Numerical Integration:

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Numerical Integration

• Approximating an integral by a sum

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Integrals as areas
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Integrals as areas
Approximate the integral by a finite sum of areas
rectangles
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Integrals as areas
Approximate the integral by a finite sum of areas
trapeziums
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Integrals as areas
Trapezium rule
Associated error
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Integrals as areas
Composite trapezium rule
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Integrals as areas
Simpsons rule
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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

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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

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Simpsons rule derivation by method of
undetermined
coefficients
Now calculate the right-hand-side
Solving these gives
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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

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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
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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
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so
Example,
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Example,
We end up with the following triangle
First approximation
Second approximation
Third approximation
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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

(prove this)