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## Lecture 27 Numerical Integration

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### Evaluate the integral. h1 = 1.5, h2 = 0.5. Richardson Extrapolation ... Choose (c1, c2, x1, x2) such that the method yields 'exact integral' for f(x) = x0, x1, x2, x3 ... – PowerPoint PPT presentation

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Title: Lecture 27 Numerical Integration

1
Lecture 27 - Numerical Integration
• CVEN 302
• October 31, 2001

2
Lectures Goals
• Trapezoidal Rule
• Simpsons Rule
• Romberg

Composite Numerical Integration
3
Composite Trapezoid Example
4
Composite Simpsons Rule
f(x)
...
x
x0
x2
x4
h
h
xn-2
h
xn
h
x3
x1
xn-1
5
Composite Simpsons Rule
• Multiple applications of Simpsons rule

6
Composite Simpsons Rule
• Evaluate the integral
• n 2, h 2
• n 4, h 1

7
Composite Simpsons Example
8
Composite Simpsons Rule with Unequal Segments
• Evaluate the integral
• h1 1.5, h2 0.5

9
Richardson Extrapolation
• Use trapezoidal rule as an example
• subintervals n 2j 1, 2, 4, 8, 16, .

10
Richardson Extrapolation
• For trapezoidal rule
• kth level of extrapolation

11
Romberg Integration
• Accelerated Trapezoid Rule

12
Romberg Integration
• Accelerated Trapezoid Rule

13
Romberg Integration Example
14
• Newton-Cotes Formulae
• use evenly-spaced functional values
• select functional values at non-uniformly
distributed points to achieve higher accuracy
• change of variables so that the interval of
integration is -1,1
• Gauss-Legendre formulae

15
x2
x1
-1
1
• Choose (c1, c2, x1, x2) such that the method
yields exact integral for f(x) x0, x1, x2, x3

16
• Exact integral for f x0, x1, x2, x3
• Four equations for four unknowns

17
x3
x1
x2
-1
1
• Choose (c1, c2, c3, x1, x2, x3) such that the
method yields exact integral for f(x) x0, x1,
x2, x3,x4, x5

18
19
• Exact integral for f x0, x1, x2, x3, x4, x5

20
• Coordinate transformation from a,b to -1,1

t2
t1
a
b
21
• Evaluate
• Coordinate transformation
• Two-point formula

22
• Three-point formula
• Four-point formula

23
Summary
• Integration Techniques
• Trapezoidal Rule Linear