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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
  • Gaussian Quadrature

Composite Numerical Integration
3
Composite Trapezoid Example
4
Composite Simpsons Rule
Piecewise Quadratic approximations
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
Gaussian Quadratures
  • Newton-Cotes Formulae
  • use evenly-spaced functional values
  • Gaussian Quadratures
  • 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
Gaussian Quadrature on -1, 1
x2
x1
-1
1
  • Choose (c1, c2, x1, x2) such that the method
    yields exact integral for f(x) x0, x1, x2, x3

16
Gaussian Quadrature on -1, 1
  • Exact integral for f x0, x1, x2, x3
  • Four equations for four unknowns

17
Gaussian Quadrature on -1, 1
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
Gaussian Quadrature on -1, 1
19
Gaussian Quadrature on -1, 1
  • Exact integral for f x0, x1, x2, x3, x4, x5

20
Gaussian Quadrature on a, b
  • Coordinate transformation from a,b to -1,1

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

22
Example Gaussian Quadrature
  • Three-point formula
  • Four-point formula

23
Summary
  • Integration Techniques
  • Trapezoidal Rule Linear
  • Simpsons 1/3-Rule Quadratic
  • Simpsons 3/8-Rule Cubic
  • Booles Rule Fourth-order
  • Gaussian Quadrature

24
Homework
  • Check the Homework webpage
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