2. Pengintegralan Numerik (lanjutan) Pertemuan 10
Matakuliah : METODE NUMERIK I
Tahun : 2008
Pengintegralan Numerik (lanjutan) Pertemuan 10

3. Integrasi IIntegrasi Proses mencari luas di bawah suatu curva. Where:
f(x) a b y x
Proses mencari luas di bawah suatu curva.
Where:
f(x) is the integrand
a= lower limit of integration
b= upper limit of integration
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4. Aturan Kuadrat Gauss
Previously, the Trapezoidal Rule was developed by the method
of undetermined coefficients. The result of that development is
summarized below.
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5. Aturan Kuadrat Gauss
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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6. Aturan Kuadrat Gauss It follows that
Equating Equations the two previous two expressions yield
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7. Aturan Kuadrat Gauss Since the constants a0, a1, a2, a3 are arbitrary
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8. Aturan Kuadrat Gauss
The previous four simultaneous nonlinear Equations have only one acceptable solution,
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9. Aturan Kuadrat Gauss Hence Two-Point Gaussian Quadrature Rule
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10. Aturan Kuadrat Gauss dengan n titik
Dinamakan kuadrat Gauss dengan 3 titik
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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11. Arguments and Weighing Factors for n-point Gauss Quadrature Formulas
Table 1: Weighting factors c and function arguments x used in Gauss Quadrature Formulas.
In handbooks, coefficients and
arguments given for n-point
Points
Weighting
Factors
Function
Arguments 2
c1 =
c2 =
x1 =
x2 = 3
c1 =
c2 =
c3 =
x1 =
x2 =
x3 = 4
c1 =
c2 =
c3 =
c4 =
x1 =
x2 =
x3 =
x4 =
Gauss Quadrature Rule are
given for integrals
as shown in Table 1.
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12. 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
Weighting
Factors
Function
Arguments 5
c1 =
c2 =
c3 =
c4 =
c5 =
x1 =
x2 =
x3 =
x4 =
x5 = 6
c1 =
c2 =
c3 =
c4 =
c5 =
c6 =
x1 =
x2 =
x3 =
x4 =
x5 =
x6 =
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13. 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
Let
If then
Such that:
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14. 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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15. Contoh 1 Solution For an integral
derive the one-point Gaussian Quadrature
Rule.
Solution
The one-point Gaussian Quadrature Rule is
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16. Solution Assuming the formula gives exact values for integrals and
Since
the other equation becomes
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17. Solution (cont.)
Therefore, one-point Gauss Quadrature Rule can be expressed as
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18. Contoh 2
Use two-point Gauss Quadrature Rule to approximate the distance
covered by a rocket from t=8 to t=30 as given by
Find the true error, for part (a).
Also, find the absolute relative true error, for part (a). a) b)
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19. Solution First, change the limits of integration from [8,30] to [-1,1]
by previous relations as follows
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20. Solution (cont)
Next, get weighting factors and function argument values from Table 1
for the two point rule, . .
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21. Solution (cont.) Now we can use the Gauss Quadrature formula
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22. Solution (cont)
since
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23. Solution (cont) b) The true error, , is c)
The absolute relative true error,
, is (Exact value = m)
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24. Integral Romberg 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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25. Integral Romberg 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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26. Integral Romberg
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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27. Integral Romberg
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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28. Contoh 2
A company advertises that every roll of toilet paper has at least 250 sheets. The probability that there are 250 or more sheets in the toilet paper is given by
Approximating the above integral as
Use Romberg’s rule to find the probability. Use the 1, 2, 4, and
8-segment Trapezoidal rule results as given.
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29. 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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30. Penyelesaian (lanjutan)
To get the first order extrapolation values,
Similarly,
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31. Penyelesaian (lanjutan)
For the second order extrapolation values,
Similarly,
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32. Penyelesaian (lanjutan)
For the third order extrapolation values,
Table 3 shows these increased correct values in a tree graph.
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33. Solution (cont.)
Table 3: Improved estimates of the integral value using Romberg Integration
1-segment
2-segment
4-segment
8-segment
1.2042
1.2711
1.2881
1st Order
2nd Order
3rd Order
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34. Soal Latihan Hitunglah Menggunakan
Aturan Trapezoidal pada segmen (n) = 2, 4, 6,dan 8
Kuadratur Gauss untuk 2 dan 4 titik
Integral Romberg (gunakan hasil a)
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