Exercise Adaptive Integration

  1. Implement a recursive adaptive integrator which estimates the integral of a given function on a given interval with a required absolute (acc) or relative (eps) accuracy.
  2. Calculate ∫01  dx (ln(x)/√(x)) = -4 with acc=eps=0.001 and estimate the number of integrand evaluations.
  3. Calculate ∫01  dx 4√(1-(1-x)2) = π with as many significant digits as possible and estimate the number of integrand evaluations.
  4. Test your implementation on some other interesting integrals.
  5. Generalise your integrator such that it can accept infinite limits.
  6. [this one is for later comparison with ordinary diferential equation solver] A definite integral ∫abf(x)dx can be reformulated as an ordinary differential equation, y'=f(x), y(a)=0, y(b)=?, which can be solved with your adaptive ODE solver. Pick an interesing f(x) and compare the effectiveness of your ODE drivers with your adaptive integrator.

Projects

  1. Devise some higher order classical quadratures and implement them in your adaptive integrator.
  2. Generalise your integrator such that it can integrate complex functions along a given path in the complex plane.