lecture 19 - numerical integration cven 302 july 22, 2002
TRANSCRIPT
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Lecture 19 - Numerical Integration Lecture 19 - Numerical Integration
CVEN 302
July 22, 2002
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Lecture’s GoalsLecture’s Goals
• Trapezoidal Rule• Simpson’s Rule
– 1/3 Rule– 3/8 Rule
• Midpoint• Gaussian Quadrature
Basic Numerical Integration
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Basic Numerical IntegrationBasic Numerical Integration
We want to find integration of functions of various forms of the equation known as the Newton Cotes integration formulas.
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Basic Numerical IntegrationBasic Numerical IntegrationWeighted sum of function values
)x(fc)x(fc)x(fc
)x(fcdx)x(f
nn1100
i
n
0ii
b
a
x0 x1 xnxn-1x
f(x)
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0
2
4
6
8
10
12
3 5 7 9 11 13 15
Numerical IntegrationNumerical IntegrationIdea is to do integral in small parts, like the way you first learned integration - a summation
Numerical methods just try to make it faster and more accurate
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Numerical IntegrationNumerical Integration• Newton-Cotes Closed Formulae --
Use both end points– Trapezoidal Rule : Linear– Simpson’s 1/3-Rule : Quadratic– Simpson’s 3/8-Rule : Cubic
– Boole’s Rule : Fourth-order
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Numerical IntegrationNumerical Integration• Newton-Cotes Open Formulae -- Use
only interior points– midpoint rule
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Trapezoid RuleTrapezoid RuleStraight-line approximation
)x(f)x(f2
h
)x(fc)x(fc)x(fcdx)x(f
10
1100i
1
0ii
b
a
x0 x1x
f(x)
L(x)
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Trapezoid RuleTrapezoid Rule
Lagrange interpolation
010 1
0 1 1 0
0 1
( ) ( ) ( )
dxlet a x , b x , , d ;
h
0( ) (1 ) ( ) ( ) ( )
1
x xx xL x f x f x
x x x x
x ah b a
b a
x aL f a f b
x b
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Trapezoid RuleTrapezoid Rule
Integrate to obtain the rule
1
0
1 1
0 0
1 12 2
0 0
( ) ( ) ( )
( ) (1 ) ( )
( ) ( ) ( ) ( ) ( )2 2 2
b b
a af x dx L x dx h L d
f a h d f b h d
hf a h f b h f a f b
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Example:Trapezoid RuleExample:Trapezoid RuleEvaluate the integral
• Exact solution
• Trapezoidal Rule
926477.5216)1x2(e4
1
e4
1e
2
xdxxe
1
0
x2
4
0
x2x24
0
x2
dxxe4
0
x2
%12.357926.5216
66.23847926.5216
66.23847)e40(2)4(f)0(f2
04dxxeI 84
0
x2
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Simpson’s 1/3-RuleSimpson’s 1/3-RuleApproximate the function by a parabola
)x(f)x(f4)x(f3
h
)x(fc)x(fc)x(fc)x(fcdx)x(f
210
221100i
2
0ii
b
a
x0 x1x
f(x)
x2h h
L(x)
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Simpson’s 1/3-RuleSimpson’s 1/3-Rule
1 xx
0 xx
1 xxh
dxd ,
h
xx ,
2
abh
2
ba x ,bx ,ax let
)x(f)xx)(xx(
)xx)(xx(
)x(f)xx)(xx(
)xx)(xx( )x(f
)xx)(xx(
)xx)(xx()x(L
2
1
0
1
120
21202
10
12101
200
2010
21
)x(f2
)1()x(f)1()x(f
2
)1()(L 21
20
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Simpson’s 1/3-RuleSimpson’s 1/3-Rule
1
1
23
2
1
1
3
1
1
1
23
0
1
12
1
0
21
1
10
1
1
b
a
)2
ξ
3
ξ(
2
h)f(x
)3
ξ(ξ)hf(x)
2
ξ
3
ξ(
2
h)f(x
)dξ1ξ(ξ2
h)f(x)dξξ1()hf(x
)dξ1ξ(ξ2
h)f(xdξ)(Lhf(x)dx
)f(x)4f(x)f(x3
hf(x)dx 210
b
a
Integrate the Lagrange interpolation
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Simpson’s 3/8-RuleSimpson’s 3/8-RuleApproximate by a cubic polynomial
