Assignment -a

Write a code to evaluate an integral of a Lorentzian from A to B – use Richardson.htm.

-- Note that the analytical anti-derivative is In G/R p.60

Trap rule error in periodic functions

A periodic function can always be expanded as

Note that c_j is complex and that

Since the first exponential is always 1. The integral from 0 to T of f is

That is the actual value of the integral is given by the c₀ term alone. All other terms in the expansion integrate to zero.

Let so that the mid point trap rule approximation to this integral is

note interchange of summation orders. Let , so that

The last term can be written as a sum of z^k and we can use the familiar relation for z¹1

So that

Note that so that the numerator in is always zero. The denominator is also zero for j=mN for which . In this case the last sum in is N so that

Note that the terms that enter are not uniform. If for some accidental reason c₁₀ = 0, 10 point integration will seem exact up to c₂₀. Changing N from 10 to 20 will give a false accuracy reading. The correct change is from 10 to 11 or 13, Do not simply double the points. --- Press pay attention.

Assignment -b

Evaluate Using various integration points ranging from N=m/2 to N=4m. Plot the integration error as a function of 1/N². If the method “breaks” plot the details where it breaks.