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Composite integration(복합적분 ) : Applying one of the lower order methods presented in the previous section repeatedly on several sub intervals.
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11.3.1 Composite Trapezoid Rule
If we divide the interval of integration, [a, b], into two or more subintervals and use the trapezoid rule on each subintervals, we obtain the composite trapezoid rule.
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11 1
1 1
( ) ( ) ( ) [ ( ) ( )] [ ( ) ( )]2 2
[ ( ) 2 ( ) ( )] [ ( ) 2 ( ) ( )]2 4
b x b
a a x
h hf x dx f x dx f x dx f a f x f x f b
h b af a f x f b f a f x f b
2
b ah
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11.3.1 Composite Trapezoid Rule
If we divide the interval into n subintervals, we get
Using Composite Simpson’s Rule and the length between day 0 and 10 (n=20) is 0.88952. (Trapezoid=0.889567, Text=0.8556)Using Composite Simpson’s Rule and the length between day 60 and 70 (n=20) is 0.382108. (Trapezoid=0.382109, Text=0.3702)The former is 2.3279 times faster than the latter.
Get the definite integration of f(x) on [-1,1] using linear combinations of coefficient ck and evaluated function value f(xk) at the point xk
Appropriate values of the points xk and ck depend on the choice of n
By choosing the quadrature point x1 ,… xn as the n zeros of the nth-degree Gauss-Legendre polynomial, and by using the appropriate coefficients, the integration formular is exact for polynomials of degree up to 2n-1
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11.4.1 Gaussian Quadrature on [-1,1]
Gaussian Quadrature Formular (cont.)
n=2
n=3
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11.4.1 Gaussian Quadrature on [-1,1]
Example 11.13 integral of exp(-x2) Using G.Q
n Xi ci
2
3
4
±0.557753
0
±0.77459
±0.861136
±0.339981
1
8/9
5/9
0.34785
0.652145
Table 11.2 parameters of Gaussian quadrature
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Gaussian-Legendre Polynomials
11.4.1 Gaussian Quadrature on [-1,1]
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Extends Gaussian Quadrature for f(t) on [a, b] by Transformation f(t) on [a, b] to f(x) on [-1,1]
For the given integral
change interval of t by using next formular
so the interval
11.4.2 Gaussian Quadrature on [a,b]
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Extends Gaussian Quadrature for f(t) on [a, b] (cont.) f(t) rewrite for variable x
remark the factor (b-a)/2 (∵td convert to dx)
Apply f(x) to the integral
11.4.2 Gaussian Quadrature on [a,b]
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Example 11.14 integral of exp(-x2) on [0,2] using G.Q with n = 2
Consider again the integral
Transform f(t) on [0,2] to f(x) on [-1,1] using next formular
11.4.2 Gaussian Quadrature on [a,b]
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Example 11.14 (cont) So we can get
Apply Gaussian Quadrature to the integral with n = 2