Transcript Gauss Quadrature Rule of Integration
Gauss Quadrature Rule of Integration
4/25/2020 Chemical Engineering Majors Authors: Autar Kaw, Charlie Barker http://numericalmethods.eng.usf.edu
Transforming Numerical Methods Education for STEM Undergraduates http://numericalmethods.eng.usf.edu
1
Gauss Quadrature Rule of Integration
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3
What is Integration?
Integration
The process of measuring the area under a curve.
y
b a
f ( x ) dx
f(x)
I
a
b f ( x ) dx
Where: f(x) is the integrand a= lower limit of integration b= upper limit of integration a b x http://numericalmethods.eng.usf.edu
4
Two-Point Gaussian Quadrature Rule
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5
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 a
f
(
x
)
dx
c
1
f
(
a
)
c
2
f
(
b
)
b
a
2
f
(
a
)
b
a
2
f
(
b
) http://numericalmethods.eng.usf.edu
6
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 x 1 and x 2 . In the two-point Gauss Quadrature Rule, the integral is approximated as
I
a
b f ( x ) dx
c
1
f ( x
1
)
c
2
f ( x
2
)
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7
Basis of the Gaussian Quadrature Rule
The four unknowns x the formula gives exact results for integrating a general third order polynomial,
f
1
(
, x
x
2
)
, c 1
a
and c 0
a
2 1
x
are found by assuming that
a
2
x
2
a
3
x
3
.
Hence
a
b f ( x ) dx
a
b
a
a
0 0
x
a
1
x a
1
x
2 2
a
2
x
2
a
2
a
3
x
3
dx x
3 3
a
3
x
4 4
b a
a
0
b
a
a
1
b
2
a
2 2
a
2
b
3
a
3 3
a
3
b
4
a
4 4 http://numericalmethods.eng.usf.edu
Basis of the Gaussian Quadrature Rule
8
a b
It follows that
f ( x ) dx
c
1
a
0
a
1
x
1
a
2
x
1 2
a
3
x
1 3 2
a
0
a
1
x
2
a
2
x
2 2
a
3
x
2 3 Equating Equations the two previous two expressions yield
a
0
b
a
a
1
b
2
a
2 2
a
2
b
3
a
3 3
a
3
b
4
a
4 4
c
1
a
0
a
1
x
1
a
2
x
1 2
a
3
x
1 3 2
a
0
a
1
x
2
a
2
x
2 2
a
3
x
2 3
a
0
c
1
c
2
a
1
c
1
x
1
c
2
x
2
a
2
c
1
x
1 2
c
2
x
2 2
a
3
c
1
x
1 3
c
2
x
2 3 http://numericalmethods.eng.usf.edu
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Basis of the Gaussian Quadrature Rule
Since the constants a 0 , a 1 , a 2 , a 3 are arbitrary
b
a
c
1
c
2
b
3
a
3
c
1
x
1 2
c
2
x
2 2 3
b
2
a
2
c
1
x
1
c
2
x
2 2
b
4
a
4
c
1
x
1 3
c
2
x
2 3 4 http://numericalmethods.eng.usf.edu
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Basis of Gauss Quadrature
The previous four simultaneous nonlinear Equations have only one acceptable solution,
x
1 2
a
1 3
b
2
a x
2 2
a
1 3
b
a
2
c
1
b
a
2
c
2
b
a
2 http://numericalmethods.eng.usf.edu
11
Basis of Gauss Quadrature
Hence Two-Point Gaussian Quadrature Rule
a
b f
(
x
)
dx
b
a
2
c
1
f
1
c
2
f
2
f
b
2
a
1 3
b
a
2
b
a
2
f
b
a
2 1 3
b
2
a
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12
Higher Point Gaussian Quadrature Formulas
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Higher Point Gaussian Quadrature Formulas
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b a
f
(
x
)
dx
c
1
f
(
x
1 )
c
2
f
(
x
2 )
c
3
f
(
x
3 ) is called the three-point Gauss Quadrature Rule. The coefficients c 1 , c 2 , and c 3 , and the functional arguments x 1 , x 2 , and x 3 are calculated by assuming the formula gives exact expressions for integrating a fifth order polynomial
a
b
a
0
a
1
x
a
2
x
2
a
3
x
3
a
4
x
4
a
5
x
5
dx
General n-point rules would approximate the integral
a b
f ( x ) dx
c
1
f ( x
1
)
c
2
f ( x
2
)
.
.
.
.
.
.
.
c n f ( x n )
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14 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 1
g
(
x
)
dx
i n
1
c i g
(
x i
) as shown in Table 1.
Table 1: Weighting factors c and function arguments x used in Gauss Quadrature Formulas.
