Transcript Document 7651947

Numerical Integration:
Approximating an integral by a
sum
Integrals as areas
Integrals as areas
Approximate the integral by a finite sum of areas
rectangles
Integrals as areas
Approximate the integral by a finite sum of areas
trapeziums
Integrals as areas
Trapezium rule:
Associated error:
Integrals as areas
Composite trapezium rule:
Integrals as areas
Simpson’s rule:
Integrals as areas
• the rectangle approximation takes the function to be
constant in the interval
• the trapezium rule uses linear interpolation between points
• Simpson’s 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…
Simpson’s 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
Simpson’s rule: derivation by method of undetermined
coefficients
Now calculate the right-hand-side:
Solving these gives
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
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
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
Example,
so
Example,
We end up with the following triangle
First approximation
Second approximation
Third approximation
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)