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

so Example,

We end up with the following triangle 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)