Computational Methods CMSC/AMSC/MAPL 460 Quadrature: Integration Ramani Duraiswami, Dept. of Computer Science
Dec 20, 2015
Computational MethodsCMSC/AMSC/MAPL 460
Quadrature: Integration
Ramani Duraiswami, Dept. of Computer Science
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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
Basic Numerical IntegrationBasic Numerical Integration• Weighted sum of function values
)x(fc)x(fc)x(fc
)x(fcdx)x(f
nn1100
i
n
0ii
b
a
+++=
≈ ∑∫=
LL
x0 x1 xnxn-1x
f(x)
Numerical IntegrationNumerical Integration• Characterized by where the function is
evaluated• 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
Trapezoid RuleTrapezoid Rule• Straight-line approximation
[ ])x(f)x(f2h
)x(fc)x(fc)x(fcdx)x(f
10
1100i
1
0ii
b
a
+=
+=≈ ∑∫=
x0 x1x
f(x)
L(x)
Example:Trapezoid RuleExample:Trapezoid RuleEvaluate the integral• Exact solution (integration by parts)`
• Trapezoidal Rule
926477.5216)1x2(e41
e41e
2xdxxe
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
−=−
=
=+=+−
≈= ∫ε
4
SimpsonSimpson’’s 1/3s 1/3--RuleRuleApproximate the function by a parabola
[ ])x(f)x(f4)x(f3h
)x(fc)x(fc)x(fc)x(fcdx)x(f
210
221100i
2
0ii
b
a
++=
++=≈ ∑∫=
x0 x1x
f(x)
x2h h
L(x)
SimpsonSimpson’’s 3/8s 3/8--RuleRuleApproximate by a cubic polynomial
[ ])x(f)x(f3)x(f3)x(f8h3
)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
Example: SimpsonExample: Simpson’’s Ruless 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
2 0 4(2 ) 4 8240.41135216.926 8240.411 57.96%
5216.926
x hI xe dx f f f
e e
ε
= ≈ + +
⎡ ⎤= + + =⎣ ⎦
−= = −
∫
[ ]
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 6819.2098
5216.926 6819.209 30.71%5216.926
x hI xe dx f f f f
ε
⎡ ⎤= ≈ + + +⎢ ⎥⎣ ⎦
= + + + =
−= = −
∫
Midpoint RuleMidpoint RuleNewton-Cotes Open Formula
)(f24
)ab()2
ba(f)ab(
)x(f)ab(dx)x(f3
m
b
a
η′′−+
+−=
−≈∫
a b x
f(x)
xm
TwoTwo--point Newtonpoint Newton--Cotes Open Cotes Open FormulaFormula
Approximate by a straight line
[ ] )(f108
)ab()x(f)x(f2
abdx)x(f3
21
b
aη′′−
++−
≈∫
x0 x1x
f(x)
x2h h x3h
ThreeThree--point Newtonpoint Newton--Cotes Open Cotes Open FormulaFormula
Approximate by a parabola
[ ]
)(f23040
)ab(7
)x(f2)x(f)x(f23
abdx)x(f5
321
b
a
η′′′′−+
+−−
≈∫
x0 x1x
f(x)
x2h h x3h h x4
• F_1*( x^3/3-5h*x^2/2+6*h^2*x)/(2h^2)• F_1*((4h)^3/3-5h*(4h)^2/2+6h^2*4*h))/(2h^2)• F_1*(4h)(8/3-5+3)• F_1*(4h/3)(8+9-15)• (b-a)/3*F_1*2
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 segments0
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(x2h
)f(x)f(x2h)f(x)f(x
2h)f(x)f(x
2h
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
++++++=
++++++=
+++=
−
−
∫∫∫∫−
LL
L
LL
x0 x1x
f(x)
x2h h x3h h x4
nabh −
=
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(f2hI25.0h,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(f2hI5.0h,8n
%71.39 79.7288)4(f)3(f2
)2(f2)1(f2)0(f2hI1h,4n
%75.132 23.12142)4(f)2(f2)0(f2hI2h,2n
%12.357 66.23847)4(f)0(f2hI4h,1n
−==+++
+++=⇒==
−==++++++
++=⇒==
−==++
++=⇒==
−==++=⇒==
−==+=⇒==
ε
ε
ε
ε
ε
L
Composite Trapezoid Rule with Unequal Composite Trapezoid Rule with Unequal SegmentsSegments
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
0.5322120
22
)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
−=⇒=++
+++++=
++++
+++=
+++= ∫∫∫∫
εee
eeee
ffhffh
ffhffh
dxxfdxxfdxxfdxxfI
Composite SimpsonComposite Simpson’’s Rules Rule
x0 x2x
f(x)
x4h h xn-2h xn
nabh −
=
…...
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(x3h
)f(x)4f(x)f(x3h
)f(x)f(x4)f(x3h)f(x)f(x4)f(x
3h
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
+++
++++
+++++=
++++
+++++=
+++=
−−
+
−−
∫∫∫∫−
L
L
L
L
Composite SimpsonComposite Simpson’’s Rules RuleMultiple applications of Simpson’s rule
Composite SimpsonComposite Simpson’’s Rules RuleEvaluate the integral• n = 2, h = 2
• n = 4, h = 1
dxxeI4
0
x2∫=
[ ]
[ ]%70.8 975.5670
e4)e3(4)e2(2)e(4031
)4(f)3(f4)2(f2)1(f4)0(f3hI
8642
−=⇒=
++++=
++++=
ε
[ ]
[ ] %96.57 411.8240e4)e2(4032
)4(f)2(f4)0(f3hI
84 −=⇒=++=
++=
ε
Composite SimpsonComposite Simpson’’s Rule with 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(4335.03)5.1(40
35.1
)4(2)5.3(4)3(3
)3(2)5.1(4)0(3
)()(
87663
2
1
4
3
3
0
−=⇒=
+++++=
+++
++=
+= ∫∫
ε
eeeee
fffh
fffh
dxxfdxxfI