Points Weighting Factors Function Arguments
2 3 4 c 1 c 2 c 3 c 1 c 2 c 3 c 4 c 1 c 2 = 1.000000000
= 1.000000000
= 0.555555556
= 0.888888889
= 0.555555556
= 0.347854845
= 0.652145155
= 0.652145155
= 0.347854845
x 1 x 2 = -0.577350269
= 0.577350269
x 1 x 2 x 3 = -0.774596669
= 0.000000000
= 0.774596669
x 1 x 2 x 3 x 4 = -0.861136312
= -0.339981044
= 0.339981044
= 0.861136312
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15 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
5 6 c 1 c 2 c 3 c 4 c 5 c 1 c 2 c 3 c 4 c 5 c 6
Weighting Factors
= 0.236926885
= 0.478628670
= 0.568888889
= 0.478628670
= 0.236926885
= 0.171324492
= 0.360761573
= 0.467913935
= 0.467913935
= 0.360761573
= 0.171324492
Function Arguments
x 1 x 2 x 3 x 4 x 5 = -0.906179846
= -0.538469310
= 0.000000000
= 0.538469310
= 0.906179846
x 1 x 2 x 3 x 4 x 5 x 6 = -0.932469514
= -0.661209386
= -0.2386191860
= 0.2386191860
= 0.661209386
= 0.932469514
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16 Arguments and Weighing Factors for n-point Gauss Quadrature Formulas
b a
So if the table is given for
f ( x ) dx
1 1
g ( x ) dx
integrals, how does one solve ? The answer lies in that any integral with limits of can be converted into an integral with limits 1
,
1 Let
x
b x
mt
c t t
1 1 Such that:
m
b
a
2 http://numericalmethods.eng.usf.edu
17 Arguments and Weighing Factors for n-point Gauss Quadrature Formulas Then
c
b
a
2
x
b
2
a t
b
a
2 Hence
dx
b
a dt
2 Substituting our values of x, and dx into the integral gives us
a
b f
(
x
)
dx
1 1
f b
a t
2
b
a
2
b
a dt
2 http://numericalmethods.eng.usf.edu
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Example 1
For an integral Rule.
a b
f ( x ) dx ,
derive the one-point Gaussian Quadrature
Solution
The one-point Gaussian Quadrature Rule is
a b
f ( x ) dx
c
1
f
1 http://numericalmethods.eng.usf.edu
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Solution
The two unknowns x polynomial, 1 , and c 1 are found by assuming that the formula gives exact results for integrating a general first order
f
(
x
)
a
0
a
1
x
.
a
b f
(
x
)
dx
a
b
a
0
a
1
x
dx
a
0
x
a
1
x
2 2
b a
a
0
b
a
a
1
b
2
a
2 2 http://numericalmethods.eng.usf.edu
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Solution
It follows that
a b
f
(
x
)
dx
c
1
a
0
a
1
x
1 Equating Equations, the two previous two expressions yield
a
0
b
a
a
1
b
2
a
2 2
c
1
a
0
a
1
x
1
a
0 (
c
1 )
a
1 (
c
1
x
1 ) http://numericalmethods.eng.usf.edu
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Basis of the Gaussian Quadrature Rule
Since the constants a 0 , and a 1 are arbitrary
b
a
c
1
b
2
a
2
c
1
x
1 2 giving
c
1
b
a x
1
b
a
2 http://numericalmethods.eng.usf.edu
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Solution
Hence One-Point Gaussian Quadrature Rule
a
b f
(
x
)
dx
c
1
f
1 (
b
a
)
f
b
2
a
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23
Example 2
In an attempt to understand the mechanism of the depolarization process in a fuel cell, an electro-kinetic model for mixed oxygen-methanol current on platinum was developed in the laboratory at FAMU. A very simplified model of the reaction developed suggests a functional relation in an integral form. To find the time required for 50% of the oxygen to be consumed, the time, T (s) is given by
T
0 .
61 10 1 .
22 10 6 6 6 .
73
x
4 .
3025 2 .
316 10 11 10
x
7
dx
a) Use two-point Gauss Quadrature Rule to find the time required for 50% of the oxygen to be consumed.
E t
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24
Solution
a) to [-1,1] by previous relations as follows 1 .
22 10 6 , 0 .
61 6 0 .
61 10 6 1 .
22 10
f
6 (
x
)
dx
0 .
61 10 6 1 .
22 10 6 2 1 1
f
0 .
61 10 6 1 .
22 10 6 2
x
0 .
61 10 6 1 .
22 10 6 2
dx
0 .
305 10 6 1 1
f
3 .
05 10 6
x
0 .
915 10 6
dx
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25
Solution (cont)
Next, get weighting factors and function argument values from Table 1 for the two point rule,
c
1 1 .
0000
x
1 0 .
57735
c
2
x
2 1 .
0000 0 .
57735 http://numericalmethods.eng.usf.edu
Solution (cont.)
26 Now we can use the Gauss Quadrature formula 0 .
305 10 6 1 1
f
0 .
305 10 6
x
0 .
915 10 6
dx
0 .
305 10 6
c
1
f
0 .
305 10 6
x
1 0 .
915 10 6
c
2
f
0 .
305 10 6
x
2 0 .
915 10 6 0 .
305 10 6
f
0 .
305 10 6 ( 0 .
57735 ) 0 .
915 10 6 0 .
305 10 6 ( 0 .
57735 ) 0 .
915 10 6 0 .
305 10 6
f
( 1 .
0911 10 6 )
f
( 0 .
73891 10 6 ) 0 .
305 10 6 ( 3 .
0761 10 11 ) ( 3 .
1573 10 11 ) 190120
s
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27
Solution (cont)
since
f
1 .
0911 10 6 6 .
73 1 .
0911 2 .
316 10 10 11 6 4 .
3025 1 .
0911 10 6 10 7 3 .
0761 10 11
f
0 .
73891 10 6 6 .
73 0 .
73891 2 .
316 10 6 10 11 4 .
3025 0 .
73891 10 6 10 7 3 .
1573 10 11 http://numericalmethods.eng.usf.edu
28 c) b)
Solution (cont)
E t E t
True Value
Approximat e Value
1 .
90140 10 5 1 .
90120 10 5 15 .
595 The absolute relative true error,
t
, is (Exact value = 190140s )
t
1 .
90140 10 5 190120 1 .
90140 10 5 100 0 .
0082023 % http://numericalmethods.eng.usf.edu
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 http://numericalmethods.eng.usf.edu/topics/gauss_qua drature.html
THE END
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