)x(f)x(f3)x(f3)x(f8
h3
)f(xc)f(xc)f(xc)f(xc)x(fcdx)x(f
3210
33221100i
3
0ii
b
a
x0 x1x
f(x)
x2h h
L(x)
x3h
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Simpson’s 3/8-RuleSimpson’s 3/8-Rule
)x(f)xx)(xx)(xx(
)xx)(xx)(xx(
)x(f)xx)(xx)(xx(
)xx)(xx)(xx(
)x(f)xx)(xx)(xx(
)xx)(xx)(xx(
)x(f)xx)(xx)(xx(
)xx)(xx)(xx()x(L
3231303
210
2321202
310
1312101
320
0302010
321
)x(f)x(f3)x(f3)x(f8
h33
a-bh ; L(x)dxf(x)dx
3210
b
a
b
a
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Example: Simpson’s RulesExample: Simpson’s RulesEvaluate the integral• Simpson’s 1/3-Rule
• Simpson’s 3/8-Rule
dxxe4
0
x2
4 2
0
4 8
(0) 4 (2) (4)3
20 4(2 ) 4 8240.411
35216.926 8240.411
57.96%5216.926
x hI xe dx f f f
e e
4 2
0
3 4 8(0) 3 ( ) 3 ( ) (4)
8 3 3
3(4/3)0 3(19.18922) 3(552.33933) 11923.832 6819.209
85216.926 6819.209
30.71%5216.926
x hI xe dx f f f f
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Midpoint RuleMidpoint RuleNewton-Cotes Open Formula
)(f24
)ab()
2
ba(f)ab(
)x(f)ab(dx)x(f3
m
b
a
a b x
f(x)
xm
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Two-point Newton-Cotes Two-point Newton-Cotes Open FormulaOpen Formula
Approximate by a straight line
)(f108
)ab()x(f)x(f
2
abdx)x(f
3
21
b
a
x0 x1x
f(x)
x2h h x3h
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Three-Point Newton-Cotes Open Three-Point Newton-Cotes Open FormulaFormula
Approximate by a parabola
)(f23040
)ab(7
)x(f2)x(f)x(f23
abdx)x(f
5
321
b
a
x0 x1x
f(x)
x2h h x3h h x4
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Better Numerical IntegrationBetter Numerical Integration
• Composite integration
– Composite Trapezoidal Rule
– Composite Simpson’s Rule
• Richardson Extrapolation
• Romberg integration
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Apply trapezoid rule to multiple segments over Apply trapezoid rule to multiple segments over integration limitsintegration limits
0
1
2
3
4
5
6
7
3 5 7 9 11 13 15
Two segments
0
1
2
3
4
5
6
7
3 5 7 9 11 13 15
Four segments
0
1
2
3
4
5
6
7
3 5 7 9 11 13 15
Many segments
0
1
2
3
4
5
6
7
3 5 7 9 11 13 15
Three segments
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Composite Trapezoid RuleComposite Trapezoid Rule
)x(f)x(f2)2f(x)f(x2)f(x2
h
)f(x)f(x2
h)f(x)f(x
2
h)f(x)f(x
2
h
f(x)dxf(x)dxf(x)dxf(x)dx
n1ni10
n1n2110
x
x
x
x
x
x
b
a
n
1n
2
1
1
0
x0 x1x
f(x)
x2h h x3h h x4
n
abh
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Composite Trapezoid RuleComposite Trapezoid RuleEvaluate the integral dxxeI
4
0
x2
%66.2 95.5355
)4(f)75.3(f2)5.3(f2
)5.0(f2)25.0(f2)0(f2
hI25.0h,16n
%50.10 76.5764)4(f)5.3(f2 )3(f2)5.2(f2)2(f2)5.1(f2
)1(f2)5.0(f2)0(f2
hI5.0h,8n
%71.39 79.7288)4(f)3(f2
)2(f2)1(f2)0(f2
hI1h,4n
%75.132 23.12142)4(f)2(f2)0(f2
hI2h,2n
%12.357 66.23847)4(f)0(f2
hI4h,1n
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Composite Trapezoid Composite Trapezoid ExampleExample
2
1 1
1dx
x
x f(x)1.00 0.50001.25 0.44441.50 0.40001.75 0.36362.00 0.3333
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Composite Trapezoid Rule with Composite Trapezoid Rule with Unequal SegmentsUnequal Segments
Evaluate the integral
• h1 = 2, h2 = 1, h3 = 0.5, h4 = 0.5 dxxeI
4
0
x2
%45.14 58.5971 45.3 2
0.5
5.33e 2
0.532
2
120
2
2
)4()5.3(2
)5.3()3(2
)3()2(2
)2()0(2
)()()()(
87
76644
43
21
4
5.3
5.3
3
3
2
2
0
ee
eeee
ffh
ffh
ffh
ffh
dxxfdxxfdxxfdxxfI
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Composite Simpson’s RuleComposite Simpson’s Rule
x0 x2x
f(x)
x4h h xn-2h xn
n
abh
…...
