Gauss Quadrature Rule of Integration
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Transcript Gauss Quadrature Rule of Integration
Gauss Quadrature Rule of
Integration
Major: All Engineering Majors
Authors: Autar Kaw, Charlie Barker
http://numericalmethods.eng.usf.edu
Transforming Numerical Methods Education for STEM
Undergraduates
7/17/2015
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1
Gauss Quadrature Rule of
Integration
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What is Integration?
b
Integration
The process of measuring
the area under a curve.
f ( x )dx
y
a
f(x)
b
I f ( x )dx
a
Where:
f(x) is the integrand
a= lower limit of integration
b= upper limit of integration
3
a
b
x
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Two-Point Gaussian
Quadrature Rule
4
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Basis of the Gaussian
Quadrature Rule
Previously, the Trapezoidal Rule was developed by the method
of undetermined coefficients. The result of that development is
summarized below.
b
f ( x)dx c f (a) c
1
2
f (b)
a
ba
ba
f (a)
f (b)
2
2
5
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Basis of the Gaussian
Quadrature Rule
The two-point Gauss Quadrature Rule is an extension of the
Trapezoidal Rule approximation where the arguments of the
function are not predetermined as a and b but as unknowns
x1 and x2. In the two-point Gauss Quadrature Rule, the
integral is approximated as
b
I f ( x )dx c1 f ( x1 ) c2 f ( x2 )
a
6
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Basis of the Gaussian
Quadrature Rule
The four unknowns x1, x2, c1 and c2 are found by assuming that
the formula gives exact results for integrating a general third
order polynomial, f ( x ) a a x a x 2 a x 3 .
0
1
2
3
Hence
2
3
f ( x )dx a0 a1 x a2 x a3 x dx
b
b
a
a
b
x
x
x
a0 x a1 a 2
a3
2
3
4 a
2
3
4
b2 a2
b3 a3
b4 a4
a2
a3
a0 b a a1
2
3
4
7
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Basis of the Gaussian
Quadrature Rule
It follows that
b
f ( x )dx c1 a0 a1 x1 a2 x1 a3 x1 c2 a0 a1 x2 a2 x2 a3 x2
2
3
2
3
3
a
Equating Equations the two previous two expressions yield
b2 a2
b3 a3
b4 a4
a2
a3
a0 b a a1
2
3
4
c x a c x
c1 a0 a1 x1 a2 x1 a3 x1 c2 a0 a1 x2 a2 x2 a3 x2
2
a0 c1 c2 a1 c1 x1
8
3
2 2
2
1 1
2
2
3
c2 x2 a3 c1 x1 c2 x2
2
3
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Basis of the Gaussian
Quadrature Rule
Since the constants a0, a1, a2, a3 are arbitrary
b a c1 c2
b3 a 3
2
2
c1 x1 c2 x2
3
9
b2 a2
c1 x1 c2 x2
2
b4 a4
3
3
c1 x1 c2 x2
4
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Basis of Gauss Quadrature
The previous four simultaneous nonlinear Equations have
only one acceptable solution,
b a 1 b a
x1
3
2
2
ba
c1
2
10
b a 1 b a
x2
2
2 3
c2
ba
2
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Basis of Gauss Quadrature
Hence Two-Point Gaussian Quadrature Rule
b
f ( x)dx
c1 f x1 c2 f x2
a
ba
2
11
ba 1 ba ba
f
2 3 2 2
ba 1 ba
f
2 3 2
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Higher Point Gaussian
Quadrature Formulas
12
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Higher Point Gaussian
Quadrature Formulas
b
f ( x)dx c f ( x ) c
1
1
2
f ( x2 ) c3 f ( x3 )
a
is called the three-point Gauss Quadrature Rule.
The coefficients c1, c2, and c3, and the functional arguments x1, x2, and x3
are calculated by assuming the formula gives exact expressions for
integrating a fifth order polynomial
2
3
4
5
a0 a1 x a2 x a3 x a4 x a5 x dx
b
a
General n-point rules would approximate the integral
b
f ( x )dx c1 f ( x1 ) c2 f ( x2 ) . . . . . . . cn f ( xn )
a
13
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Arguments and Weighing Factors
for n-point Gauss Quadrature
Formulas
In handbooks, coefficients and
arguments given for n-point
Gauss Quadrature Rule are
given for integrals
1
n
1
i 1
Table 1: Weighting factors c and function
arguments x used in Gauss Quadrature
Formulas.
Points
Weighting
Factors
2
c1 = 1.000000000
c2 = 1.000000000
14
x1 = -0.577350269
x2 = 0.577350269
3
c1 = 0.555555556
c2 = 0.888888889
c3 = 0.555555556
x1 = -0.774596669
x2 = 0.000000000
x3 = 0.774596669
4
c1
c2
c3
c4
x1 = -0.861136312
x2 = -0.339981044
x3 = 0.339981044
x4 = 0.861136312
g( x )dx ci g( xi )
as shown in Table 1.
