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Numerical Integration:

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Composite trapezium rule: Integrals as areas. Simpson's rule: ... the trapezium rule uses linear interpolation ... column are given by the trapezium rule ... – PowerPoint PPT presentation

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Title: Numerical Integration:


1
Numerical Integration
  • Approximating an integral by a sum

2
Integrals as areas
3
Integrals as areas
Approximate the integral by a finite sum of areas
rectangles
4
Integrals as areas
Approximate the integral by a finite sum of areas
trapeziums
5
Integrals as areas
Trapezium rule
Associated error
6
Integrals as areas
Composite trapezium rule
7
Integrals as areas
Simpsons rule
8
Integrals as areas
  • the rectangle approximation takes the function
    to be
  • constant in the interval
  • the trapezium rule uses linear interpolation
    between points
  • Simpsons rule uses polynomial (quadratic)
    interpolation
  • between the points and has an associated error
  • these should be compared to a Taylor expansion,
    the first
  • term is a constant, the second term is linear,
    the third is
  • quadratic

9
Simpsons rule derivation by method of
undetermined
coefficients
  • express the integral as
  • this must hold for polynomials of degree two or
    less.
  • In particular, it must hold for
  • but we already know the left hand side for
    these

10
Simpsons rule derivation by method of
undetermined
coefficients
Now calculate the right-hand-side
Solving these gives
11
Romberg integration integration by iteration
  • make repeated use of the trapezium rule

etc.
  • this is equivalent to

etc.
  • this gives a series of approximations which can
    be used to
  • extrapolate to give the answer

12
Romberg integration doing the extrapolation
  • the algorithm takes the form of a triangle, Rjk,
    where we start with R00 and work down

R00 R10 R11 R20 R21 R22 R30 R31 R32 R33

are the approximations, repeat until
13
Romberg integration doing the extrapolation
The left hand column are given by the trapezium
rule
starting with
To work from the left to the right for a given
column use
14
so
Example,
15
Example,
We end up with the following triangle
First approximation
Second approximation
Third approximation
16
Infinite ranged integrals
  • evaluate an integral over an infinite range
  • the previous methods would lead to an infinite
    sum
  • so cannot be used
  • transform the integral into a finite ranged
    integral, e.g.
  • change the variable in the second integral

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