integration

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Integration
Topic: Trapezoidal Rule
Major: General Engineering
Author: Autar Kaw, Charlie Barker
http://numericalmethods.eng.usf.edu
1
What is Integration
Integration:
The process of measuring
the area under a function
plotted on a graph.
Where:
f(x) is the integrand
a= lower limit of integration
b= upper limit of integration
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Basis of Trapezoidal Rule
Trapezoidal Rule is based on the Newton-Cotes
Formula that states if one can approximate the
integrand as an nth order polynomial…
where
and
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Basis of Trapezoidal Rule
Then the integral of that function is approximated
by the integral of that nth order polynomial.
Trapezoidal Rule assumes n=1, that is, the area
under the linear polynomial,
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Derivation of the Trapezoidal Rule
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Method Derived From Geometry
The area under the
curve is a trapezoid.
The integral
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Example 1
The vertical distance covered by a rocket from t=8 to
t=30 seconds is given by:
a) Use single segment Trapezoidal rule to find the distance covered.
b) Find the true error,
for part (a).
c) Find the absolute relative true error,
for part (a).
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Solution
a)
8
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Solution (cont)
a)
b) The exact value of the above integral is
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Solution (cont)
b)
c)
10
The absolute relative true error,
, would be
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Multiple Segment Trapezoidal Rule
In Example 1, the true error using single segment trapezoidal rule was
large. We can divide the interval [8,30] into [8,19] and [19,30] intervals
and apply Trapezoidal rule over each segment.
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Multiple Segment Trapezoidal Rule
With
Hence:
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Multiple Segment Trapezoidal Rule
The true error is:
The true error now is reduced from -807 m to -205 m.
Extending this procedure to divide the interval into equal
segments to apply the Trapezoidal rule; the sum of the
results obtained for each segment is the approximate
value of the integral.
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Multiple Segment Trapezoidal Rule
Divide into equal segments
as shown in Figure 4. Then
the width of each segment is:
The integral I is:
Figure 4: Multiple (n=4) Segment Trapezoidal Rule
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Multiple Segment Trapezoidal Rule
The integral I can be broken into h integrals as:
Applying Trapezoidal rule on each segment gives:
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Example 2
The vertical distance covered by a rocket from to seconds is
given by:
a) Use two-segment Trapezoidal rule to find the distance covered.
b) Find the true error,
for part (a).
c) Find the absolute relative true error,
for part (a).
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Solution
a) The solution using 2-segment Trapezoidal rule is
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Solution (cont)
Then:
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Solution (cont)
b) The exact value of the above integral is
so the true error is
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Solution (cont)
c)
The absolute relative true error,
20
, would be
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Solution (cont)
Table 1 gives the values
obtained using multiple
segment Trapezoidal rule
for:
n
Value
Et
1
11868
-807
7.296
---
2
11266
-205
1.853
5.343
3
11153
-91.4
0.8265
1.019
4
11113
-51.5
0.4655
0.3594
5
11094
-33.0
0.2981
0.1669
6
11084
-22.9
0.2070
0.09082
7
11078
-16.8
0.1521
0.05482
8
11074
-12.9
0.1165
0.03560
Table 1: Multiple Segment Trapezoidal Rule Values
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Example 3
Use Multiple Segment Trapezoidal Rule to find
the area under the curve
.
from
Using two segments, we get
22
to
and
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Solution
Then:
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Solution (cont)
So what is the true value of this integral?
Making the absolute relative true error:
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Solution (cont)
Table 2: Values obtained using Multiple Segment
Trapezoidal Rule for:
25
n
Approximate
Value
1
0.681
245.91
99.724%
2
50.535
196.05
79.505%
4
170.61
75.978
30.812%
8
227.04
19.546
7.927%
16
241.70
4.887
1.982%
32
245.37
1.222
0.495%
64
246.28
0.305
0.124%
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Error in Multiple Segment
Trapezoidal Rule
The true error for a single segment Trapezoidal rule is given by:
where
is some point in
What is the error, then in the multiple segment Trapezoidal rule? It will
be simply the sum of the errors from each segment, where the error in
each segment is that of the single segment Trapezoidal rule.