Function
Arguments
=
=
=
=
0.347854845
0.652145155
0.652145155
0.347854845
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Arguments and Weighing Factors
for n-point Gauss Quadrature
Formulas
Table 1 (cont.) : Weighting factors c and function arguments x used in
Gauss Quadrature Formulas.
Points
15
Weighting
Factors
Function
Arguments
5
c1
c2
c3
c4
c5
=
=
=
=
=
0.236926885
0.478628670
0.568888889
0.478628670
0.236926885
x1 = -0.906179846
x2 = -0.538469310
x3 = 0.000000000
x4 = 0.538469310
x5 = 0.906179846
6
c1
c2
c3
c4
c5
c6
=
=
=
=
=
=
0.171324492
0.360761573
0.467913935
0.467913935
0.360761573
0.171324492
x1 = -0.932469514
x2 = -0.661209386
x3 = -0.2386191860
x4 = 0.2386191860
x5 = 0.661209386
x6 = 0.932469514
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Arguments and Weighing Factors
for n-point Gauss Quadrature
Formulas
So if the table is given for
1
g( x )dx
integrals, how does one solve
1
b
f ( x )dx ?
The answer lies in that any integral with limits of
a
can be converted into an integral with limits
1, 1
a , b
Let
x mt c
If
x a,
then
t 1
If
x b,
then
t 1
Such that:
ba
m
2
16
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Arguments and Weighing Factors
for n-point Gauss Quadrature
Formulas
ba
c
2
Then
ba ba
x
t
2
2
Hence
dx
ba
dt
2
Substituting our values of x, and dx into the integral gives us
b
a
17
baba
ba
f
t
dt
1 2
2 2
1
f ( x )dx
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Example 1
For an integral
Rule.
b
f ( x )dx,
derive the one-point Gaussian Quadrature
a
Solution
The one-point Gaussian Quadrature Rule is
b
f ( x )dx c1 f x1
a
18
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Solution
The two unknowns x1, and c1 are found by assuming that the
formula gives exact results for integrating a general first order
polynomial,
f ( x) a0 a1 x.
b
b
f ( x )dx a
a
0
a1 x dx
a
b
x2
a0 x a1
2 a
b2 a 2
a0 b a a1
2
19
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Solution
It follows that
b
f ( x)dx c a
1
0
a1 x1
a
Equating Equations, the two previous two expressions yield
b2 a 2
c1 a0 a1 x1
a0 b a a1
2
20
a0 (c1 ) a1 (c1 x1 )
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Basis of the Gaussian
Quadrature Rule
Since the constants a0, and a1 are arbitrary
b a c1
b2 a 2
c1 x1
2
giving
c1 b a
ba
x1
2
21
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Solution
Hence One-Point Gaussian Quadrature Rule
b
a
22
ba
f ( x )dx c1 f x1 (b a ) f
2
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Example 2
a)
Use two-point Gauss Quadrature Rule to approximate the distance
covered by a rocket from t=8 to t=30 as given by
140000
x 2000 ln
9
.
8
t
dt
140000 2100t
8
30
23
b)
Find the true error,
Et for part (a).
c)
Also, find the absolute relative true error, a for part (a).
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Solution
First, change the limits of integration from [8,30] to [-1,1]
by previous relations as follows
30 8 1 30 8
30 8
f
(
t
)
dt
f
x
dx
2 1 2
2
8
30
1
11 f 11x 19dx
1
24
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Solution (cont)
Next, get weighting factors and function argument values from Table 1
for the two point rule,
c1 1.000000000
x1 0.577350269
c2 1.000000000
x2 0.577350269
25
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Solution (cont.)
Now we can use the Gauss Quadrature formula
1
11 f 11x 19dx 11c1 f 11x1 19 11c2 f 11x2 19
1
11 f 11( 0.5773503) 19 11 f 11( 0.5773503) 19
11 f ( 12.64915 ) 11 f ( 25.35085 )
11( 296.8317 ) 11( 708.4811)
11058.44 m
26
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Solution (cont)
since
140000
f ( 12.64915 ) 2000 ln
9.8( 12.64915 )
140000 2100( 12.64915 )
296.8317
140000
f ( 25.35085 ) 2000 ln
9.8( 25.35085 )
140000 2100( 25.35085 )
708.4811
27
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Solution (cont)
b) The true error,
Et
, is
Et True Value Approximate Value
11061 .34 11058 .44
2.9000 m
c) The absolute relative true error,
t
28
t
, is (Exact value = 11061.34m)
11061.34 11058.44
100%
11061.34
0.0262%
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Additional Resources
For all resources on this topic such as digital audiovisual
lectures, primers, textbook chapters, multiple-choice
tests, worksheets in MATLAB, MATHEMATICA, MathCad
and MAPLE, blogs, related physical problems, please
visit
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drature.html
THE END
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