Post on 18-Dec-2015
USC
Numerical Integration Basic Numerical Integration
We want to find integration of functions of various forms of the equation known as the Newton Cotes integration formulas.
Trapezoidal Rule Simpson’s Rule
1/3 Rule 3/8 Rule
Midpoint Gaussian Quadrature
USC
Basic Numerical Integration• Weighted 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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2
4
6
8
10
12
3 5 7 9 11 13 15
Numerical 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
USC
Numerical 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
Newton-Cotes Open Formulae -- Use only interior points midpoint rule
USC
Trapezoid Rule Straight-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 Rule Lagrange interpolation
010 1
0 1 1 0
0 1
( ) ( ) ( )
dx a x , b x , , d ;
h
0( ) (1 ) ( ) ( ) ( )
1
x xx xL x f x f x
x x x x
x alet h b a
b a
x aL f a f b
x b
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Trapezoid 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 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-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-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-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-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
USC
Simpson’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 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
USC
Midpoint 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
USC
Two-point Newton-Cotes Open 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 Formula
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 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 integration 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 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
USC
Composite 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 Example
2
1 1
1dx
x
x f(x)1.00 0.50001.25 0.44441.50 0.40001.75 0.36362.00 0.3333
USC
Composite Trapezoid Rule with Unequal Segments
Evaluate the integral h1 = 2, h2 = 1, h3 = 0.5, h4 = 0.5
dxxeI4
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 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 RuleMultiple applications of Simpson’s rule
USC
Composite 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 Example
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 Unequal 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
USC
Richardson 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
USC
Richardson 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
USC
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 IntegrationAccelerated Trapezoid Rule
USC
Romberg IntegrationAccelerated Trapezoid Rule
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
USC
Romberg Integration Example
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 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
USC
Gaussian Quadrature on [-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
USC
Gaussian Quadrature on [-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 [-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
USC
Gaussian Quadrature on [-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
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Gaussian Quadrature on [-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
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Gaussian Quadrature on [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
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Example: 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
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Example: 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