The error in each segment is
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Error in Multiple Segment
Trapezoidal Rule
Similarly:
It then follows that:
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Error in Multiple Segment
Trapezoidal Rule
Hence the total error in multiple segment Trapezoidal rule is
.
The term
is an approximate average value of the
Hence:
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Error in Multiple Segment
Trapezoidal Rule
Below is the table for the integral
as a function of the number of segments. You can visualize that as the number
of segments are doubled, the true error gets approximately quartered.
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n
Value
2
11266
-205
1.854
5.343
4
11113
-51.5
0.4655
0.3594
8
11074
-12.9
0.1165
0.03560
16
11065
-3.22
0.02913
0.00401
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Integration
Topic: Simpson’s 1/3rd Rule
Major: General Engineering
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Basis of Simpson’s 1/3rd Rule
Trapezoidal rule was based on approximating the integrand by a first
order polynomial, and then integrating the polynomial in the interval of
integration. Simpson’s 1/3rd rule is an extension of Trapezoidal rule
where the integrand is approximated by a second order polynomial.
Hence
Where
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is a second order polynomial.
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Basis of Simpson’s 1/3rd Rule
Choose
and
as the three points of the function to evaluate a0, a1 and a2.
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Basis of Simpson’s 1/3rd Rule
Solving the previous equations for a0, a1 and a2 give
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Basis of Simpson’s 1/3rd Rule
Then
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Basis of Simpson’s 1/3rd Rule
Substituting values of a0, a1, a 2 give
Since for Simpson’s 1/3rd Rule, the interval [a, b] is broken
into 2 segments, the segment width
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Basis of Simpson’s 1/3rd Rule
Hence
Because the above form has 1/3 in its formula, it is called Simpson’s 1/3rd Rule.
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Example 1
The distance covered by a rocket from t=8 to t=30 is given by
a) Use Simpson’s 1/3rd Rule to find the approximate value of x
b) Find the true error,
c) Find the absolute relative true error,
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Solution
a)
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Solution (cont)
b) The exact value of the above integral is
True Error
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Solution (cont)
a)c)
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Absolute relative true error,
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Multiple Segment Simpson’s
1/3rd Rule
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Multiple Segment Simpson’s 1/3rd
Rule
Just like in multiple segment Trapezoidal Rule, one can subdivide the interval
[a, b] into n segments and apply Simpson’s 1/3rd Rule repeatedly over
every two segments. Note that n needs to be even. Divide interval
[a, b] into equal segments, hence the segment width
where
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Multiple Segment Simpson’s 1/3rd
Rule
f(x)
.
. . .
x
.
x0
x2
xn-2
xn
Apply Simpson’s 1/3rd Rule over each interval,
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Multiple Segment Simpson’s 1/3rd
Rule
Since
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Multiple Segment Simpson’s 1/3rd
Rule
Then
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Multiple Segment Simpson’s 1/3rd
Rule
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Example 2
Use 4-segment Simpson’s 1/3rd Rule to approximate the distance
covered by a rocket from t= 8 to t=30 as given by
Use four segment Simpson’s 1/3rd Rule to find the approximate
value of x.
b)
Find the true error,
for part (a).
c) Find the absolute relative true error,
for part (a).
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a)
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Solution
a)
Using n segment Simpson’s 1/3rd Rule,
So
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Solution (cont.)
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Solution (cont.)
cont.
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Solution (cont.)
b)
In this case, the true error is
c)
The absolute relative true error
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Solution (cont.)