Piecewise Quadratic approximations
hx3x1 xn-1
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)x(f)x(f4)x(f2
)4f(x)x(f2)f(x4
)2f(x)f(x4)2f(x)f(x4)f(x3
h
)f(x)4f(x)f(x3
h
)f(x)f(x4)f(x3
h)f(x)f(x4)f(x
3
h
f(x)dxf(x)dxf(x)dxf(x)dx
n1n2n
12ii21-2i
43210
n1n2n
432210
x
x
x
x
x
x
b
a
n
2n
4
2
2
0
Composite Simpson’s RuleComposite Simpson’s RuleMultiple applications of Simpson’s rule
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Composite Simpson’s RuleComposite Simpson’s RuleEvaluate the integral
• n = 2, h = 2
• n = 4, h = 1
dxxeI4
0
x2
%70.8 975.5670
e4)e3(4)e2(2)e(403
1
)4(f)3(f4)2(f2)1(f4)0(f3
hI
8642
%96.57 411.8240e4)e2(403
2
)4(f)2(f4)0(f3
hI
84
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Composite Simpson’s Composite Simpson’s ExampleExample
2
1 1
1dx
x
x f(x)1.00 0.50001.25 0.44441.50 0.40001.75 0.36362.00 0.3333
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Composite Simpson’s Rule with Composite Simpson’s Rule with Unequal SegmentsUnequal Segments
Evaluate the integral
• h1 = 1.5, h2 = 0.5
dxxeI4
0
x2
%76.3 23.5413
4)5.3(433
5.03)5.1(40
3
5.1
)4(2)5.3(4)3(3
)3(2)5.1(4)0(3
)()(
87663
2
1
4
3
3
0
eeeee
fffh
fffh
dxxfdxxfI
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Richardson ExtrapolationRichardson ExtrapolationUse trapezoidal rule as an example
– subintervals: n = 2j = 1, 2, 4, 8, 16, ….
j2
1jjn1n10
b
ahc)x(f)x(f2)f(x2)f(x
2
hf(x)dx
)()()()(
)()()()(
)()()()()(
)()()(
)()(
bfxf2xf2af2
hI2j
bfxf2xf2af16
hI83
bfxf2xf2xf2af8
hI42
bfxf2af4
hI21
bfaf2
hI10
Formulanj
1n1jjj
713
3212
11
0
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Richardson ExtrapolationRichardson ExtrapolationFor trapezoidal rule
– kth level of extrapolation
)h(B)2
h(B16
15
1)h(C
)2
h(b)
2
h(BA
hb)h(BA
hb)h(B h4
c)h(A)
2
h(A4
3
1A
)2
h(c)
2
h(c)
2
h(AA
hchc)h(AA
hc)h(Adx)x(fA
42
42
42
42
42
21
42
21
21
b
a
14
)h(Ch/2)(C4)h(D
k
k
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255
II256
63
II64
15
II16
3
II4I16h
II8h
III4h
IIII2h
IIIIIh
hOhOhOhOhO
4k3k2k1k0k
sBoolesSimpsonTrapezoid
3j31j2j21j1j11j0j01j
04
1303
221202
31211101
4030201000
108642
,,,,,,,,
,
,,
,,,
,,,,
,,,,,
/
/
/
/
)()()()()(
''
3, 2,1,k ;14
II4I
k
k,jk,1jk
k,j
Romberg IntegrationRomberg IntegrationAccelerated Trapezoid Rule
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Romberg IntegrationRomberg Integration
926477.5216dxxeI4
0
x2
%00050.0%00168.0%0053.0%0527.0%66.2
95.535525.0h
68.521976.57645.0h
20.521775.525679.72881h
01.521714.522998.56702.121422h
95.521684.522468.549941.82407.238474h
)h(O)h(O)h(O)h(O)h(O
4k3k2k1k0k
s'Booles'SimpsonTrapezoid
108642
Accelerated Trapezoid Rule
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Romberg Integration Romberg Integration ExampleExample
2
1 1
1dx
x
x f(x)1.00 0.50001.25 0.44441.50 0.40001.75 0.36362.00 0.3333
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Gaussian QuadraturesGaussian Quadratures• Newton-Cotes Formulae
– use evenly-spaced functional values
• Gaussian Quadratures– select functional values at non-uniformly distributed
points to achieve higher accuracy
– change of variables so that the interval of integration is [-1,1]
– Gauss-Legendre formulae
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Gaussian Quadrature on Gaussian Quadrature on [-1, 1][-1, 1]
• Choose (c1, c2, x1, x2) such that the method yields “exact integral” for f(x) = x0, x1, x2, x3
