lecture 19 - numerical integration cven 302 july 22, 2002
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Lecture 19 - Numerical Integration Lecture 19 - Numerical Integration
CVEN 302
July 22, 2002
Lecture’s GoalsLecture’s Goals
• Trapezoidal Rule• Simpson’s Rule
– 1/3 Rule– 3/8 Rule
• Midpoint• Gaussian Quadrature
Basic Numerical Integration
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.
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)
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
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
Numerical IntegrationNumerical Integration• Newton-Cotes Open Formulae -- Use
only interior points– midpoint rule
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)
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
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
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
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)
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
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
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
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
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
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
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
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
Better Numerical IntegrationBetter Numerical Integration
• Composite integration
– Composite Trapezoidal Rule
– Composite Simpson’s Rule
• Richardson Extrapolation
• Romberg integration
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
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
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
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
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
Composite Simpson’s RuleComposite Simpson’s Rule
x0 x2x
f(x)
x4h h xn-2h xn
n
abh
…...
Piecewise Quadratic approximations
hx3x1 xn-1
)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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
HomeworkHomework
• Check the Homework webpage
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