Table 1: Values of Simpson’s 1/3rd Rule for Example 2 with multiple segments
52
n
Approximate Value
Et
2
4
6
8
10
11065.72
11061.64
11061.40
11061.35
11061.34
4.38
0.30
0.06
0.01
0.00
|Єt |
0.0396%
0.0027%
0.0005%
0.0001%
0.0000%
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Error in the Multiple Segment
Simpson’s 1/3rd Rule
The true error in a single application of Simpson’s 1/3rd Rule is given as
In Multiple Segment Simpson’s 1/3rd Rule, the error is the sum of the errors
in each application of Simpson’s 1/3rd Rule. The error in n segment Simpson’s
1/3rd Rule is given by
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Error in the Multiple Segment
Simpson’s 1/3rd Rule
.
.
.
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Error in the Multiple Segment
Simpson’s 1/3rd Rule
Hence, the total error in Multiple Segment Simpson’s 1/3rd Rule is
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Error in the Multiple Segment
Simpson’s 1/3rd Rule
The term
is an approximate average value of
Hence
where
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Integration
Topic: Romberg Rule
Major: General Engineering
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What is The Romberg Rule?
Romberg Integration is an extrapolation formula of
the Trapezoidal Rule for integration. It provides a
better approximation of the integral by reducing the
True Error.
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Error in Multiple Segment
Trapezoidal Rule
The true error in a multiple segment Trapezoidal
Rule with n segments for an integral
Is given by
where for each i,
domain ,
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is a point somewhere in the
.
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Error in Multiple Segment
Trapezoidal Rule
The term
can be viewed as an
approximate average value of
in
.
This leads us to say that the true error, Et
previously defined can be approximated as
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Error in Multiple Segment
Trapezoidal Rule
Table 1 shows the results
obtained for the integral
using multiple segment
Trapezoidal rule for
n
Value
Et
1
11868
807
7.296
---
2
11266
205
1.854
5.343
3
11153
91.4
0.8265
1.019
4
11113
51.5
0.4655
0.3594
5
11094
33.0
0.2981
0.1669
6
11084
22.9
0.2070
0.09082
7
11078
16.8
0.1521
0.05482
8
11074
12.9
0.1165
0.03560
Table 1: Multiple Segment Trapezoidal Rule Values
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Error in Multiple Segment
Trapezoidal Rule
The true error gets approximately quartered as
the number of segments is doubled. This
information is used to get a better approximation
of the integral, and is the basis of Richardson’s
extrapolation.
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Richardson’s Extrapolation for
Trapezoidal Rule
The true error,
is estimated as
in the n-segment Trapezoidal rule
where C is an approximate constant of
proportionality. Since
Where TV = true value and
63
= approx. value
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Richardson’s Extrapolation for
Trapezoidal Rule
From the previous development, it can be shown
that
when the segment size is doubled and that
which is Richardson’s Extrapolation.
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Example 1
The vertical distance covered by a rocket from 8 to 30
seconds is given by
a) Use Richardson’s rule to find the distance covered.
Use the 2-segment and 4-segment Trapezoidal
rule results given in Table 1.
b) Find the true error, Et for part (a).
c) Find the absolute relative true error,
for part (a).
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Solution
a)
Using Richardson’s extrapolation formula
for Trapezoidal rule
and choosing n=2,
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Solution (cont.)
b) The exact value of the above integral is
Hence
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Solution (cont.)
c) The absolute relative true error
would then be
.
Table 2 shows the Richardson’s extrapolation
results using 1, 2, 4, 8 segments. Results are
compared with those of Trapezoidal rule.
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Solution (cont.)
Table 2: The values obtained using Richardson’s
extrapolation formula for Trapezoidal rule for
.
n
Trapezoidal
Rule
1
2
4
8
11868
11266
11113
11074
for Trapezoidal
Rule
7.296
1.854
0.4655
0.1165
Richardson’s
Extrapolation
-11065
11062
11061
for Richardson’s
Extrapolation
-0.03616
0.009041
0.0000
Table 2: Richardson’s Extrapolation Values
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Romberg Integration
Romberg integration is same as Richardson’s
extrapolation formula as given previously. However,
Romberg used a recursive algorithm for the
extrapolation. Recall
This can alternately be written as
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Romberg Integration
Note that the variable TV is replaced by
as the
value obtained using Richardson’s extrapolation formula.