)x(fc)x(fc)x(fc)x(fcdx)x(f nn2211i
1
1
n
1ii
)f(xc)f(xc
f(x)dx :2n
2211
1
1
x2x1-1 1
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Gaussian Quadrature on Gaussian Quadrature on [-1, 1][-1, 1]
Exact integral for f = x0, x1, x2, x3
– Four equations for four unknowns
)f(xc)f(xcf(x)dx :2n 2211
1
1
3
1x
3
1x
1c1c
xcxc0dxx xf
xcxc3
2dxx xf
xcxc0xdx xf
cc2dx1 1f
2
1
2
1
322
31
1
1 133
222
21
1
1 122
221
1
1 1
2
1
1 1
)3
1(f)
3
1(fdx)x(fI
1
1
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Gaussian Quadrature on Gaussian Quadrature on [-1, 1][-1, 1]
• Choose (c1, c2, c3, x1, x2, x3) such that the method yields “exact integral” for f(x) = x0, x1, x2, x3,x4, x5
)x(fc)x(fc)x(fcdx)x(f :3n 332211
1
1
x3x1-1 1x2
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Gaussian Quadrature on Gaussian Quadrature on [-1, 1][-1, 1]
533
522
511
1
1
55
433
422
411
1
1
44
333
322
311
1
1
33
233
222
211
1
1
22
332211
1
1
321
1
1
0
5
2
0
3
2
0
21
xcxcxcdxxxf
xcxcxcdxxxf
xcxcxcdxxxf
xcxcxcdxxxf
xcxcxcxdxxf
cccxdxf
5/3
0
5/3
9/5
9/8
9/5
3
2
1
3
2
1
x
x
x
c
c
c
![Page 42: Lecture 19 - Numerical Integration CVEN 302 July 22, 2002](https://reader031.vdocument.in/reader031/viewer/2022032106/56649e175503460f94b0291e/html5/thumbnails/42.jpg)
Gaussian Quadrature on Gaussian Quadrature on [-1, 1][-1, 1]
Exact integral for f = x0, x1, x2, x3, x4, x5
)5
3(f
9
5)0(f
9
8)
5
3(f
9
5dx)x(fI
1
1
![Page 43: Lecture 19 - Numerical Integration CVEN 302 July 22, 2002](https://reader031.vdocument.in/reader031/viewer/2022032106/56649e175503460f94b0291e/html5/thumbnails/43.jpg)
Gaussian Quadrature on Gaussian Quadrature on [a, b][a, b]
Coordinate transformation from [a,b] to [-1,1]
t2t1a b
1
1
1
1
b
adx)x(gdx)
2
ab)(
2
abx
2
ab(fdt)t(f
bt 1xat1x2
abx
2
abt
![Page 44: Lecture 19 - Numerical Integration CVEN 302 July 22, 2002](https://reader031.vdocument.in/reader031/viewer/2022032106/56649e175503460f94b0291e/html5/thumbnails/44.jpg)
Example: Gaussian QuadratureExample: Gaussian QuadratureEvaluate
Coordinate transformation
Two-point formula
33.34%)( 543936.3477376279.3468167657324.9
e)3
44(e)
3
44()
3
1(f)
3
1(fdx)x(fI 3
44
3
441
1
926477.5216dtteI4
0
t2
1
1
1
1
4x44
0
t2 dx)x(fdxe)4x4(dtteI
2dxdt ;2x22
abx
2
abt
![Page 45: Lecture 19 - Numerical Integration CVEN 302 July 22, 2002](https://reader031.vdocument.in/reader031/viewer/2022032106/56649e175503460f94b0291e/html5/thumbnails/45.jpg)
Example: Gaussian QuadratureExample: Gaussian QuadratureThree-point formula
Four-point formula
4.79%)( 106689.4967
)142689.8589(9
5)3926001.218(
9
8)221191545.2(
9
5
e)6.044(9
5e)4(
9
8e)6.044(
9
5
)6.0(f9
5)0(f
9
8)6.0(f
9
5dx)x(fI
6.0446.04
1
1
%)37.0( 54375.5197 )339981.0(f)339981.0(f652145.0
)861136.0(f)861136.0(f34785.0dx)x(fI1
1
![Page 46: Lecture 19 - Numerical Integration CVEN 302 July 22, 2002](https://reader031.vdocument.in/reader031/viewer/2022032106/56649e175503460f94b0291e/html5/thumbnails/46.jpg)
SummarySummary
• Integration Techniques– Trapezoidal Rule : Linear
– Simpson’s 1/3-Rule : Quadratic
– Simpson’s 3/8-Rule : Cubic
– Boole’s Rule : Fourth-order
• Gaussian Quadrature
![Page 47: Lecture 19 - Numerical Integration CVEN 302 July 22, 2002](https://reader031.vdocument.in/reader031/viewer/2022032106/56649e175503460f94b0291e/html5/thumbnails/47.jpg)
HomeworkHomework
• Check the Homework webpage