,Let
00)(
such that 1n degree of polynomial nontrivial a be Let
0
210
+≤
=≈
≤≤=
+
∫∑∫
∫
=
dxxcwherexfcdxxf
xqxxxx
nkdxxqx
q
i
b
ai
n
iii
b
a
n
kb
a
l
SE301_Topic 6 Al-Amer2005 ٤٩
Determining The Weights and NodesDetermining The Weights and Nodes
?5order of spolynomial allfor exact is formula that theso
weights theand nodes select the wedo How
)()()()( 221100
1
1
≤
++=∫− xfcxfcxfcdxxf
SE301_Topic 6 Al-Amer2005 ٥٠
Determining The Weights and NodesDetermining The Weights and NodesSolutionSolution
5,3,0solution possible one
0)(
0)(
0)(
satisfy must )()(
3120
1
1
2
1
1
1
1
33
2210
=−===
=
=
=
+++=
∫∫∫
−
−
−
aaaa
dxxxq
xdxxq
dxxq
xqxaxaxaaxqLet
SE301_Topic 6 Al-Amer2005 ٥١
Theorem Theorem
12norder of spolynomial allfor exact be willformula The
)()()(
then)( of zeros theare,...,,,Let
00)(
such that 1n degree of polynomial nontrivial a be Let
0
210
+≤
=≈
≤≤=
+
∫∑∫
∫
=
dxxcwherexfcdxxf
xqxxxx
nkdxxqx
q
i
b
ai
n
iii
b
a
n
kb
a
l
SE301_Topic 6 Al-Amer2005 ٥٢
Determining The Weights and NodesDetermining The Weights and NodesSolutionSolution
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
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SE301_Topic 6 Al-Amer2005 ٥٣
Determining The Weights and NodesDetermining The Weights and NodesSolutionSolution
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SE301_Topic 6 Al-Amer2005 ٥٤
626
Gaussian Gaussian QuadratureQuadratureSee more in Table 22.1 (page 626)See more in Table 22.1 (page 626)
236926.0,478628.0,568889.0,478628.0,236926.0906179.0,538469.0,00000.0,538469.0,906179.04
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SE301_Topic 6 Al-Amer2005 ٥٥
627
Error Analysis for Gauss Error Analysis for Gauss QuadratureQuadrature
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SE301_Topic 6 Al-Amer2005 ٥٦
ExampleExample
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SE301_Topic 6 Al-Amer2005 ٥٧
ExampleExample
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SE301_Topic 6 Al-Amer2005 ٥٨
627
Improper IntegralsImproper Integrals
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SE301_Topic 6 Al-Amer2005 ٥٩
QuizQuiz
x 1.0 1.5 2.0 2.5 3.0
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