Note also that the sign
is replaced by = sign.
Hence the estimate of the true value now is
Where Ch4 is an approximation of the true error.
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Romberg Integration
Determine another integral value with further halving
the step size (doubling the number of segments),
It follows from the two previous expressions
that the true value TV can be written as
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Romberg Integration
A general expression for Romberg integration can be
written as
The index k represents the order of extrapolation.
k=1 represents the values obtained from the regular
Trapezoidal rule, k=2 represents values obtained using the
true estimate as O(h2). The index j represents the more and
less accurate estimate of the integral.
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Example 2
The vertical distance covered by a rocket from
to
seconds is given by
Use Romberg’s rule to find the distance covered. Use
the 1, 2, 4, and 8-segment Trapezoidal rule results as
given in the Table 1.
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Solution
From Table 1, the needed values from original
Trapezoidal rule are
where the above four values correspond to using 1, 2,
4 and 8 segment Trapezoidal rule, respectively.
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Solution (cont.)
To get the first order extrapolation values,
Similarly,
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Solution (cont.)
For the second order extrapolation values,
Similarly,
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Solution (cont.)
For the third order extrapolation values,
Table 3 shows these increased correct values in a tree
graph.
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Solution (cont.)
Table 3: Improved estimates of the integral value using Romberg Integration
First Order
1-segment
Second Order
Third Order
11868
11065
2-segment
1126
11062
11062
4-segment
11113
11061
11061
11061
8-segment
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11074
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Integration
Topic: Gauss Quadrature Rule of
Integration
Major: General Engineering
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Two-Point Gaussian
Quadrature Rule
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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.
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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
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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,
Hence
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Basis of the Gaussian
Quadrature Rule
It follows that
Equating Equations the two previous two expressions yield
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Basis of the Gaussian
Quadrature Rule
Since the constants a0, a1, a2, a3 are arbitrary
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Basis of Gauss Quadrature
The previous four simultaneous nonlinear Equations have
only one acceptable solution,
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Basis of Gauss Quadrature
Hence Two-Point Gaussian Quadrature Rule
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Higher Point Gaussian
Quadrature Formulas
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Higher Point Gaussian Quadrature
Formulas
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
General n-point rules would approximate the integral
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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
as shown in Table 1.
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Table 1: Weighting factors c and function
arguments x used in Gauss Quadrature
Formulas.
Points
Weighting
Factors
2
c1 = 1.000000000
c2 = 1.000000000
Function
Arguments
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
=
=
=
=
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
92
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
?
integrals, how does one solve
The answer lies in that any integral with limits of
can be converted into an integral with limits
93
If
then
If
then
Let
Such that:
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Arguments and Weighing Factors
for n-point Gauss Quadrature
Formulas
Then
Hence
Substituting our values of x, and dx into the integral gives us
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Example 1
For an integral
derive the one-point Gaussian Quadrature
Rule.
Solution
The one-point Gaussian Quadrature Rule is
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Solution
Assuming the formula gives exact values for integrals
and
Since
96
the other equation becomes
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Solution (cont.)
Therefore, one-point Gauss Quadrature Rule can be expressed as
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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
b)
Find the true error,
c)
Also, find the absolute relative true error,
98
for part (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
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Solution (cont)
Next, get weighting factors and function argument values from Table 1
.
for the two point rule,
.
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Solution (cont.)
Now we can use the Gauss Quadrature formula
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Solution (cont)
since
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Solution (cont)
b) The true error,
, is
c) The absolute relative true error,
103
, is (Exact value = 11061.34m)
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